Spinor Bose-Einstein condensates

Masahito Uedaa,b, Yuki Kawaguchia

aDepartment of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, JapanbERATO Macroscopic Quantum Control Project, JST, Tokyo 113-8656, Japan

Abstract

An overview of the physics of spinor and dipolar Bose-Einstein condensates (BECs) is pre-sented. Mean-field ground states, Bogoliubov spectra, and many-body ground and excited statesof spinor BECs are discussed. Properties of spin-polarized dipolar BECs and those of spinor-dipolar BECs are reviewed. Some of the unique features of the vortices in spinor BECs, such asfractional vortices and non-Abelian vortices, are delineated. The symmetry of the order parame-ter is classified using group theory, and various topological excitations are investigated based onhomotopy theory. Some of the more recent developments such as the Kibble-Zurek mechanismof defect formation and the Berezinskii-Kosterlitz-Thouless transition in a spinor BEC are alsodiscussed.

Key words:spinor BEC, dipolar BEC, vortices, topological excitations, low-dimensional systems

Contents

1 Introduction 5

2 General Theory 72.1 Single-particle Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Symmetry considerations and irreducible operators . . . . . . . . . . . . 72.2.2 Operator relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Interaction Hamiltonian of spin-1 BEC . . . . . . . . . . . . . . . . . . 102.2.4 Interaction Hamiltonian of spin-2 BEC . . . . . . . . . . . . . . . . . . 102.2.5 Interaction Hamiltonian of spin-3 BEC . . . . . . . . . . . . . . . . . . 11

3 Mean-Field Theory of Spinor Condensates 113.1 Number-conserving theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Mean-field theory of spin-1 BECs . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Mean-field theory of spin-2 BECs . . . . . . . . . . . . . . . . . . . . . . . . . 17

Email addresses: ueda@phys.s.u-tokyo.ac.jp (Masahito Ueda), yuki@cat.phys.s.u-tokyo.ac.jp (YukiKawaguchi)

Preprint submitted to Physics Report March 16, 2010

4 Dipolar BEC 254.1 Dipole–dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Scattering properties of dipole interaction . . . . . . . . . . . . . . . . . 254.1.2 Dipolar systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.3 Tuning of dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . 274.1.4 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Spin-polarized Dipolar BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.1 Equilibrium shape and instability . . . . . . . . . . . . . . . . . . . . . 304.2.2 Dipolar collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.3 Roton-maxon excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.4 Two-dimensional solitons . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.5 Supersolid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.6 Ferrofluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Spinor-dipolar BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.1 Einstein-de Haas effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3.2 Ground-state spin textures at zero magnetic field . . . . . . . . . . . . . 384.3.3 Dipole-dipole interaction under a magnetic field . . . . . . . . . . . . . . 40

5 Bogoliubov Theory 435.1 Spin-1 BECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.1 Ferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.1.2 Antiferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1.3 Domain formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Spin-2 BECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2.1 Ferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.2 Antiferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2.3 Cyclic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Dipolar BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Vortices in Spinor BECs 586.1 Mass and spin supercurrents in spinor BECs . . . . . . . . . . . . . . . . . . . . 596.2 Spin-1 BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2.1 Ferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.2.2 Polar phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2.3 Stability of vortex states . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3 Spin-2 BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3.1 Ferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.3.2 Uniaxial nematic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.3.3 Biaxial nematic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3.4 Cyclic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.4 Rotating spinor BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.5 Spin-polarized dipolar BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.6 Non-Abelian vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2

7 Topological Excitations 747.1 Symmetry classification of broken symmetry states . . . . . . . . . . . . . . . . 74

7.1.1 Order parameter manifold . . . . . . . . . . . . . . . . . . . . . . . . . 747.1.2 Symmetry of order parameter and stationary states . . . . . . . . . . . . 767.1.3 Procedure to find ground states . . . . . . . . . . . . . . . . . . . . . . . 787.1.4 Symmetry and order parameter structure of spin-2 spinor BECs . . . . . 80

7.2 Homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2.1 Classification of topological excitations . . . . . . . . . . . . . . . . . . 837.2.2 Relative homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.3.1 Line defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.3.2 Point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.3.3 Action of one type of defect on another . . . . . . . . . . . . . . . . . . 917.3.4 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 Many-Body Theory 948.1 Many-body states of spin-1 BECs . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.1.1 Eigenspectrum and eigenstates . . . . . . . . . . . . . . . . . . . . . . . 958.1.2 Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.2 Many-body states of spin-2 BECs . . . . . . . . . . . . . . . . . . . . . . . . . 998.2.1 Eigenspectrum and eigenstates . . . . . . . . . . . . . . . . . . . . . . . 1008.2.2 Magnetic response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.2.3 Symmetry considerations on possible phases . . . . . . . . . . . . . . . 104

9 Special Topics 1059.1 Quenched BEC: Kibble-Zurek mechanism . . . . . . . . . . . . . . . . . . . . . 105

9.1.1 Instantaneous quench . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.1.2 Finite-time quench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.1.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.2 Low-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.2.1 Berezinskii-Kosterlitz-Thouless transition in a single-component Bose gas 1109.2.2 2D spinor gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10 Summary and Future Prospects 115

3

symbol definitionAFM(r, r′) annihilation operator for atomic pairs in |F ,M〉A pair amplitude of the spin-singlet pair, 〈A00〉0aF s-wave scattering length of the total spin-F channeld unit vector indicating the spin quantization axisF ,M total spin of two colliding atomsf ,m atomic spin and its projectionf(r) = (fx, fy, fz) vector of f × f spin matricesF(r) =

∑mn ψ

†mfmnψn spin operator

F(r) =∑

mn ψ∗mfmnψn spin expectation value

f (r) =∑

mn ζ∗mfmnζn spin expectation value per atom

i, j, k indices for x, y, and z in the coordinate spaceM atomic massMz total magnetization of the condensatem, n indices for the magnetic subleveln(r) =

∑m ψ∗mψm number density

p linear Zeeman energyq quadratic Zeeman energys = F/|F| unit vector spin along the spin expectation valuet, u unit vector satisfying s = t × uψm(r) field operatorψm(r) order parameterζm ≡ ψm/

√n normalized spinor order parameter

µ, ν, λ indices for x, y, and z in the spin space

Table 1: Table of symbols

abbreviation definitionAF antiferromagneticBA broken axisymmetryBEC Bose-Einstein condensateBN biaxial nematicC cyclicCSV chiral spin vortexDDI dipole-dipole interactionDW density waveF ferromagneticFL flowerGP Gross-PitaevskiiMI Mott insulatorP polarPCV polar-core vortexSF superfluidSS supersolidUN uniaxial nematic

Table 2: Table of abbreviations

4

1. Introduction

Most Bose-Einstein condensates (BECs) of dilute atomic gases realized thus far have internaldegrees of freedom arising from spins. When a BEC is trapped in a magnetic potential, the spinof each atom is oriented along the direction of the local magnetic field. The spin degrees offreedom are therefore frozen and the BEC is described by a scalar order parameter.

On the other hand, when the BEC is confined in an optical trap, the direction of the spin canchange dynamically due to the interparticle interaction. Consequently, the order parameter of aspin- f BEC has 2 f + 1 components that can vary over space and time, producing a very richvariety of spin textures. A BEC with spin degrees of freedom is referred to as a spinor BEC. Thisarticle provides an overview of the physics of spinor and dipolar BECs.

Bose-Einstein condensation is a genuinely quantum-mechanical phase transition in that itoccurs without the help of interaction. However, in the case of spinor BECs, several phasesare possible below the transition temperature Tc, and which phase is realized does depend onthe nature of the interaction. With the spin and gauge degrees of freedom, the full symmetryof the system above Tc is SO(3) × U(1). Below the transition temperature, the symmetry isspontaneously broken in several different ways; the number of possible broken symmetry statesis four for the spin-1 case, at least five for the spin-2 case, and the situation has yet to be fullyunderstood for the spin-3 case. This wealth of possible phases makes a spinor BEC a fascinatingarea of research for quantum gases.

The spin is accompanied by a magnetic moment that gives rise to magnetism. Dependingon the phases of individual systems, spinor BECs exhibit a rich variety of magnetic phenomena.Moreover, because the systems are superfluid, we can expect an interplay between superfluidityand magnetism. For example, a ferromagnetic BEC has spin-gauge symmetry; if you rotate thespin locally, the system tries to undo it by generating a supercurrent. The basic properties of themean-field spinor condensates are reviewed in Sec. 3.

The magnetic moment of the atom leads to the magnetic dipole-dipole interaction. Althoughthe strength of this interaction is the smallest among all the energy scales involved in the system,it plays a pivotal role in producing the spin texture—the spatial variation of the spin direction.The magnetic dipole-dipole interaction plays key roles in determining the static and dynamicproperties of spin-polarized dipolar BECs. It also serves as a key parameter in determining thequantum phases in an ultralow magnetic field (below 10 µG). The physics of dipolar BECs isdiscussed in Sec. 4.

The Bose-Einstein condensed system responds to an external perturbation in a very uniquemanner. Even if the perturbation is weak, the response may be nonperturbative; for example,the phonon velocity depends on the scattering length in a nonanalytic manner. The low-lyingexcitations of spinor and dipolar BECs are described by the Bogoliubov theory, as discussed inSec. 5.

One of the hallmarks of superfluidity manifests itself in its response to an external rotation.In a scalar BEC, the system hosts vortices that are characterized by the quantum of circulation,κ = h/M, where h is the Planck constant and M, the mass of the atom. The origin of thisquantization is the single-valuedness of the order parameter. As mentioned above, however, inthe spinor BEC, the gauge degree of freedom is coupled with the spin degrees of freedom. Thisspin-gauge coupling gives rise to some of the unique features of the spinor BEC. For example,the fundamental unit of circulation can be a rational fraction of κ, and when two vortices collide,they may not reconnect unlike the case of the scalar BEC. Vortices of spinor BECs are discussedin Sec. 6.

5

The direction of the spin can vary rather flexibly over space and time. Nonetheless, the globalconfiguration of the spin texture must satisfy certain topological constraints. This is because theorder parameter in each phase belongs to a particular order parameter manifold whose symmetryleads to a conserved quantity called a topological charge. Such a topological constraint deter-mines the nature of topological excitations such as line defects, point defects, skyrmions, andknots. The order parameter can be classified using group theory, and possible topological excita-tions are determined based on homotopy theory. Elements of group theory and homotopy theorywith applications to spinor BECs are reviewed in Sec. 7.

Efforts are underway to magnetically shield the system so that the ambient magnetic fieldis reduced to below 10 µG. In such an ultralow magnetic field, the spin degeneracy, which isusually lifted by Zeeman effects, sets in significantly, and we have to confront the question ofhow the degeneracy can be lifted by minute spin-exchange interactions. It is in this region wherethe many-body spin correlations dramatically alter the ground states of spinor BECs and theirresponse to an external magnetic field. The present understanding of this problem is described inSec. 8.

Spinor BECs offer topics that are of broad interest beyond ultracold atomic gases. Primeexamples are the dynamics of a quenched BEC and the Berezinskii-Kosteritz-Thouless transitionin two dimensions. These subjects are discussed in Sec. 9.

It seems almost certain that we will face more interesting questions than have been answeredso far. In fact, several new possibilities are emerging both experimentally and theoretically: mag-netic crystalization and non-Abelian vortices, to cite only a few examples. Moreover, under anambient magnetic field below 10 µG, not only do manybody spin correlations become significantbut the magnetic dipole-dipole interaction also plays an equally important role. We thereforehave good reasons to expect that new symmetry breakings will occur in this regime. In viewof these ongoing developments, it now seems appropriate to consolidate the knowledge that hasbeen accumulated over the past few years. In this paper, we provide an overview of the basicsand recent developments on the physics of spinor and dipolar BECs. Some of the remaining andemerging problems are listed in Sec. 10.

One of the major topics that is not treated in this review is fictitious spin systems such asa binary mixture of hyperfine spin states. Such systems do not possess rotational symmetry inspace nor is the fictitious spin associated with the magnetic moment. However, they have someintrinsic interest such as interlaced vortex lattices and vortex molecules caused by the Josephsoncoupling. A comprehensive review of this subject is given in Ref. [1].

This paper is organized as follows. Section 2 describes the fundamental Hamiltonian ofthe spinor BEC. Section 3 develops the mean-field theory of spinor condensates and discussesthe ground-state properties of the spin-1, 2, and 3 BECs. Section 4 provides an overview ofthe dipole BEC and the spinor-dipolar BEC. Section 5 develops the Bogoliubov theory of thespinor and spinor-dipolar BEC. Section 6 discusses various types of vortices that can be createdin spinor BECs. In particular, fractional vortices and non-Abelian vortices are discussed. Sec-tion 7 examines the topological aspects of spinor BECs. The symmetry of the order parameteris classified based on group theory, and possible topological excitations are investigated usinghomotopy theory. Section 8 reviews the many-body aspects of spinor BECs. Section 9 discussesthe Kibble-Zurek mechanism and the Berezinskii-Kosterlitz-Thouless transition in spinor BECs.Section 10 summarizes the main results of this paper and discusses possible future developments.

6

2. General Theory

2.1. Single-particle HamiltonianThe fundamental Hamiltonian of a spinor BEC can be constructed quite generally based on

the symmetry argument. We consider a system of identical bosons with mass M and spin f thatare described by the field operators ψm(r), where m = f , f − 1, · · · ,− f denotes the magneticquantum number. The field operators are assumed to satisfy the canonical commutation relations

[ψm(r), ψ†n(r′)] = δmnδ(r − r′),

[ψm(r), ψn(r′)] = [ψ†m(r), ψ†n(r′)] = 0, (1)

where δmn is the Kronecker delta.The noninteracting part of the Hamiltonian comprises the kinetic term, one-body potential

U(r), and linear and quadratic Zeeman terms:

H0 =

∫dr

f∑m,n=− f

ψ†m

[−~2∇2

2M+ U(r) − p(fz)mm + q(f2

z )mn

]ψn, (2)

where fz is the z-component of the spin matrix whose matrix elements are given by (fz)mn = mδmn,and hence, (f2

z )mn = m2δmn; and p = −gµBB is the product of the Lande hyperfine g-factor g,Bohr magneton µB = e~/2me (me is the electron mass, and e > 0 is the elementary charge), andexternal magnetic field B that is assumed to be applied in the z-direction. The coefficient q of thequadratic Zeeman term is calculated using the second-order perturbation theory to be

q =(gµBB)2

Ehf, (3)

where Ehf = Em − Ei is the hyperfine splitting and it is given by the difference between theinitial (Ei) and intermediate (Em) energies. For example, for the case of 87Rb, g = −1/2 andEhf ' 6.8 GHz for the f = 1 hyperfine manifold and g = 1/2 and Ehf ' −6.8 GHz for the f = 2hyperfine manifold. The value of q can be tuned in the negative direction by using a linearlypolarized microwave field due to the AC Stark shift [2, 3].

2.2. Interaction Hamiltonian2.2.1. Symmetry considerations and irreducible operators

Next, we consider the interaction Hamiltonian. We discuss the case of the magnetic dipole-dipole interaction later as it does not conserve the total spin of the system, and hence, it requiresa separate treatment. Because the system is very dilute, we consider only the binary interaction.Upon the exchange of two identical particles of spin f , the many-body wave function changesby the phase factor (−1)2 f . By the same operation, the spin and orbital parts of the wave functionchange by (−1)F+2 f and (−1)L, respectively, where F is the total spin of the two particles and L,is the relative orbital angular momentum between them. To be consistent, (−1)2 f must be equalto (−1)F+2 f × (−1)L; hence, (−1)F+L = 1. Thus, F +L must be even, regardless of the statisticsof the particles. In the following, we consider only the s-wave scattering (L = 0), and therefore,the total spin of two interacting particles must be even. The interaction Hamiltonian of the binarycollision can therefore be written as

V =∑

F=0,2,··· ,2 f

V (F ), (4)

7

where V (F ) is the interaction Hamiltonian between two bosons whose total spin is F .The interaction V (F ) in Eq. (4) can be constructed from the irreducible operator AFM(r, r′)

that annihilates a pair of bosons at positions r and r′. The irreducible operator is related to a pairof field operators via the Clebsch-Gordan coefficients 〈F ,M| f ,m1; f ,m2〉 as follows:

AFM(r, r′) =f∑

m1,m2=− f

〈F ,M| f ,m1; f ,m2〉ψm1 (r)ψm2 (r′). (5)

Because the boson field operators ψm1 (r) and ψm2 (r′) commute, AFM(r) vanishes identically forodd F . In fact, when F = 1 and f = 1, the Clebsch-Gordan coefficient is given by

〈1,M|1,m1; 1,m2〉 =(−1)1−m1

√2

δm1+m2,M

× [δM,1(δm1,1 + δm1,0) + δM,0 m1 − δM,−1(δm1,0 + δm1,−1)

]. (6)

Substituting this into Eq. (5), we find that A1M(r) = 0. Similarly, we can show that AFM(r) = 0if F is odd. Incidentally, for the case of fermions, we obtain AFM(r) = 0 for even F because thefermion field operators anticommute.

Because the interaction is scalar, it must take the following form:

V (F ) =12

∫dr

∫dr′v(F )(r, r′)

F∑M=−F

A†FM(r, r′)AFM(r, r′), (7)

where v(F )(r, r′) describes the dependence of the interaction on the positions of the particles.When the system is dilute and the range of interaction is negligible as compared to the inter-particle spacing, the interaction kernel v(F ) may be approximated by the delta function with aneffective coupling constant gF :

v(F )(r, r′) = gF δ(r − r′), (8)

where gF is related to the s-wave scattering length of the total spin-F channel, aF , as

gF =4π~2

MaF . (9)

2.2.2. Operator relationsIt follows from the completeness relation∑

F ,M|F ,M〉〈F ,M| = 1 (10)

that ∑F=0,2,··· ,2 f

F∑M=−F

A†FM(r, r′)AFM(r, r′) =: n(r)n(r′) :, (11)

where

n(r) ≡f∑

m=− f

ψ†m(r)ψm(r) (12)

8

is the total density operator and :: denotes normal ordering that places annihilation operators tothe right of creation operators.

Another useful relation can be derived from the composition law of the angular momentum:

F1 · F2 =12

[(F1 + F2)2 − F2

1 − F22

]=

12

F2tot − f ( f + 1), (13)

where Ftot = F1 + F2 is the total spin angular momentum vector. Operating this on the com-pleteness relation (10), we have

F1 · F2 =∑F ,M

[12F (F + 1) − f ( f + 1)

]|F ,M〉〈F ,M|. (14)

Applying this formula to the case of f = 1, we obtain

F1 · F2 =

2∑M=−2

|2,M〉〈2,M| − 2|0, 0〉〈0, 0| = 1 − 3|0, 0〉〈0, 0|, (15)

where we have used the completeness relation (10) to derive the second equality. It follows fromthis that the following operator identity holds:

3A†00(r, r′)A00(r, r′)+ : F(r) · F(r′) :=: n(r)n(r′) :, (16)

where F = (Fx, Fy, Fz) is the spin vector operator defined by

Fµ(r) =f∑

m,n=− f

(fµ)mnψ†m(r)ψn(r) (µ = x, y, z). (17)

Here, (fµ)mn (µ = x, y, z) are the (m, n)-components of spin matrices fµ.For the case of f = 2, we have

F1 · F2 = −6|0, 0〉〈0, 0| − 3∑M|2,M〉〈2,M| + 4

∑M|4,M〉〈4,M|. (18)

The last term can be eliminated by using the completeness relation (10), giving the followingoperator identity:

17

: F(r) · F(r′) : +2∑

M=−2

A†2M(r, r′)A2M(r, r′) +107

A†00(r, r′)A00(r, r′) =47

: n(r)n(r′) : . (19)

Similarly, for the case of f = 3, we obtain

111

: F(r) · F(r′) : +2111

A†00(r, r′)A00(r, r′)

+1811

2∑M=−2

A†2M(r, r′)A2M(r, r′) +4∑

M=−4

A†4M(r, r′)A4M(r, r′) =911

: n(r)n(r′) : . (20)

9

2.2.3. Interaction Hamiltonian of spin-1 BECWe derive the interaction Hamiltonians for the f = 1 case. For f = 1, the total spin F of two

colliding bosons must be 0 or 2, and the corresponding interaction Hamiltonian (7) gives

V (0) =g0

2

∫drA†00(r)A00(r) (21)

V (2) =g2

2

∫dr

2∑M=−2

A†2M(r)A2M(r)

=g2

2

∫dr

[: n2(r) : −A†00(r)A00(r)

], (22)

where AFM(r) ≡ AFM(r, r), and Eq. (11) and A1M(r) = 0 are used to derive the last equality.Combining Eqs. (21) and (22), we obtain

V =∫

dr[g2

2: n2(r) : +

g0 − g2

2A†00(r)A00(r)

]. (23)

Equation (23) is the interaction Hamiltonian of the spin-1 BEC. We use the operator identity (16)to rewrite Eq. (23) as

V =12

∫dr

[c0 : n2(r) : +c1 : F2(r) :

], (24)

where

c0 =g0 + 2g2

3, c1 =

g2 − g0

3. (25)

In Eq. (23), A00 is the spin-singlet pair operator. The explicit form of this operator can befound using the Clebsch-Gordan coefficient for F =M = 0:

〈0, 0| f ,m1; f ,m2〉 = δm1+m2,0(−1) f−m1√

2 f + 1. (26)

Substituting this into Eq. (5), we obtain

A00(r, r′) =1√

2 f + 1

f∑m=− f

(−1) f−mψm(r)ψ−m(r′). (27)

2.2.4. Interaction Hamiltonian of spin-2 BECFor f = 2, F must be 0, 2, or 4, and the interaction Hamiltonian (7) for F = 4 gives

V (4) =g4

2

∫dr

4∑M=−4

A†4M(r)A4M(r)

=g4

2

∫dr

: n2(r) : −A†00(r)A00(r) −2∑

M=−2

A†2M(r)A2M(r)

, (28)

10

where Eq. (11) and A1M = A3M = 0 are used to obtain the last equality. Summing Eqs. (21),(22), and (28), we obtain

V = V (0) + V (2) + V (4)

=

∫dr

g4

2: n2(r) : +

g0 − g4

2A†00(r)A00(r) +

g2 − g4

2

2∑M=−2

A†2M(r)A2M(r)

. (29)

We can eliminate the last term by using the operator identity (19), obtaining

V =12

∫dr

[c0 : n2(r) : +c1 : F2(r) : +c2A†00(r)A00(r)

], (30)

where

c0 =4g2 + 3g4

7, c1 =

g4 − g2

7, c2 =

7g0 − 10g2 + 3g4

7. (31)

We use the same notations c0 and c1 for both spin-1 and spin-2 cases because there is no fear ofconfusion.

2.2.5. Interaction Hamiltonian of spin-3 BECIn a similar manner, the interaction Hamiltonian of the spin-3 spinor BEC is given by

V =12

∫dr

c0 : n2(r) : +c1 : F2(r) : +c2A†00(r)A00(r) + c3

2∑M=−2

A†2M(r)A2M(r)

, (32)

where

c0 =9g4 + 2g6

11, c1 =

g6 − g4

11, c2 =

11g0 − 21g4 + 10g6

11, c3 =

11g2 − 18g4 + 7g6

11. (33)

3. Mean-Field Theory of Spinor Condensates

3.1. Number-conserving theoryThe mean-field theory is usually obtained by replacing the field operator with its expectation

value 〈ψm〉. This recipe, though widely used and technically convenient, has one conceptualdifficulty; that is, it breaks the global U(1) gauge invariance, which implies that the number ofatoms is not conserved. However, in reality, the number of atoms is strictly conserved, as are thebaryon (proton and neutron) and lepton (electron) numbers. In fact, it is possible to construct themean-field theory without breaking the U(1) gauge symmetry [4, 5].

To construct a number-conserving mean-field theory, we first expand the field operator interms of a complete orthonormal set of basis functions ϕmi(r):

ψm(r) =∑

i

amiϕmi(r) (m = f , f − 1, · · · ,− f ), (34)

where ϕmi(r) describes the basis function for the magnetic quantum number m and spatial modei, and ami’s are the corresponding annihilation operators that satisfy the canonical commutationrelations

[ami, a†n j] = δmnδi j, [ami, an j] = [a†mi, a

†n j] = 0. (35)

11

The basis functions are assumed to satisfy the orthonormality conditions∫dr ϕ∗mi(r) ϕm j(r) = δi j, (36)

and the completeness relation ∑i

ϕmi(r)ϕ∗mi(r′) = δ(r − r′). (37)

Then, the field operators (34) satisfy the field commutation relations (1).In the mean-field approximation, it is assumed that all Bose-condensed bosons occupy a

single spatial mode, say i = 0, and a single spin state specified by a linear superposition ofmagnetic sublevels. Then, the state vector is given by

|ζ〉 = 1√

N!

f∑m=− f

ζma†m0

N

|vac〉, (38)

where |vac〉 denotes the particle vacuum and ζm’s are assumed to satisfy the following normal-ization condition:

f∑m=− f

|ζm|2 = 1. (39)

It is straightforward to show that

〈ψm(r)〉0 = 〈ψ†m(r)〉0 = 0, (40)

〈ψ†m(r)ψn(r′)〉0 = ψ∗m(r)ψn(r′), (41)

〈ψ†m(r)ψ†n(r′)ψk(r′′)ψ`(r′′′)〉0 =(1 − 1

N

)ψ∗m(r)ψ∗n(r′)ψk(r′′)ψ`(r′′′), (42)

where 〈· · · ·〉0 ≡ 〈ζ | · · · · |ζ〉 and

ψm(r) =√

Nζmϕm0(r). (43)

As shown in Eq. (40), the expectation value of the field operator vanishes as it should in a number-conserving theory. Nevertheless, all experimentally observable physical quantities, which areexpressed in terms of the correlation function of the field operators such as (41) and (42), canhave nonzero values. These values agree with those obtained by the U(1) symmetry-breakingapproach within the factor of 1/N.

3.2. Mean-field theory of spin-1 BECsWe use Eqs. (41) and (42) to evaluate the expectation value of the Hamiltonian H = H0 + V

over the state (38) with f = 1, where H0 and V are given in Eqs. (2) and (24). By ignoring theterms of the order of 1/N in Eq. (42), we obtain

E[ψ] ≡ 〈H〉0 =∫

dr 1∑

m=−1

ψ∗m

[−~2∇2

2M+ U(r) − pm + qm2

]ψm +

c0

2n2 +

c1

2|F|2

, (44)

12

where

n(r) ≡ 〈n(r)〉0 =1∑

m=−1

|ψm(r)|2 (45)

is the particle density and F = (Fx, Fy, Fz) is the spin density vector defined by

Fµ(r) ≡ 〈Fµ(r)〉0 =1∑

m,n=−1

ψ∗m(r)(fµ)mnψn(r) (µ = x, y, z). (46)

Because the spin-1 matrices are given by

fx =1√

2

0 1 01 0 10 1 0

, fy =i√

2

0 −1 01 0 −10 1 0

, fz =

1 0 00 0 00 0 −1

, (47)

the components of the spin vector F are explicitly written as

Fx =1√

2

[ψ∗1ψ0 + ψ

∗0(ψ1 + ψ−1) + ψ∗−1ψ0

], (48)

Fy =i√

2

[−ψ∗1ψ0 + ψ

∗0(ψ1 − ψ−1) + ψ∗−1ψ0

], (49)

Fz = |ψ1|2 − |ψ−1|2. (50)

From the energy functional in Eq. (44), we find that c0 must be nonnegative; otherwise, thesystem collapses. This condition also guarantees that the Bogoliubov excitation energy corre-sponding to density fluctuations is real, as shown in Sec. 5.

It is easy to identify the ground-state magnetism of a uniform system when the Zeeman termsare negligible (p = q = 0). From the energy functional (44), one can see that the ground stateis ferromagnetic (|F| = n) when c1 < 0 and antiferromagnetic or polar (|F| = 0) when c1 > 0.Here, the order parameter of the antiferromagnetic phase [6] is given by

√n/2(1, 0, 1)T and that

of the polar state [7] is√

n(0, 1, 0)T . The former state is obtained by rotating the latter aboutthe y-axis by π/2, and therefore, these two states are degenerate in the absence of the magneticfield. Moreover, all states obtained by rotating these states are degenerate. In the presence of anexternal magnetic field, however, the ground state becomes more complicated due to the linearand quadratic Zeeman effects, as shown below.

The time evolution of the mean field is governed by

i~∂ψm(r)∂t

=δE

δψ∗m(r). (51)

Substituting Eq. (44) into the right-hand side gives

i~∂ψm

∂t=

[−~2∇2

2M+ U(r) − pm + qm2

]ψm

+ c0nψm + c1

1∑n=−1

F · fmnψn (m = 1, 0,−1), (52)

13

which are the multicomponent Gross-Pitaevskii equations (GPEs) that describe the mean-fieldproperties of spin-1 Bose-Einstein condensates [6, 7]. In a stationary state, we substitute

ψm(r, t) = ψm(r) e−i~ µt (53)

in Eq. (52), where µ is the chemical potential, to obtain[−~2∇2

2M+ U(r) − pm + qm2

]ψm + c0nψm + c1

1∑n=−1

F · fmnψn = µψm. (54)

Writing down the three components m = 1, 0,−1 explicitly, we obtain[−~2∇2

2M+ U(r) − p + q + c0n + c1Fz − µ

]ψ1 +

c1√2

F−ψ0 = 0, (55)

c1√2

F+ψ1 +

[−~2∇2

2M+ U(r) + c0n − µ

]ψ0 +

c1√2

F−ψ−1 = 0, (56)

c1√2

F+ψ0 +

[−~2∇2

2M+ U(r) + p + q + c0n − c1Fz − µ

]ψ−1 = 0, (57)

where F± ≡ Fx± iFy. By solving the set of equations (55)–(57), we can investigate the propertiesof the ground and excited states of the spin-1 BEC.

Note that because the system is suspended in a vacuum chamber, the projected spin angularmomentum in the direction of the external magnetic field is conserved for a long time (≥ 1 s) [8].Because the atomic interaction given by Eq. (7) conserves the total spin of two colliding atoms,the spin dynamics obeying the GPEs conserves the total magnetization. Therefore, when wesearch for the ground state for a given magnetization Mz =

∫drFz(r), we need to replace E in

Eq. (51) with E − λMz, where λ is a Lagrange multiplier. Then, p in Eqs. (55)–(57) is replacedwith p = p + λ, which is determined as a function of Mz.

Here, we solve Eqs. (55)–(57) and investigate the ground-state phase diagram [9, 10] in aparameter space of (q, p). We assume a uniform system for a given number density n. Theground state of a uniform system is obtained if we set the kinetic energy and U(r) to be zero. Wealso note that c0n can be absorbed in µ by defining µ ≡ µ − c0n. Furthermore, we may assumeFy = 0 without loss of generality, provided that the system is symmetric about the z-axis. Then,by performing a gauge transformation to make ψ0 real, all ψ1,0,−1 become real. (After obtainingthe real solution, we can recover the general complex order parameter by performing a globalgauge transformation eiχ0 and spin rotation e− fzχz about the z-axis.) Then, Eqs. (55)–(57) reduceto

(−p + q + c1Fz − µ)ψ1 + c1(ψ1 + ψ−1)ψ20 = 0, (58)

[µ − c1(ψ1 + ψ−1)2]ψ0 = 0, (59)

c1(ψ1 + ψ−1)ψ20 + (p + q − c1Fz − µ)ψ−1 = 0. (60)

The energy per particle is given by

ε =1n

∑m

(−pm + qm2)|ψm|2 +12

c0n2 +12

c1|F|2 . (61)

14

From Eq. (59), we have either (i) ψ0 = 0 or (ii) µ = c1(ψ1 + ψ−1)2. In case (i), we have threestationary states:

I :(eiχ1√

n, 0, 0)T, εI = −p + q +

12

c0n +12

c1n, (62)

II :(0, 0, eiχ−1

√n)T, εII = p + q +

12

c0n +12

c1n, (63)

III :

eiχ1

√n + p/c1

2, 0, eiχ−1

√n − p/c1

2

T

, εIII = q +12

c0n − p2

2c1n, (64)

where we have recovered the phases of the order parameter with χ±1 = χ0 ∓ χz. States I and IIare fully polarized (Fz = ±n), and therefore, ferromagnetic, whereas state III has a longitudinalmagnetization that depends on p because Fz = p/c1. State III is often referred to as an antifer-romagnetic state. In case (ii), we may solve Eqs. (58) and (60), together with the normalizationcondition

∑1m=−1 |ψm|2 = n, to obtain the following two stationary states: one, referred to as a

polar state, is given by

IV :(0, eiχ0

√n, 0

)T, εIV =

12

c0n, (65)

and it has no magnetization in any direction; the other is given by

V : ψ±1 = ei(χ0∓χz) q ± p2q

√−p2 + q2 + 2c1nq

2c1q, (66)

ψ0 = eiχ0

√(q2 − p2)(−p2 − q2 + 2c1nq)

4c1q3 , (67)

εV =(−p2 + q2 + 2qc1n)2

8c1nq2 +12

c0n, (68)

and it exists only when the argument of the square root for every ψm is nonnegative. State Voccurs for c1 < 0 (ferromagnetic case). When

p2 − q2 + 2|c1|nq > 0 and q2 > p2, (69)

all three components ψ± and ψ0 in Eqs. (66) and (67) are nonzero and the magnetization tiltsagainst the external magnetic field:

Fz =p(p2 − q2 + 2q|c1|n)

2|c1|q2 , F+ = eiχz

√(q2 − p2)(p2 + 2|c1|nq)2 − q4

2|c1|q2 . (70)

The polar angle [arctan(|F+|/Fz)] is determined by the interaction and the magnetic field, whereasthe azimuthal angle (χz) is spontaneously chosen in each realization of the system. Because theaxial symmetry with respect to the applied magnetic field axis is broken, this phase is referred toas the broken-axisymmetry phase [10].

Comparing the energies of Eqs. (62)–(68), we obtain the phase diagram in a parameter spaceof (q, p), as shown in Fig. 1 [9].

15

q/|c1|n

p/|c1|n

q

p

q/c1n

p/c1n

1

−1

1/2

p=q+c1n/2

p=−q+c1n/2

p=q

p=−q

p2=2c1nq

p=q

p=−q

p2=q

2−2|c1|nqI

II

I

II

I

II

III IV IVV

IV

(a) c1>0 (b) c1=0 (c) c1<0

2

Figure 1: Phase diagram of spin-1 BEC for (a) c1 > 0, (b) c1 = 0, and (c) c1 < 0. The rotation symmetry about themagnetic field is broken in the shaded region.

phase order parameter ψT Fz µ = µ − c0n ε = ε − 12 c0n

(I) F (eiχ1√

n, 0, 0) n −p + q + c1n −p + q + 12 c1n

(II) F (0, 0, eiχ−1√

n) −n p + q + c1n p + q + 12 c1n

(III) AF (eiχ1

√n+p/c1

2 , 0, eiχ1

√n−p/c1

2 ) pc1

q q − p2

2c1n

(IV) P (0, eiχ0√

n, 0) 0 0 0

(V) BA Eqs. (66), (67) p(−p2+q2+2qc1n)2c1q2 q + 2c1n − p2

q(−p2+q2+2qc1n)2

8c1nq2

Table 3: Possible ground-state phases of a spin-1 BEC, where F, AF, P, and BA denote ferromagnetic, antiferromagnetic,polar, and broken-axisymmetry phases, respectively.

Note that the rotation symmetry about the magnetic-field axis is broken in state III, despitethe fact that the magnetization is parallel to the magnetic field in this phase. To understand theunderlying physics, we introduce the spin nematic tensor defined by

Nµν ≡1n

⟨fµfν + fνfµ

2

⟩0

µ, ν = x, y, z. (71)

The nematic tensor for the ferromagnetic (I, II) and polar (IV) phases is independent of χ0 andχz, and given as

N I,II =

12 0 00 1

2 00 0 1

, N IV =

1 0 00 1 00 0 0

, (72)

respectively. Here, Nxx = Nyy indicates that the order parameter has rotational symmetry about

16

the z-axis. On the other hand, the nematic tensor for state III for χz = 0 is given by

N III =

12 (1 + α) 0 0

0 12 (1 − α) 0

0 0 1

, α =

√1 −

(p

c1n

)2

. (73)

Here, Nxx , Nyy indicates the anisotropy on the xy plane in the spin space. This is the physicalorigin of the axisymmetry breaking of state III. In fact, it can be shown that off-diagonal elementssuch as Nxy appear when χz , 0.

In general, the symmetry of the order parameter can be visualized by drawing the wavefunction in spin space. Because an integer-spin state can be described in terms of the sphericalharmonics Ym

f (s), where s is a unit vector in spin space, the order parameter for a spin- f systemcan be expressed in terms of a complex wave function given by

Ψ(s) =∑

m

ψmYmf (s). (74)

Figure 2 shows the order parameter for states I–V. From Fig. 2, one can see that the order pa-rameters for states III and V have the same morphology and that the magnetization is parallel tothe magnetic field in one phase (III) but not in the other (V). Additional symmetry properties arediscussed in Sec. 7.1.1.

I II III IV VF//BFz=n

Fz=¡nF//B

|F |=0

2¼0 phase

Figure 2: Order parameter for spin-1 stationary states. Surface plots of |Ψ(s)|2 are shown, where the gray scale on thesurface represents argΨ(s).

3.3. Mean-field theory of spin-2 BECs

We take the expectation value of the Hamiltonian H = H0 + V over the state in Eq. (38) withf = 2, where H0 and V are given in Eqs. (2) and (30):

E[ψ] ≡ 〈H〉0

=

∫dr

2∑m=−2

ψ∗m

[−~2∇2

2M+ U(r) − pm + qm2

]ψm +

c0

2n2 +

c1

2|F|2 + c2

2|A|2

. (75)

17

Here, the spin-2 matrices are given by

fx =

0 1 0 0 0

1 0√

32 0 0

0√

32 0

√32 0

0 0√

32 0 1

0 0 0 1 0

, fy =

0 −i 0 0 0

i 0 −i√

32 0 0

0 i√

32 0 −i

√32 0

0 0 i√

32 0 −i

0 0 0 i 0

,

fz =

2 0 0 0 00 1 0 0 00 0 0 0 00 0 0 −1 00 0 0 0 −2

. (76)

As compared to the energy functional (44) of the spin-1 BEC, the new term c2|A|2/2 appears inEq. (75), where

A ≡ 〈A00(r)〉0 =1√

5

(2ψ2ψ−2 − 2ψ1ψ−1 + ψ

20

)(77)

is the amplitude of the spin-singlet pair.For the spin-1 case, the spin-singlet amplitude is uniquely related to the magnetization by

relation (16), and only one of them appears in the energy functional. However, for the spin-2case, they can change independently, giving rise to new ground-state phases. This can be bestillustrated when the system is uniform. In this case, the kinetic-energy term can be ignored andthe Hartree term c0n2/2 is constant; therefore, the ground state is determined by the remainingterms in Eq. (75).

Let us first consider the ground state in the zero magnetic field. The phase diagram in thiscase is determined by the last two terms in Eq. (75), as shown in Fig. 3. When c1 < 0 and c2 > 0,the energy of the system is lowered as magnetization increases. Therefore, the ground state isferromagnetic, and the order parameter is given by ψferro = (

√n, 0, 0, 0, 0)T or its rotated state in

the spin space.When c1 > 0 and c2 < 0, the energy of the system is lowered as the amplitude of the

spin-singlet pair increases. Therefore, the ground state is uniaxial nematic (UN) or biaxialnematic (BN), and the order parameters are given by ψuniax = (0, 0,

√n, 0, 0)T (uniaxial) or

ψbiax =√

n/2(1, 0, 0, 0, 1)T (biaxial). While these states are often referred to as polar or antifer-romagnetic [4, 11, 5], respectively, here, we adopt the liquid-crystal terminology which impliesthe symmetry of the order parameter (see the insets in Fig. 3). In this review, we refer to C4phase discussed below as antiferromagnetic, in order to distinguish it from the UN state. Theorder parameter

√n/2(0, 1, 0, 1, 0) also represents the BN state, which is obtained by rotating

ψbiax as ie−ifyπ/2e−ifzπ/4ψbiax, while ψuniax cannot be obtaind by rotating ψbiax since the symmetryof the order parameters are different. Note that the symmetries of the UN and BN states differ;however, the two states are degenerate at the mean-field level. Moreover, the superposition ofthese two states

√n/2(cos ξ, 0,

√2 sin ξ, 0, cos ξ) (ξ is an arbitrary real) is also degenerate. It has

been shown that zero-point fluctuations lift the degeneracy [12, 13].When both c1 and c2 are positive, neither the ferromagnetic nor UN/BN state is energetically

favorable and frustration arises, resulting in a new phase referred to as the cyclic phase [4, 11, 5].18

The order parameter is given by ψcyclic =√

n/2(1, 0, i√

2, 0, 1)T , which possesses tetrahedralsymmetry, as shown in the inset of Fig. 3. In the many-body ground state of the cyclic state, threebosons form a spin-singlet trimer and the boson trimers undergo Bose-Einstein condensation, asshown in Sec. 8.

c2n

c1n

cyclicferromagnetic

biaxial nematic

uniaxial

nematic

c 2n=20c 1n

Figure 3: Phase diagram of the spin-2 BEC at zero magnetic field. In each phase, the profile of the order parameterΨ(s) =

∑m ψmYm

2 (s) is shown.

The time-dependent GPEs for the spin-2 case can be obtained by substituting Eq. (75) inEq. (51):

i~∂ψ±2

∂t=

[−~2∇2

2M+ U(r) ∓ 2p + 4q + c0n ± 2c1Fz − µ

]ψ±2

+ c1F∓ψ±1 +c2√

5Aψ∗∓2, (78)

i~∂ψ±1

∂t=

[−~2∇2

2M+ U(r) ∓ p + q + c0n ± c1Fz − µ

]ψ±1

+ c1

√62

F∓ψ0 + F±ψ±2

− c2√5

Aψ∗∓1, (79)

i~∂ψ0

∂t=

[−~2∇2

2M+ U(r) + c0n − µ

]ψ0 +

√6

2c1 (F+ψ1 + F−ψ−1) +

c2√5

Aψ∗0, (80)

19

where

F+ = F∗− = 2(ψ∗2ψ1 + ψ

∗−1ψ−2

)+√

6(ψ∗1ψ0 + ψ

∗0ψ−1

), (81)

Fz = 2(|ψ2|2 − |ψ−2|2

)+ |ψ1|2 − |ψ−1|2. (82)

The time-independent GPEs can be obtained by substituting ψm(r, t) = ψm(r)e−iµt/~ in Eqs. (78)–(80). If we assume that the system is uniform (i.e., U(r) = 0), then Eqs. (78)–(80) reduce to

(4q + 2γ0 − µ) ψ2 + aψ∗−2 = −γ−ψ1, (83)

a∗ψ2 + (4q − 2γ0 − µ) ψ∗−2 = −γ−ψ∗−1, (84)

γ+ψ2 + (q + γ0 − µ) ψ1 − aψ∗−1 = −√

62

γ−ψ0, (85)

−a∗ψ1 + (q − γ0 − µ) ψ∗−1 + γ+ψ∗−2 = −

√6

2γ−ψ

∗0, (86)

µψ0 − aψ∗0 =

√6

2(r+ψ1 + γ−ψ−1), (87)

where µ ≡ µ − c0n, γ± ≡ c1F±, γ0 ≡ c1Fz, γ0 ≡ γ0 − p, and a ≡ c2A/√

5. We may use the degreeof gauge transformation (i.e., the U(1) phase) to make ψ0 real. Furthermore, because the physicalproperties of the system are invariant under a rotation about the quantization axis (i.e., the z-axis),we may choose the coordinate system to make Fy = 0 such that γ+ = γ− ≡ γ. A mean-fieldground state can be obtained as a solution of Eqs. (83)–(87) under such a simplification.

Here, we consider the case of γ = 0, i.e., the case of no transverse magnetization. Then,Eqs. (83)–(87) reduce to

(4q + 2γ0 − µ)ψ2 + aψ∗−2 = 0, (88)a∗ψ2 + (4q − 2γ0 − µ)ψ∗−2 = 0, (89)

(q + γ0 − µ)ψ1 − aψ∗−1 = 0, (90)−a∗ψ1 + (q − γ0 − µ)ψ∗−1 = 0, (91)

(µ − a)ψ0 = 0. (92)

In this case, the free energy per particle is given by

ε =1n

2∑m=−2

(−pm + qm2) |ψm|2 +c0n2+

c1

2nF2

z +c2

2n|A|2. (93)

Because Eqs. (88)–(92) are decoupled into three parts, the solutions can be classified accord-ing to the determinant of the coefficient matrix of Eqs. (88) and (89),

D2 ≡ (4q − µ)2 − 4γ20 − |a|2, (94)

and the determinant of the coefficient matrix of Eqs. (90) and (91),

D1 ≡ (q − µ)2 − γ20 − |a|2. (95)

20

• If D1 , 0 and D2 , 0, then we have ψ1 = ψ−1 = 0 and ψ2 = ψ−2 = 0, so that the solutionis the UN state, that is, the order parameter is given by

UN: (0, 0,√

n, 0, 0)T . (96)

The chemical potential is determined from (92) to be µ = 15 c2n and Fz = 0. The energy

per particle is found from Eq. (93) to be ε = 12 (c0 +

15 c2)n.

• If D1 = 0 and D2 , 0, ψ2 = ψ−2 = 0. From the assumption that there is no transversemagnetization, ψ±1 , 0 leads to ψ0 = 0. Clearly,

F1+ : (0, eiχ1√

n, 0, 0, 0)T (97)

and

F1− : (0, 0, 0, eiχ−1√

n, 0)T (98)

are the solutions of Eqs. (90) and (91) with Fz = n, µ = −p + q + c1n and ε = −p + q +12 (c0 + c1)n, and Fz = −n, µ = p + q + c1n and ε = p + q + 1

2 (c0 + c1)n, respectively. When|p| < |c1 − 1

5 c1|n, a solution with nonzero ψ±1 exists, which is given by

C2 :

0, eiχ1

√n + Fz

2, 0, eiχ−1

√n − Fz

2, 0

T

, (99)

where Fz = p/(c1 − 15 c2). The chemical potential and the energy per particle are µ =

q + 15 c2n and ε = q + 1

2 (c0 +15 c2)n − p2/[2(c1 − 1

5 c2)n]. At p = 0, this state continuouslybecoms the BN state.

• If D1 , 0 and D2 = 0, ψ1 = ψ−1 = 0. Moreover, if µ , a, we have ψ0 = 0. In a mannersimilar to the previous case, Eqs. (88) and (89) have three solutions. Two of them areferromagnetic and they are given by

F2+ : (eiχ2√

n, 0, 0, 0, 0)T (100)

with Fz = 2n, µ = −2p + 4q + 4c1n and ε = −2p + 4q + 12 (c0 + 4c1)n, and by

F2− : (0, 0, 0, 0, eiχ−2√

n)T (101)

with Fz = −2n, µ = 2p+4q+4c1n and ε = 2p+4q+ 12 (c0+4c1)n. When |p| < |2c1− 1

10 c2|n,the other solution is given by

C4 :

eiχ2

√12

(n +

Fz

2

), 0, 0, 0, eiχ−2

√12

(n − Fz

2

)T

, (102)

where Fz = p/(c1 − 120 c2), µ = 4q + 1

5 c2n and ε = 4q + 12 (c0 +

15 c2)n − p2/[2(c1 − 1

20 c2)n].This state becomes the BN state at p = 0.

21

• If D1 , 0 and D2 = 0 and µ = a, we have ψ1 = ψ−1 = 0; however, ψ±2 and ψ0 can benonzero. They obey |ψ2|2 + |ψ−2|2 + |ψ0|2 = n and |ψ2|2 − |ψ−2|2 = 1

2 Fz, and hence,

C2′ :

eiχ2

√n + Fz/2 − ψ2

0

2, 0, ψ0, 0, eiχ−2

√n − Fz/2 − ψ2

0

2

T

. (103)

Because a = (c2/5)[ei(χ2+χ−2)

√(n − ψ0)2 − F2

z /4 + ψ20

]= µ is real, we have χ2 + χ−2 = 0

or π. Here, µ and Fz are determined from Eqs. (88) and (89) to be µ = 2q − γ20/(2q) and

Fz = (γ0 + p)/c1, respectively, where γ0 is a real solution of the following equation:

γ30 + pγ2

0 + 4q[q + 2c1(n − ψ20)]γ0 + 4pq2 = 0, (104)

and ψ0 is determined so as to minimize the energy per particle given by

ε = 4q1 − ψ2

0

n

+ 12

c0n +γ0

2 − p2

2c1n+

c2

10n

∣∣∣∣∣∣∣∣√

(n − ψ20)2 −

F2z

4e−i(χ2+χ−2) + ψ2

0

∣∣∣∣∣∣∣∣2

. (105)

In particular, when p = 0 and |q| > 2c1(n − ψ20), γ0 is determined to be zero, leading to

Fz = 0. Moreover, when c2 > 0, Eq. (105) is minimized at χ2+χ−2 = π and ψ20 =

12 n+5q/c2

to be ε = 2q + 12 c0n − 10q2/(c2n). The corresponding order parameter is given by

D2′ :

ieiχ

√n − 10q/c2

2, 0,

√n − 10q/c2

2, 0, ie−iχ

√n − 10q/c2

2

T

, (106)

where we set χ±2 =π2 ± χ. This state continuously becomes the cyclic state at q = 0. On

the other hand when c2 < 0, Eq. (105) is minimized at χ2+χ−2 = 0 and the order parameteris obtained to be UN for q > 0 and BN for q < 0.

• If D1 = D2 = 0, we solve the simultaneous equations D1 = 0 and D2 = 0 to obtain

µ =5q2 − γ2

0

2q= ±

√4q2 + |a|2. (107)

When q , 0, µ , a, we find from Eq. (92) that ψ0 = 0. Then, γ = 2 (ψ∗2ψ1 + ψ∗−1ψ−2). We

may use Eqs. (88)–(91) and (107) to show that

γ = 2c1ψ∗2ψ1

(1− γ0 + 3q

γ0 − 3q

)= 2c1ψ

∗−1ψ−2

(1− γ0 − 3q

γ0 + 3q

)=0. (108)

Thus, we must, in general, have ψ2ψ1 = ψ−1ψ−2 = 0. To be consistent with Eqs. (88)–(91),we find that either ψ1 = ψ−2 = 0 or ψ2 = ψ−1 = 0 should hold. In the former case, theorder parameter is given by

C3+ :

eiχ2

√n + Fz

3, 0, 0, eiχ−1

√2n − Fz

3, 0

T

(109)

22

with Fz = (p − q)/c1 and ε = 2q − 12 c0n − (p − q)2/(2c1n). In the latter case, the solution

is given by

C3− :

0, eiχ1

√2n + Fz

3, 0, 0, eiχ−2

√n − Fz

3

T

(110)

with Fz = (p + q)/c1 and ε = 2q − 12 c0n − (p + q)2/(2c1n). At zero magnetic field

(i.e., p = q = 0), these two states become the cyclic state, which are related to ψcyclic =√n/2(1, 0, i

√2, 0, 1)T as

√n3

100√2

0

= −i exp(ifzπ

4

)exp

(−i

fx − fy√2

arccos1√

3

)ψcyclic, (111)

√n3

0√2

001

= −i exp(ifz

3π4

)exp

(−i

fx + fy√2

arccos1√

3

)ψcyclic. (112)

The above results are summarized in Table 4. By comparing the energy of the obtained state,we find the ground state phase diagram of a spin-2 BEC, as shown in Fig. 4 for the special caseof p = 0 and q < 0.

23

stat

eor

derp

aram

eterψ

TF

zµ=µ−

c 0n

ε=ε−

1 2c 0

n

F2+

(eiχ

2√ n,

0,0,

0,0)

2n−2

p+

4q+

4c1n

−2p+

4q+

2c1n

F2−

(0,0,0,0,e

iχ−2√ n)

−2n

2p+

4q+

4c1n

2p+

4q+

2c1n

F1+

(0,e

iχ1√ n,

0,0,

0)n

−p+

q+

c 1n

−p+

q+

1 2c 1

n

F1−

(0,0,0,e

iχ−1√ n,

0)−n

p+

q+

c 1n

p+

q+

1 2c 1

n

UN

(0,0,e

iχ0√ n,

0,0)

01 5c 2

n1 10

c 2n

C4

( eiχ2

√ n+F

z/2

2,0,0,0,e

iχ−2

√ n−F

z/2

2

)p

c 1−c

2/20

4q+

1 5c 2

n4q+

1 10c 2

n−

p2

2(c 1−c

2/20

)n

C3 +

( eiχ2

√ n+F

z3,0,0,e

iχ−1

√ 2n−F

z3,0

)p−

qc 1

2q2q−

(p−q

)2

2c1n

C3 −

( 0,eiχ

1

√ 2n+

Fz

3,0,0,e

iχ−2

√ n−F

z3

)p+

qc 1

2q2q−

(p+

q)2

2c1n

C2

( 0,eiχ

1

√ n+F

z2,0,e

iχ−1

√ n−F

z2,0

)p

c 1−c

2/5

q+

1 5c 2

nq+

1 10c 2

n−

p2

2(c 1−c

2/5)

n

C2′

( eiχ2

√ n+F

z/2+ψ

2 02

,0,e

iχ0ψ

0,0,

eiχ−2

√ n−F

z/2−ψ

2 02

)p+γ

0c 1

2q−

γ2 0 2q

4q(1−

ψ2 0 n)+

γ2 0−p

2

2c1n

+c 2 10

n

∣ ∣ ∣ ∣ ∣√ (n−ψ

2 0)2−

F2 z 4ei(

χ2+χ−2−2χ

0)+ψ

2 0∣ ∣ ∣ ∣ ∣2

Tabl

e4:

Poss

ible

stat

esfo

rthe

grou

ndst

ate

ofa

spin

-2B

EC

.Eac

hst

ates

isas

soci

ated

with

apa

rtic

ular

sym

met

ry,a

sid

entifi

edin

Sec.

7.1.

1.In

the

C2′

stat

e,γ

0is

are

also

lutio

nof

Eq.

(104

),an

dχ

2+χ−2−

2χ0=π

forc

2>

0an

dχ

2+χ−2−

2χ0=

0fo

rc2<

0.A

tp=

0th

eC

4an

dC

2st

ates

cont

inuo

usly

beco

me

the

BN

stat

e,an

dth

eC

3 ±st

ate

beco

mes

the

cycl

icst

ate.

Forc

2>

0,th

eC

2′st

ate

beco

mes

the

D2′

stat

eat

p=

0an

dth

ecy

clic

stat

eat

p=

q=

0.

24

c2n

c1n

D2'

BN

c 2n=20c 1n

C3

|q|/2

10|q|

F2

Figure 4: Ground-state phase diagram of a spin-2 BEC for p = 0 and q < 0, where F2± states and C3± states aredegenerate in the region of F2 and C3, respectively. The M state and C state in Ref. [14] correspond to C3 and D2′ inthis phase diagram, respectively.

4. Dipolar BEC

In this section, we consider the dipole-dipole interaction (DDI) that, unlike the s-wave con-tact interaction, is long-range and anisotropic. The DDI between alkali atoms is a thousand timessmaller than the short-range interaction for the background scattering length. However, recentexperimental developments, such as the realization of a BEC of 52Cr atoms [15, 16, 17, 18,19, 20], a technique for creating cold molecular gases [21, 22, 23], and high-accuracy measure-ments [24, 25, 26], have enabled us to study several consequences of the DDI. Here, we brieflyintroduce the general properties of the DDI in cold gases (Sec. 4.1), and discuss two impor-tant cases: spin-polarized dipolar gases (Sec. 4.2) and spinor dipolar gases (Sec. 4.3). Excellentreviews on dipolar gases have been published by Baranov [27, 28] and by Lahaye et al. [29].

4.1. Dipole–dipole interaction

4.1.1. Scattering properties of dipole interactionThe dipole-dipole interaction (DDI) between two dipole moments with relative position r

[see Fig. 5 (a)] is given by

Vdd(r) = cddd1 · d2 − 3(d1 · r)(d2 · r)

r3 = cdd

∑µ,ν

d1µQµν(r)d2ν, (113)

where d1,2 are unit vectors indicating the directions of dipole moments, r = |r|, r = r/r, andQµν(r) (µ, ν = x, y, z) is a rank-2 traceless symmetric tensor that can be expressed in terms of

25

rank-2 spherical harmonics Ym2 (r) as

Qµν(r) ≡δµν − 3rµrν

r3 (114)

=

√6π5

1r3

√

23 Y0

2 (r) − Y22 (r) − Y−2

2 (r) iY22 (r) − iY−2

2 (r) Y12 (r) − Y−1

2 (r)

iY22 (r) − iY−2

2 (r)√

23 Y0

2 (r) + Y22 (r) + Y−2

2 (r) −iY12 (r) − iY−1

2 (r)

Y12 (r) − Y−1

2 (r) −iY12 (r) − iY−1

2 (r) −2√

23 Y0

2 (r)

(115)

In a special case in which the dipole moments are polarized under an external field, the DDItakes a simple form:

Vdd(r) = cddQzz(r) = cdd1 − 3 cos2 θ

r3 = −√

16π5

cddY0

2 (r)r3 , (116)

where θ is the angle between the polarization direction and the relative position r [see Fig. 5(b)].Because Vdd is negative (positive) for θ = 0 (π/2), the DDI interaction favors a “head-to-tail”configuration [Fig. 5 (c)] as opposed to a “side-by-side” one [Fig. 5 (d)].

d1d2

rd1

d2

r

µ

(a) (b) (c) (d)

Figure 5: Interaction between two dipole moments for (a) an unpolarized case and (b) a polarized case. The dipole-dipole interaction is most attractive for (c) the head-to-tail configuration and most repulsive for (d) the side-by-sideconfiguration. The bottom figures below (c) and (d) show characteristic spin textures.

The long-range character (∼ 1/r3) of the DDI becomes prominent in low-energy scattering.When a potential decreases as 1/rn for large r, the phase shift δl behaves in the zero-energylimit as k2l+1 if l < (n − 3)/2 and as kn−2 otherwise [30]. For a potential with n > 3, such asvan der Waals potential (n = 6), the main contribution at k → 0 is an s-wave channel δ0 ∼ k,and therefore, the potential can be described with a single parameter, namely, with the s-wavescattering length. On the other hand, when n = 3, all partial waves contribute to the scatteringas δl ∼ k, and therefore, the scattering process is no longer described with a single parametersuch as the s-wave scattering length. Moreover, the anisotropy of the DDI induces couplingbetween partial waves. Because the dipole interaction is described by rank-2 spherical harmonicsand it therefore has d-wave symmetry, a partial wave of angular momentum (l,m) is coupledto (l,m + ∆m f ) and (l ± 2,m + ∆m f ), where ∆m f is the change in the projected total spin ofcolliding particles and ∆m f = 0 for the polarized case. Although there is no diagonal term forthe l = 0 channel, this coupling induces the effective potential for l = 0 that behaves as r−6 atlarge distances. Hence, the dipole interaction generates a short-range interaction as well. Yi and

26

You [31, 32] have shown that away from the shape resonance [33, 34], the effective interactionbetween two dipoles can be described with the following pseudo-potential (see also [35, 36]):

Veff(r) =4π~2a(d)

Mδ(r) + Vdd(r), (117)

where a(d) depends effectively on the strength of the dipole moment.

4.1.2. Dipolar systemsThere are several systems that undergo long-range and anisotropic DDI. The coefficient cdd

in Eq. (113) is given by cdd = d2/(4πε0) and cdd = µ0d2/(4π) for electric and magnetic dipolemoments, respectively, where ε0 and µ0 are the dielectric constant and magnetic permeabilityof vacuum, respectively, and d is the magnitude of the dipole moment. The typical order ofmagnitude of the electric dipole moment is a product of the elementary charge e and the Bohrradius a0, whereas the magnetic dipole moment of an atom is of the order of the Bohr magnetonµB. Therefore, the ratio of the energy of the magnetic DDI to that of the electric DDI is given by

µ0µ2B

(ea0)/ε0∼ α2 ∼ 10−4, (118)

where α = e2/(4πε~c) ' 1/137 is the fine structure constant.Due to the strong electric dipole moments, gases of polar molecules are ideal candidates for

dipole-dominant systems. The permanent dipole moments of the lowest 1,3Σ+ states of heteronu-clear alkali dimers are calculated to be of the order of 1 Debye (' 3.335 × 10−30 Cm) [37].However, because the ground state of the molecule is rotationally symmetric and the dipole mo-ment averages to zero, we need to apply a strong electric field, typically of the order of 104

V/cm, to polarize the dipole moments. Several research groups are now trying to cool moleculesusing techniques such as buffer-gas cooling [21, 38, 39] and Stark deceleration (see [40] for areview). Another way to achieve quantum degeneracy of heteronuclear molecules is to createweakly bound Feshbach molecules from cold atoms and to transfer the molecules to the tightlybound rovibrational ground state [22, 23, 41].

The effects of the DDI have recently been observed in several systems. The Stuttgart grouphave achieved a BEC of chromium (52Cr) [15]. The 52Cr atom has a magnetic dipole moment of6µB, which is six times larger than that of alkali atoms, and therefore, the DDI is 36 times largerthan that of alkali atoms. Moreover, the s-wave scattering length of 52Cr can be decreased bymeans of Feshbach resonance [42]. Thus, the Stuttgart group realized a dipole-dominant BECand observed magnetostriction [16, 17, 18] and anisotropic collapse [19, 20].

A dipole-dominant BEC, albeit a very weak one, has also been realized in a BEC of 7Liby almost quenching the s-wave contact interaction using Feshbach resonance [26]. Fattori etal. observed the decoherence of the interferometer of 39K due to the dipole interaction [24].The Berkeley group observed an evidence of the crystallization of spin domains in a BEC of87Rb [25].

4.1.3. Tuning of dipole-dipole interactionTunability of the DDI is crucial for systematically investigating the properties of the dipolar

BEC. The strength of the electric dipole moments can be manipulated by an external electricfield with the magnetic dipole moment kept constant. Giovanazzi et al. [43] proposed a methodto control the sign of the dipole interaction as well as its strength by using a rotating field. This

27

method is applicable to both electric and magnetic dipole interactions. Consider a magneticdipole moment under a rotating magnetic field described by

B(t) = B

sinϕ cosΩtsinϕ sinΩt

cosϕ

, (119)

where ϕ is the angle between the rotating axis and the dipole moments (see Fig. 6). The angularfrequency Ω is set to be much smaller than the Larmor frequency so that the magnetic dipolemoment adiabatically follows the direction of B. When Ω is much larger than the trap frequency,the effective dipole interaction is time-averaged over 2π/Ω. Substituting d = B/|B| in Eq. (113)and taking its time-average, we obtain the following effective interaction:

〈Vdd〉 = cdd1 − 3 cos2 θ

r3

3 cos2 ϕ − 12

. (120)

The last factor in the right-hand side of (120) can take a value between −1/2 to 1. Hence, onecan change the sign of the dipole interaction as well as its strength by varying the angle ϕ.

- µ

d1

d2

r

-

'

Figure 6: Two dipoles adiabatically following the external magnetic field (119) that rotates with angular frequency Ω.The sign and strength of the dipole-dipole interaction (120) can be controlled by varying the angle ϕ.

4.1.4. Numerical methodIn the mean-field treatment, we need to calculate the effective potential induced by the distri-

bution of the dipole moments d(r), which includes an integral of the form∑ν

∫d3r′Qµν(r − r′)dν(r′). (121)

While this integral converges, the 1/r divergence of the integrand makes it difficult to numericallycarry out the integration in real space. We therefore use the convolution theorem to rewrite (121)as [44] ∑

ν

∑k

Qµν(k)dν(k)eik·r, (122)

28

where Qµν(k) ≡∫

dre−ik·rQµν(r) and dν(k) ≡∫

dre−ik·rdν(r). Here, the Fourier componentsof the kernel Qµν can be found as follows. First, we note that the spherical harmonics is theeigenstate of the three-dimensional Fourier transformation, i.e.,∫

eik·rYml (r)dΩr = 4πil jl(kr)Ym

l (k), (123)

where k = |k|, k = k/k, jl(x) is the lth spherical Bessel function and dΩr denotes the integrationof the angular part of r. Because the radial integration is then carried out as∫ ∞

0

j2(kr)r3 r2dr =

13, (124)

we obtain

Qµν(k) ≡∫

dreik·rQµν(r) = −4π3

(δµν − 3kµkν

). (125)

For spin-polarized gases, we only require

Qzz(k) = −4π3

(1 − 3 cos2 θk), (126)

where θk is the angle between the polarization direction and the momentum k.In the numerical calculation, we evaluate dν(k) from dν(r) by means of a standard fast Fourier

transform algorithm—we multiply dν(k) by Qµν(k), sum the result over ν, and perform the in-verse Fourier transform. The result is not sensitive to the grid size ∆r in the coordinate space aslong as dν(1/∆r) is negligible. However, a small numerical error arises, depending on the systemsize R, because the discrete Fourier transform presupposes a 3D periodic lattice with a unit cellof linear dimension R. If the Fourier transform is performed in a cubic region of R3, the accuracyimproves if we introduce an infrared cutoff to calculate the Fourier transform of the integrationkernel as Qcut

µν (k) =∫|r|<R/2 drQµν(r)e−ik·r [45].

4.2. Spin-polarized Dipolar BECWhen the dipole moment of every atom is polarized in the direction of an external field, say

in the z-direction, the condensate is described with a single component order parameter ψ(r) thatobeys the GP equation

i~∂

∂tψ(r, t) =

[− ~2

2M∇2 + Vtrap(r) +

g2|ψ(r, t)|2 + Φdd(r, t)

]ψ(r, t), (127)

where Vtrap(r) is a trapping potential, g = 4π~2a/M with a being the scattering length for thiscomponent, and

Φdd(r, t) = cdd

∫dr′

1 − 3 cos2 θ

|r − r′|3 |ψ(r′, t)|2 (128)

is a mean-field effective potential due to the DDI. Here, we assume a repulsive short-range inter-action a > 0 so that the condensate is always stable in the absence of the DDI, whereas cdd canbe either positive or negative due to the tunability discussed in Sec. 4.1.3.

29

4.2.1. Equilibrium shape and instabilityDue to the anisotropy of the interaction, the equilibrium shape of dipolar BECs is highly

nontrivial [46, 47, 45]. The dipole interaction works attractively along the direction of dipolemoments and repulsively in the perpendicular direction (see Fig. 5). The attractive part makesa dipole-dominant BEC unstable in a homogeneous system. This instability can be qualitativelyunderstood from the Bogoliubov spectrum in a homogeneous system [46]:

ω =

√εk

[εk + 2ng

1 + εdd(3 cos2 θk − 1)

], (129)

where n is the number density of atoms and

εdd ≡4πcdd

3g(130)

is a parameter that expresses the relative strength of the DDI against the short-range interaction.Equation (129) shows that for εdd > 1, the BEC becomes dynamically unstable because ω isimaginary for wave numbers below kc =

√4gnM(εdd − 1)/~.

When the system is confined in a trapping potential, the quantum pressure due to the confine-ment can stabilize the condensate against the attractive interaction. Santos et al. [48] numericallyinvestigated the equilibrium shapes of pure dipolar gases with a = 0. They considered an ax-isymmetric trap with trap frequencies ωr and ωz in the radial and axial directions, respectively.When the trapping potential is prolate, i.e., when the aspect ratio l =

√ωr/ωz > 1, the dipole

interaction is always attractive. Therefore, as in the case of attractive s-wave interaction, thereexists a critical number of atoms above which the BEC becomes unstable. On the other hand, ifthe trap potential is sufficiently oblate so that l < l∗ ' 0.43 and if the DDI is not too strong (seethe last paragraph of this subsection), the dipole interaction is always positive and the BEC isstable. In the region of l∗ < l < 1, the aspect ratio of the condensate increases with the number ofatoms, and finally, it becomes unstable. Yi and You [31, 32] investigated the stability includingthe effect of repulsive short-range interaction using Gaussian variational ansatz, and obtainedresults consistent with those of Ref. [48].

In Refs. [32, 44], the low-energy excitation spectrum is calculated using time-dependentvariational methods. It is shown that the instabilities of dipolar BECs in prolate (l > 1.29) andoblate (l∗ < l < 0.75) traps originate from dynamical instabilities of the breathing and quadrupolemodes, respectively. For the intermediate region, the lowest excitation frequency corresponds toa breathing mode far below the above mentioned critical number of atoms, and to a quadrupolarone near the critical value.

In the Thomas-Fermi limit, gn 1, the GP equation (127) has an exact solution [49]. Usingthe relationship (δµν − 3rµrν)/r3 = −∇µ∇ν(1/r)− 4πδµνδ(r)/3, the mean-field dipolar potential inEq. (128) can be rewritten as

Φdd(r) = −cdd

[∂2

∂z2 φ(r) +4π3

n(r)], (131)

where φ(r) ≡∫

dr′n(r′)/|r − r′|, and therefore, φ(r) obeys Poisson’s equation, ∇2φ = −4πn(r).Because n(r) is parabolic in the Thomas-Fermi limit, Poisson’s equation is satisfied by φ =a0+a1x2+a2y2+a3z2+a4x2y2+a5y2z2+a6z2x2+a7x4+a8y4+a9z4, and hence,Φdd is also quadratic.Therefore, for a harmonically trapped BEC, all terms appearing in the GP equation (127) arequadratic, and as in the simple s-wave case, an inverted parabola for the density profile gives

30

a self-consistent solution of the dipolar hydrodynamic equations. (For an analytic form of thesolution, see Refs. [49, 50].)

Eberlein et al. [50] pointed out that the potential seen by atoms exhibits a local minimumimmediately outside the condensate, i.e., the sum of the trapping and dipole potentials is locallysmaller than the chemical potential, causing an instability that brings atoms out from the con-densate to fill the dip in the potential. The presence of this instability is in agreement with thebiconcave shape of a condensate pointed out by Ronen et al. [51]. By developing a high-precisioncode for dipolar BECs, Ronen et al. [51] have shown that the same instability causes a dipolarcondensate to become unstable in the limit of strong dipole interaction, even though the trap isstrongly oblate, in disagreement with the result of Refs. [48, 31, 32]. Interestingly, in a stronglyoblate trap, the wave function of a BEC immediately before collapse becomes biconcave, withits maximum density away from the center of the gas. They have also shown that the instabilityof a strongly oblate dipolar BEC is analogous to the “roton-maxon” instability reported for 2Ddipolar gases (see Sec. 4.2.3).

4.2.2. Dipolar collapseThe Stuttgart group observed the magnetostriction of a 52Cr BEC, where εdd = 0.16 for

the background scattering length. By reducing the s-wave scattering length using a Feshbachresonance, Lahaye et al. observed a change in the aspect ratio of the condensate [18].

The same group also observed a collapse dynamics due to the DDI [20]. In the experiment,the s-wave scattering length is adiabatically decreased to 30% of the background value, and it isthen suddenly decreased below the critical value of the collapse. After a certain hold time, theBEC is released from the trap and the TOF image is taken after expansion.

The collapse dynamics proceeds as follows. After a sudden decrease in the scattering length,atoms begin to gather at the center of the trap due to the attractive interaction; with an increase inthe atomic density at the trap center, the three-body recombination losses become predominant;as a consequence of the atom loss, the attractive interaction weakens, and the atoms are ejectedoutward due to the quantum pressure.

Figure 7 (a) shows the measured TOF image (upper panels) together with the results of nu-merical simulation (lower ones). We have numerically solved the GP equation that includes thethree-body loss using the loss rate coefficient that is determined to best fit the measured losscurve (Fig. 3 of Ref. [20]). The excellent agreement between the experiment and the theorydemonstrates the validity of the mean-field description even for dipole-dominant BECs. More-over, the numerical simulation revealed that the cloverleaf pattern in the TOF image arises due tothe creation of a pair of vortex rings, as indicated in Fig. 7 (b). When collapse occurs, the atomswere ejected in the xy plane (vertical direction to the dipole moments), whereas atoms still flowinward along the z direction, giving rise to the circulation. The velocity field shown in Fig. 7 (c)clearly shows the d-wave nature of the collapse dynamics.

4.2.3. Roton-maxon excitationSantos et al. [52] have shown that the Bogoliubov excitation spectrum exhibits roton-maxon

behavior, i.e., the excitation energy has a local maximum and minimum as a function of themomentum q, in a system that is harmonically confined in the direction of the dipole moments(i.e., the z direction) and free in the x and y directions. If in-plane momenta q are much smallerthan the inverse size L of the condensate in the z direction, excitations have a 2D character.Because the dipoles are perpendicular to the plane of the trap, particles efficiently repel eachother and the in-plane excitations are phonons. Then, the DDI increases the sound velocity. For

31

0 ms 0.1 ms 0.2 ms 0.3 ms 0.4 ms 0.5 ms

x

y

z

y

z

(a)

(b) (c)

Figure 7: (a) Series of absorption images of the collapsing condensates for different values of hold time (top) and thecorresponding results of the numerical simulations obtained without adjustable parameters (bottom). (b) Iso-densitysurface of an in-trap condensate. The locations of the topological defects are indicated by the red rings. (c) Velocity fieldof the atomic flow in the x = 0 plane. Color represents the velocity (red is faster). Reprinted from Ref. [20].

q 1/L, excitations acquire a 3D character and the interparticle repulsion is reduced due tothe attractive force in the z direction. This decreases the excitation energy with an increase in q.When εdd > 1, the excitation energy reaches a minimum (roton) and then begins increasing, andthe nature of the excitations continuously become single-particle like. As the dipole interactionbecomes stronger, the energy at the roton minimum decreases and then it reaches zero, leadingto an instability. The 3D character is essential for the appearance of the roton minimum, whichdoes not appear in the quasi-2D system where the confining potential in the z direction is strongand the BEC has no degrees of freedom in this direction [53]. The roton-maxon spectrum is alsopredicted for a 1D system with laser-induced DDI [54].

4.2.4. Two-dimensional solitonsIt is known that a nonlinear Schrodinger equation with short-range interactions admits soliton

solutions in one dimension, but it does not in higher dimensions. However, Pedri and Santos [55]showed that with a nonlocal DDI, a two-dimensional BEC can have a stable soliton.

We consider a dipolar BEC polarized in the z direction, and assume that the system is confinedonly in the z direction by a harmonic potential Vtrap(r) = 1

2 Mω2z z2, and that it is free in the xy

plane. Then, the DDI is isotropic in the xy plane. We consider the following Gaussian variationalansatz

Ψ(x, y, z) =1

π3/4l3/20 LρL1/2z

exp

− x2 + y2

2l20L2ρ

− z2

2l20L2z

, (132)

32

where l0 ≡√

~/(Mωz) and Lρ and Lz are dimensionless variational parameters that characterizethe widths in the xy plane and the z direction, respectively. Using this ansatz, the mean-fieldenergy is evaluated to give

E(Lρ, Lz) =~ωz

2

1L2ρ

+1

2L2z+

L2z

2

+ 1√

2πL2ρLzl30

g4π

[1 + εdd f (κ)

], (133)

where κ ≡ Lρ/Lz is the aspect ratio and

f (κ) ≡ 2κ2 + 1κ2 − 1

− 3κ2

(κ2 − 1)√|κ2 − 1|

arctan√κ2 − 1, κ > 1

12 ln

(1+√

1−κ2

1−√

1−κ2

), κ < 1

. (134)

For fixed Lz in the absence of the DDI (εdd = 0), E(Lρ) monotonically increases or decreasesas a function of Lρ, resulting in collapse or expansion. However, in the presence of the DDI,because f (κ) is a monotonic function of κ with f (0) = −1 and f (κ → ∞) = 2, E(Lρ) may have aminimum. When the trapping potential is strong and Lz = 1, for simplicity, the energy minimumappears for

εdd < 1 +√π

2l0

aN< −2εdd, (135)

where a is the s-wave scattering length and N, the total number of atoms (see Ref. [55]). Thiscondition holds only for εdd < 0, which can be achieved using a rotational field. On the otherhand, Tikhonenkov et al. [56] have shown that when dipole moments are polarized in the 2Dplane, stable 2D soliton waves can be generated without tuning the dipole interaction. In thiscase, the soliton is anisotropic and elongated along the direction of polarization.

Because a 2D bright soliton in a dipolar gas is stable, the roton instability discussed in thepreceding subsection does not cause a collapse of the BEC but creates 2D solitons [57]. If thedipole moments are polarized perpendicular to the 2D plane, these solitons are stable as long asthe gas remains 2D. However, if the dipoles are parallel to the 2D plane, the (anisotropic) solitonsmay become unstable even in 2D if the number of particles per soliton exceeds a critical value.

4.2.5. SupersolidThe Bose-Hubbard model with long-range interactions exhibits a rich variety of phases such

as the density wave (DW), supersolid (SS), and superfluid (SF) phases [58, 59, 60, 61]. While SFand SS phases have a nonzero superfluid density, DW and SS phases have a nontrivial crystallineorder in which the particle density modulates with a periodicity that is different from that ofthe external potential. Thus, in the SS phase, both diagonal and off-diagonal long-range orderscoexist [62, 63, 64].

A dipolar gas in an optical lattice is an ideal system to realize such exotic phases, because theinteraction parameters can be experimentally controlled. Goral et al. [65] showed that DW andSS phases, as well as SF and Mott insulator (MI) phases, can be accomplished in dipolar gases ina 2D optical lattice, and Yi et al. [66] investigated detailed phase diagrams in 2D and 3D opticallattices. A dipolar gas in an optical lattice can be described with an extended Bose-Hubbardmodel given by

HEBH = −t∑〈i, j〉

(b+i b j + b+j bi) +12

U0

∑i

ni(ni − 1) +12

∑i

U iiddni(ni − 1) +

12

∑i, j

U i jddnin j, (136)

33

where bi is the annihilation operator of a particle at the lattice site i; ni = b+i bi, the correspond-ing particle-number operator; and t, the hopping matrix element between the nearest neigh-bors, and U0 > 0 is the on-site Hubbard repulsion due to the s-wave scattering. Here, t andU can be expressed in terms of the Wannier functions w(r − ri) of the lowest energy band asU = 4πa~2/M

∫dr|w(r)|4 and t =

∫drw∗(r − ri)[− ~2

2M∇2 + V0(r)]w(r − r j), where V0 is theoptical lattice potential [67]. The last two terms in Eq. (136) describe the on-side and inter-siteDDIs with coupling parameters given by

Ui j = cdd

∫dr

∫dr′|w(r − ri)|2

1 − 3 cos2 θ

|r − r′|3 |w(r′ − r j)|2, (137)

where θ is the angle between the dipole moment and vector r− ri. Here, cdd can take both positiveand negative values by using a fast rotating field (see Sec. 4.1.3). By changing the depth of anoptical lattice, the transverse confinement, and the orientation of the dipoles, t, U0, and U i j

dd canbe tuned independently.

Let us begin by reviewing the case of U i jdd = 0, i.e., the on-site Bose-Hubbard model. At

t = 0 and commensurate filling, i.e., the average number of particles per site is an integer n, theinteraction energy is minimized by populating every lattice site with exactly n atoms; therefore,the Mott insulator (MI) phase is realized. The energy cost required to create a particle-holeexcitation in the MI phase is U0 (on-site interaction energy), and therefore, the MI state is gappedand incompressible. On the other hand, for nonzero t, the kinetic energy favors particle hopping.When t is sufficiently large to overcome the interaction-energy cost (∼ U0), the system undergoesa phase transition to the superfluid (SF) phase that is gapless and compressible.

In the presence of the long-range interaction, the ground state of the extended Bose-HubbardHamiltonian (136) in a 2D optical lattice is investigated in Refs. [65, 66] using a variationalapproach based on the Gutzwiller ansatz |Ψ〉 = Πi

∑∞n=0 f i

n|n〉i, where |n〉i denotes the state withn particles at lattice site i. The coefficients f i

n can be found by minimizing the expectation value〈Ψ|HEBH − µ

∑i ni|Ψ〉 under the constraint of a fixed chemical potential µ. In this calculation, a

cutoff length λc is introduced and the interaction between lattice sites with distance larger thanλc is ignored.

When dipole moments are polarized perpendicular to the 2D lattice, the dipole interaction isrepulsive (attractive) for cdd > 0 (cdd < 0). If site i is occupied, it is energetically favorable thatits neighboring sites are equally populated for negative cdd and less populated for positive cdd.Therefore, for negative cdd, only MI and SF phases appear as in the case of the on-site Bose-Hubbard model, and local collapse occurs for large |cdd| due to the strong attractive interaction.On the other hand, for positive cdd, the DW phase emerges as an insulating state. According tothe Landau theory of the second-order phase transition, phases with distinct symmetry breakingpatterns cannot be continuously connected with each other. Therefore, as the hopping increases,the DW state first changes to the SS state and then to the SF phase.

When dipole moments are polarized along the 2D lattice, say in the y-axis, the DDI isanisotropic. The inter-site interaction for cdd > 0 (cdd < 0) is attractive (repulsive) in the ydirection and repulsive (attractive) in the x direction. In this case, the particle density along therepulsive direction becomes periodically modulated, whereas it remains constant along the at-tractive direction, resulting in striped DW and striped SS states. Local collapse occurs for large|cdd| for both positive and negative cdd.

In Ref. [66], the phase diagram in a 3D lattice is also investigated, where dipole moments arepolarized along the z-axis. For cdd > 0, the DDI is attractive in the z direction and repulsive in the

34

xy direction. Hence, the DW phase forms a checkerboard-type pattern on the xy plane and it isuniform in the z direction. The density pattern in the SS phase reflects the same symmetry as theDW phase. On the other hand, for cdd < 0, layers of high-density sites align along the z directiondue to the repulsive interaction. Hence, layered DW and SS phases emerge perpendicular to thez-axis.

Recently, Menotti et al. [68] studied a polarized dipolar gas in a 2D optical lattice beyond theground state. They have shown that there exist many metastable states in the insulator region de-pending on the cutoff length λc. The number of metastable states and pattern variations increasesrapidly with the number of lattice sites. Metastable states also appear in magnetic domains insolid-state ferromagnets, classical ferrofulids, and quantum ferrofluids (see the next subsection),and they appear to be a characteristic consequence of long-range interactions.

4.2.6. FerrofluidLet us consider a repulsively interacting two-component BEC, in which atoms in component

1 have a magnetic dipole moment, whereas those in component 2 are nonmagnetic. The m = −3and 0 states of 52Cr can be a candidate for such a system. We apply a magnetic-field gradient inthe z direction, so that the two components phase-separate as shown in Fig. 8.

interface

component 2 without magnetic dipole

component 1 with magnetic dipole

x

z

B

Figure 8: Schematic illustration of a two-component BEC system in which atoms in component 1 have a magnetic dipolemoment and those in component 2 are nonmagnetic. The magnetic dipole is polarized in the z direction by an externalmagnetic field. The two components are separated due to the field gradient B′(z) < 0. Reprinted from Ref. [69].

The equilibrium configuration of this system can be found by solving the two-component GPequation in imaginary time [69]. We found a stable hexagonal pattern of the density distributionof component 1 as shown in Figs. 9 (a) and (b). We also found several metastable states start-ing from different initial states, such as stripe, concentric, and deformed hexagonal patterns, asshown in Figs. 9 (c)–(e). The energies for these patterns are almost degenerate; however, eachpattern is robust against small perturbations.

A striking difference between the present system and magnetic liquids is that the present sys-tem is a superfluid that supports a persistent current. We found a metastable state with a finitecirculation in component 1 and no circulation in component 2 in a stationary trap. Remarkably,the Rosensweig pattern emerges even in the presence of a superflow, indicating that the sys-tem simultaneously exhibits a diagonal and off-diagonal long-range order, although the latter isimbedded in the GP theory (see Ref. [69] for more detail).

4.3. Spinor-dipolar BECLet us now consider a dipolar BEC with spin degrees of freedom. By “spinor-dipolar,” we

imply that the the direction of the spin of the order parameter is not polarized by an external field35

(b)

x

y

(a)

-2

2

z

component 1 component 2 total

x

y

(c) (d) (e)

0 0.003

z

0

2

0 0.003

Figure 9: (a) Isodensity surface of a hexagonal pattern formed in component 1 at B′ = −1 G/cm. (b) Column densitiesfor component 1 (left), component 2 (center), and the total system (right). (c)–(d) Metastable configurations found atB′ = −0.3 G/cm, where the isodensity surface (top) and column density (bottom) for component 1 are shown. Reprintedfrom Ref. [69].

but can vary in space. In such a situation, the system develops a nontrivial spin texture due tothe DDI and self-organized magnetic patters may form. Possible quantum phases that arise dueto the DDI have been discussed by using the single mode approximation [70] in which all spincomponents are assumed to share the same spatial mode. The spin ordering and spin waves inMott states in 1D and 2D optical lattices have also been discussed in Refs. [71, 72, 73].

In the second-quantized form, the magnetic DDI for a spinor BEC is given by

Vdd =cdd

2

∫dr

∫dr′

∑µν

: Fµ(r)Qµν(r − r′)Fν(r′) :, (138)

where Fµ(r) is defined by Eq. (17) and cdd = µ0(gFµB)2/(4π), with gF being the Lande g-factorfor the atom. The dipole interaction yields a non-local term in the GP equations. In fact, the GP

36

equations for a spin-1 BEC [Eq. (52)] are given by

i~∂ψm

∂t=

[−~2∇2

2M+ U(r) − pm + qm2

]ψm

+ c0nψm + c1

1∑n=−1

F · fmnψn + cdd

1∑n=−1

b · fmnψn, (139)

and those for spin-2 BEC [Eqs. (78)–(80)] are given by

i~∂ψm

∂t=

[−~2∇2

2M+ U(r) − pm + qm2

]ψm

+ c0nψm + c1

2∑n=−2

F · fmnψn +c2√

5Aψ∗−m + cdd

2∑n=−2

b · fmnψn (140)

Here, b is the effective dipole field defined by

bµ ≡∫

dr′∑µν

Qµν(r − r′)Fν(r′), (141)

which is nonlocal, and Bdd = cddb/(gFµB) works as an effective magnetic field.

4.3.1. Einstein-de Haas effectWe first consider the dynamics of a spinor-dipolar BEC at zero external magnetic field. Sup-

pose that we prepare a spin polarized BEC in an external field. We consider what happens if wesuddenly turn off the external field. In the absence of the DDI, a spin polarized BEC is stablebecause the total spin of the system will then be conserved. However, in the presence of the DDI,the situation drastically changes because the DDI does not conserve the spin angular momentum;it only conserves the total (i.e., spin plus orbital) angular momentum of the system. This can beunderstood from the fact that the DDI (113) is not invariant under a rotation in spin space due tothe spin-orbit coupling term (d · r). On the other hand, because Vdd(r) is invariant under simul-taneous rotations in spin and coordinate spaces, the DDI conserves the total angular momentumof the system. Therefore, the DDI causes spin relaxation to occur by transferring the angularmomentum from the spin to the orbital part; as a consequence, the BEC starts rotating. This isthe Einstein-de Haas effect in atomic BECs [74].

We numerically simulated such dynamics for the case of a 52Cr BEC that is confined in anaxisymmetric trap. Figure 10 (a) shows the time evolution of the spin angular momentum Mz =∫

dr∑3

m=−3 m|ψm(r)|2 and the orbital angular momentum Lz =∫

dr∑3

m=−3 ψ∗m(r)

(−i ∂

∂ϕ

)ψm(r),

where ϕ is the azimuthal angle around the symmetry axis. The initial state is assumed to be spin-polarized in the magnetic sublevel m = −3. We find that as time progresses, the spin angularmomentum decreases with a concomitant increase in the orbital angular momentum, so that thetotal angular momentum of the system is conserved.

In the dynamics, the order parameter is given by

ψm(r, ϕ, z) = ei(Jz−m)ϕηm(r, z), (142)

where (r, ϕ, z) are the cylindrical coordinates, ηm is a complex function of r and z, and Jz isthe projected total angular momentum on the z axis (Jz = −3 in the present case). The order

37

parameter (142) is the eigenstate of the projected total angular momentum that satisfies

(fz + `z)ψm(r) =(m − i

∂

∂ϕ

)ψm(r) = Jzψm(r). (143)

The order parameters for magnetic sublevels m = −3, −2, and −1 at ωt = 2 are shown in Fig. 10(b)–(d). Each spin component except the m = −3 state is a vortex state with the winding numberJz − m, and the BEC as a whole forms a coreless vortex.

(a)

(b) (c) (d)

0

-1

-2

-3

0 2 4 6 8 10

[N~]

ωt

Figure 10: (a) Time evolution of spin and orbital angular momentum due to the Einstein-de Haas effect. (b)–(d) Isopycnicsurfaces of (b) ψ−3, (c) ψ−2, and (d) ψ−1 at ωt = 2, where the color on the surface represents the phase of the orderparameter. Reprinted from Ref. [74].

The physical origin of the spin relaxation is the Larmor precision of atomic spins around thedipole field given by Eq. (141). Figure 11 (a) shows the dipole field induced by a spin-polarizedBEC in a spherical trap, and Figs. 11 (b) and (c) show the spin textures at ωt = 2. The whirlingpatterns in the upper and lower hemispheres exhibit opposite directions, reflecting the fact thatthe xy components of the dipole field are antisymmetric with respect to the z = 0 plane.

In the case of a 52Cr BEC, the DDI compensates for the kinetic energy to form the spintexture. As discussed in the next section, a spin-1 87Rb BEC is ferromagnetic, and thereforea spin-polarized state is rather stable. In such a case, an external magnetic field applied in theopposite direction to the polarization causes a resonance between the kinetic energy and thelinear Zeeman energy, leading to the Einstein-de Haas effect [75]. In the case of a spin-1 23NaBEC, the spin-exchange interaction favors a non-magnetic state, and therefore, it tends to destroythe spin-polarized state.

4.3.2. Ground-state spin textures at zero magnetic fieldThe DDI is known to create magnetic domains in solid-state ferromagnets. Similar structures

are expected to appear in ferromagnetic BECs. Here, we consider ground-state spin structures.38

0.0013

00

0

0

0

0

0

(b)

(c)

1 mµ

(a)

0

0

0

0

0

0

4.10

0

4.1

-4.1

x [ m]µ

z [

m]

µ

0.25

0

[mG]

Figure 11: (a) Dipole field induced by a spin-polarized BEC in a spherical trap, where the color of each arrow denotesthe field strength (see the left gauge) and the solid curve indicates the periphery of the condensate. (b),(c) Spin textureson the (b) z = 2 µm plane and (c) z = −2 µm plane at ωt = 2. The length of the arrows represents the magnitude of thespin vector projected on the xy plane, and the color indicates |F|a3

ho/N with aho being the harmonic oscillator length andN, the total number of atoms. Reprinted from Ref. [74].

The mean-field energy for the DDI can be rewritten as

〈Vdd〉 =cdd

2

∫dr

∫dr′

∑µν

Fµ(r)Qµν(r − r′)Fν(r′)

=cdd

2

∑k

∑µν

Fµ(k)Qµν(k)Fν(−k)

=cdd

24π3

∑k

[3| k · F(k)|2 − |F(k)|2

], (144)

where F(k) is the Fourier transformation of F(r) and Qµν(k) is given by Eq. (125). When theBEC is fully magnetized, i.e., |F(r)| = f n(r), with f being the atomic spin angular momentumand n(r), the number density, the last term in Eq. (144) makes a constant contribution determinedby the density distribution. Then, the dipole-favored structure satisfies k · F(k) = 0 for all k, orin real space, ∇ · F(r) = 0. This condition is the same as that for solid-state ferromagnets [76],and the corersponding stable domain structure is known as a flux-closure structure. In solid-stateferromagnets, the magnitude of magnetization is constant and the domain structure is constrainedby boundary conditions such that the magnetization is parallel to the surface. In contrast, aspatial variation of |F(r)| in proportion to the number density yields a spin texture in trappedcondensates.

As compared to solid-state ferromagnets, the unique feature of ferromagnetic BECs is thatthey exhibit the spin-gauge symmetry that generate a supercurrent by developing spin textures(see, Sec. 6.2.1). The supercurrent, defined by Eq. (264), is generated by a spatial variation of thedirection of magnetization. Therefore, even in the ground state in a stationary trap, a supercurrentcan flow due to a dipole-induced spin texture.

39

Figure. 12 (a) shows the numerically determined phase diagram for a spin-1 ferromagneticBEC [77] as a function of RTF/ξsp and RTF/ξdd, where RTF is the Thomas-Fermi radius; ξsp =

~/√

2M|c1|n, the spin healing length; and ξdd = ~/√

2Mcddn, the dipole healing length. We findthree phases—flower (FL), chiral spin-vortex (CSV), and polar-core vortex (PCV)—whose spinconfigurations are shown in Fig. 12 (b), (c), and (d), respectively. The FL and CSV phases areeigenstates of the projected total angular momentum Jz = 1, whose order parameter is given byEq. (142). The orbital angular momentum for the order parameter (142) is calculated as

Lz =

∫dr

∑m=0,±1

ψ∗m(r)(−i

∂

∂ϕ

)ψm(r) =

∫dr

∑m=0,±1

(Jz − m)|ψm(r)|2. (145)

Therefore, when Jz = 1, the BEC can have a nonzero orbital angular momentum depending onthe population in the m = 0 and −1 states. When the system size is smaller than ξdd, the kineticenergy, which hinders the spin texture from developing, is dominant, and the spin is almostpolarized. Most of the atoms therefore reside in the magnetic sublevel m = 1 and the orbitalangular momentum is quite small. As the BEC becomes larger, the spin texture develops toform a flux-closure structure. Consequently, the number of atoms in the m = 0 and −1 magneticsublevels increases, leading to an increase in the orbital angular momentum. The differencebetween FL and CSV phases is the symmetry of spin textures: the spin texture has chirality inthe latter phase. Furthermore, an increase in the size of the BEC results in the phase transition tothe polar-core vortex state with Jz = 0, because the Jz = 1 state costs a large kinetic energy whenthe population in the m = −1 state increases. In the Jz = 0 phase, a polar-core vortex appears atthe symmetry axis.

All three phases in Fig. 12 can be realized in spin-1 87Rb BECs, which traces on the dottedline shown in Fig. 12 (a). The orbital angular momentum along this line is plotted in Fig. 13 asa function of the total number of atoms N and trap frequency ω. The orbital angular momentumin the ground state has a significant value, and it increases up to approximately 0.4~ in the CSVphase.

For higher-spin BECs, similar spin textures are expected to appear in the ferromagneticphase, although the mass current depends on the value of the atomic spin and the core structurein the polar-core vortex region is nontrivial. Spin textures in the limit of strong ferromagneticinteraction are discussed in Ref. [78]. On the other hand, even in the case of a non-ferromagneticphase such as a spin-1 23Na BEC, a spin texture may appear if the dipole interaction is effectivelyenhanced in a pancake trap with a large aspect ratio [79].

4.3.3. Dipole-dipole interaction under a magnetic fieldNow, we consider the effect of an external magnetic field on the DDI. Suppose that a ho-

mogeneous external magnetic field B is applied in the z direction. Because the linear Zeemanterm gFzB rotates the atomic spin around the z-axis at the Larmor frequency ωL = gFµBB/~,it is convenient to describe the system in the rotating frame of reference in spin space. Thespin vector operators F(rot) in the rotating frame are related to those in the laboratory frame byF(rot)± ≡ F(rot)

x ± iF(rot)y = e±iωLtF± and F(rot)

z = Fz. Then, the dipolar interaction (138) can be

40

(c) FL phase (d) CSV phase(b) PCV phase

0 1

x z− plane

x y plane−

0

1

2

3

4

0 2 4 6 8 10 12 14

RT

F /

dd

RTF /

sp

PCV

CSV

FLRb line

(a)

x z− plane

x y plane−

x z− plane

x y plane−

x

zy

Figure 12: (a) Phase diagram of a ferromagnetic BEC in a spherical trap. A BEC of spin-1 87Rb atoms traces on thedotted line. (b)–(d) Cross sections of spin textures in the xy (upper) and xz (bottom) planes in the (b) flower phase, (c)chiral spin-vortex phase, and (d) polar-core vortex phase, where the color and length of each arrow indicate |F|a3

ho/N andthe spin projection onto the planes. Reprinted from Ref. [77].

rewritten in terms of F(rot) as

Vdd = −√

6π5

cdd

2

∫dr

∫dr′

1|r − r′|3

× :[Y0

2 (e)√

6

4F(rot)

z (r)F(rot)z (r′) − F(rot)

+ (r)F(rot)− (r′) − F(rot)

− (r)F(rot)+ (r′)

+ e−iωLtY−1

2 (e)F(rot)+ (r)F(rot)

z (r′) + F(rot)z (r)F(rot)

+ (r′)

− eiωLtY12 (e)

F(rot)− (r)F(rot)

z (r′) + F(rot)z (r)F(rot)

− (r′)

+ e−2iωLtY−22 (e)F(rot)

+ (r)F(rot)+ (r′) + e2iωLtY2

2 (e)F(rot)− (r)F(rot)

− (r′)]

:, (146)

where e ≡ (r−r′)/|r−r′|. When the Zeeman energy is much larger than the DDI, i.e., ~ωL cddn,the spin dynamics due to the dipolar interaction is much slower than the Larmor precession. Ina large magnetic field, we may therefore use an effective DDI that is time-averaged over the

41

0

0.1

0.2

0.3

0.4

0.5

0 1000 2000

L z /

N

( [rad/s] N2 )1/5

PCVCSVFL

Figure 13: Orbital angular momentum for a spin-1 87 Rb BEC in a spherical trap with trap frequency ω. Reprinted fromRef. [77].

Larmor precession period 2π/ωL:

¯Vdd = −cdd

4

∫dr

∫dr′|r − r′|2 − 3(z − z′)2

|r − r′|5

× :F(rot)(r) · F(rot)(r′) − 3F(rot)

z (r)F(rot)z (r′)

: . (147)

More generally, an effective DDI under an external field applied in the B direction is given by

¯Vdd = −cdd

4

∑µν

∫dr

∫dr′ : F(rot)(r)Qµν(r − r′)F(rot)(r′) :, (148)

where the time-averaged integration kernel

Qµν(r) =1 − 3(r · B)2

r3 (δµν − 3BµBν) (149)

corresponds to Eq. (120). In contrast to Eq. (138), the time-averaged interaction (147) separatelyconserves the projected total spin angular momentum and the projected relative orbital angularmomentum. However, the long-range and anisotropic nature of the dipolar interaction is stillmaintained in Eq. (147). While the total longitudinal magnetization is conserved in the presenceof a magnetic field, the anisotropic dipole interaction can induce spin textures in the transversemagnetization [80].

Equation (149) provides information about how the dipole moments tend to align in theground state. Longitudinal magnetization tends to become parallel in the direction of the mag-netic field, whereas it tends to be anti-parallel in the direction perpendicular to the magnetic field[Figs. 14 (a) and (b)]. On the other hand, transverse magnetization tends to be anti-parallel alongthe magnetic field and parallel in the perpendicular direction [Figs. 14 (c) and (d)]. Compar-ing the dipole energy for longitudinal and transverse magnetizations, the energy for longitudinalmagnetization is twice as large (in negative sign) as that for transverse magnetization.

The quadratic Zeeman effect also contributes to the ground state. As discussed in Sec. 3,the quadratic Zeeman effect in a spin-1 87Rb BEC induces transverse magnetization. A stable

42

(a) (b)

B

(c) (d)

Figure 14: Alignments of (a)(b) longitudinal and (c)(d) transverse magnetization that are favored by the time-averageddipole interaction. Due to the Larmor precession, transverse magnetization tends to be anti-parallel along the magneticfield and parallel perpendicular to the magnetic field.

spin texture is determined by the balance between the quadratic Zeeman effect and the dipolareffect [81, 82, 83].

For simplicity, we consider a quasi-2D trap where the cloud size in the strongly confined di-rection is smaller than the spin healing length so that spin texture is uniform in this direction. Wealso assume that the magnetic field is applied parallel to the 2D plane and the total magnetizationalong the magnetic field is set to zero [25].

When the quadratic Zeeman energy is stronger than the DDI, the magnetization is restrictedperpendicular to the external field. Then, the transverse magnetization forms a helix structure, asin the case of Fig. 14 (c). On the other hand, when the DDI is stronger than the quadratic Zeemanenergy, the longitudinal magnetization forms a domain structure to minimize the DDI [Fig. 14(b)]. However, these ground states differ from the magnetic patterns observed by the Berkeleygroup [25, 84]. The equilibrium pattern for this system has yet to be understood.

A further topic is the rotating spinor dipolar BEC in the presence of an external field. Whenthe rotating frequency is close to the Larmor precession, the third or fourth line in Eq. (146)remains in the time-averaged dipolar interaction instead of the second line in Eq. (146). For ex-ample, when a spin-polarized BEC is rotated, the spin relaxation is expected to occur resonantlywith the Larmor frequency.

5. Bogoliubov Theory

Quantum and thermal fluctuations as well as external perturbations induce excitations fromthe mean-field ground state. When the excitations are weak, they can be described by the Bo-goliubov theory. We express the field operator ψm as a sum of its mean-field value ψm and thedeviation from it δψm:

ψm = ψm + δψm (m = f , f − 1, · · · ,− f ). (150)

Here, the mean-field part can be calculated from the Gross-Pitaevskii theory described in Sec. 3.The basic concept of the Bogoliubov theory is to substitute Eq. (150) in the second-quantizedHamiltonian and retain the terms up to the second order in δψm. The resulting Hamiltonian canbe diagonalized by means of the Bogoliubov transformations. The Bogoliubov transformationsare canonical transformations that couple particle-like excitations with hole-like ones via a Bose-Einstein condensate and they are non-perturbative. The resultant dispersion relation thereforedepends on the coupling constant in a non-analytic manner.

43

We consider a spatially uniform system (U(r) = 0). In this case, it is convenient to analyzethe system in Fourier space. We obtain the Fourier expansion of the field operator:

ψm =1√Ω

∑k

akmeikr, (151)

where Ω is the volume of the system. Then, the noninteracting part of the Hamiltonian is writtenas

H0 =∑km

(εk − pm + qm2)nkm, (152)

where εk = ~2 k2/2M and nkm = a†kmakm. Because the interaction Hamiltonian comprises theparticle density, spin density, and spin-singlet pair operators, it is convenient to introduce thecorresponding Fourier-transformed operators:

ρk =

∫drn(r)e−ikr =

∑qm

a†qmaq+km (153)

Fk =

∫drF(r)e−ikr =

∑qmn

fmna†qmaq+kn (154)

Ak =√

2 f + 1∫

drA00(r)e−ikr =∑qm

(−1) f−maqmaq−k,−m, (155)

where it is understood that m runs over m = f , f − 1, · · · ,− f .We assume that the BEC occurs in the k = 0 state, and decompose the terms in Eqs. (152)–

(155) into the k = 0 component and the rest. Then, Eq. (152) is written as

H0 =∑

m

(−pm + qm2)n0m +∑

k,0,m

(εk − pm + qm2)nkm. (156)

As shown below, the interaction Hamiltonians involves quadratic forms of the Fourier-transformedterms in Eqs. (153)–(155). They can be decomposed similarly as∑

k

: ρ†kρk : = N(N − 1) +∑k,0

: ρ†kρk :, (157)∑k

: F†kFk : = : F†0F0 : +∑k,0

: F†kFk :, (158)∑k

: A†kAk : = : A†0A0 : +∑k,0

: A†kAk : . (159)

In Eq. (157), the dominant terms in ρk,0 =∑

qm a†qmaq+km are the terms with q = 0 andq = −k:

ρk,0 '∑

m

(a†0makm + a†−kma0m) = Dk + D†−k, (160)

where Dk =∑

m a†0makm describes the density fluctuations from the condensate. SubstitutingEq. (160) into Eq. (157), we obtain∑

k

: ρ†kρk :' N(N − 1) +∑k,0

: (2D†kDk + DkD−k + D†kD†−k) : . (161)

44

In Eq. (158), the term F0 can be further decomposed into

F0 = F +∑

q,0,mn

fmna†qmaq+kn, (162)

where F =∑

mn fmna†0ma0n is the zeroth-order term and the last term is of the second order. Onthe other hand, the term Fk,0 can be approximated as

Fk,0 '∑mn

fmn(a†0makn + a†−kma0n). (163)

This operator describes the spin fluctuations from the condensate. Substituting Eqs. (162) and(163) into Eq. (158) and retaining the terms up to the second order, we obtain∑

k

: F†k Fk :'F2 + 2F∑

q,0,mn

fmna†qmaq+kn +∑

k,0,mnm′n′fmnfm′n′

× (2a†0ma0n′ a†km′ akn + a†0ma†0m′ akna−kn′ + a†0na0n′ a

†kma†−kn′). (164)

Similarly, Eq. (159) can be approximated as

∑k

: A†kAk : = 4S +S − + 2

S + ∑k,0,m

(−1) f−makma−k,−m + H.c.

+4

∑k,0,mn

(−1)m+na†0ma0,na†k,−mak,−n, (165)

where H.c. denotes the Hermitian conjugate of the preceding term and

S − = (S +)† =12

∑m

(−1) f−ma0ma0,−m. (166)

In the Bogoliubov approximation, we replace a0m by√

Nζm, which satisfies the followingnormalization condition: ∑

m

|ζm|2 = 1 − 1N

∑k,0,m

nkm. (167)

By subtracting the last term, we can construct a number-conserving Bogoliubov theory withoutintroducing the chemical potential. In keeping with the Bogoliubov theory, we retain the lastterms in operator form.

5.1. Spin-1 BECs

The interaction Hamiltonian for the spin-1 BEC is given by Eq. (23) or (24). In Fourier space,it is expressed as

V =1

2Ω

∑k

[(c0 + c1) : ρ†kρk : −c1A†kAk

], (168)

45

where c0 ≡ (g0 + 2g2)/3 and c1 ≡ (g2 − g0)/3. Using Eqs. (156), (161), and (159), we obtain theeffective Hamiltonian for the spin-1 BEC:

Heff =c0 + c1

2ΩN(N − 1) − 2c1

ΩS +S − − (p − q)n0,1 + (p + q)n0,−1

+∑k,0m

(εk − pm + qm2

)nkm

+c0 + c1

2Ω

∑k,0

: (2D†kDk + DkD−k + D†kD†−k) :

−c1

Ω

S + ∑k,0m

(−1) f−makma−k,−m + H.c.

−2c1

Ω

∑kmn

(−1)m+na†0ma0nak,−mak,−n. (169)

The first three terms on the right-hand side were studied in Refs. [85, 4, 86]; the third andfourth lines describe the density and spin fluctuations, respectively, and the last line describes thecoupling between them. In the following discussions, we shall discuss the case for q = 0 unlessotherwise stated. The case for q , 0 is discussed in Ref. [10].

5.1.1. Ferromagnetic phaseWhen c1 < 0, the ground-state phase is ferromagnetic, and we have

|ζ1|2 = 1 − 1N

∑k,0,m

nkm, ζ0 = ζ−1 = 0. (170)

Substituting this into Eq. (169), we obtain

HF =12

(c0 + c1)n(N − 1) − pN

+∑k,0

[εknk1 +

12

(c0 + c1)n(ak1a−k1 + nk1 + H.c.)

+(εk + p)nk0 + (εk + 2p − 2c1n) nk−1], (171)

where n = N/Ω. The Hamiltonian (171) shows that the m = 0 and m = −1 modes describethe massive transverse spin and quadrupole spin excitations, respectively, with the dispersionrelations given by

EFk,0 = εk + p, (172)

EFk,−1 = εk + 2p − 2c1n. (173)

To diagonalize the Hamiltonian of the m = 1 mode, we perform the Bogoliubov transforma-tion:

ak = bk cosh θ − b†−k sinh θ. (174)

46

Substituting this into Eq. (171) and requiring that the coefficients of the off-diagonal terms suchas bkb−k vanish, we obtain

sinh(2θ) =gn√

ε2k − (gn)2

, cosh(2θ) =εk√

ε2k − (gn)2

, (175)

where

g ≡ c0 + c1 =4π~2

Ma2 (176)

and εk ≡ εk + (c0 + c1)n. The diagonalized Hamiltonian then becomes

HF =gn2

(N − 1) − pN +12

∑k

( √εk(εk + 2gn) − εk − gn

)+

∑k,0

[EF

k,1b†kbk + EFk,0nk0 + EF

k,−1nk,−1

], (177)

where

EFk,1 =

√εk[εk + 2(c0 + c1)n] (178)

describes the coupled density and longitudinal spin excitations.The sum over k in the first line of Eq. (177) diverges for large k—an artifact that arises due

to the use of a delta function that does not correctly describe the short-range behavior of the realpotential. For large k, the summand becomes

√εk(εk + 2gn) − εk − gn = − (gn)2

2εk+ · · · . (179)

Subtracting this term, the sum converges, giving

12

∑k

(√εk(εk + 2gn) − εk − gn +

(gn)2

2εk

)=

12815

gn

√na3

2

πN. (180)

Thus, the ground-state energy of the system is

EF0 =

gnN2

1 + 12815

gn

√na3

2

π

− pN. (181)

It follows from this that the sound velocity is given by

c =

√Ω2

MN∂2EF

0

∂Ω2 =

√g2nM

1 + 8

√na3

2

π

, (182)

where the leading term is the Bogoliubov phonon velocity and the last term is called the Lee-Yang correction to it.

47

The Bogoliubov approximation takes into account virtual pair excitations from the mean-field, and therefore, the properties of the ground state are very different from those of the meanfield. A mean-field ground state in the ferromagnetic phase is

(a†0,1)N

√N!|vac〉. (183)

On the other hand, the Bogoliubov ground state is given by

|Φ〉 ∼ exp

φ0a†01 −∑

kx>0,ky,kz

νka†k1a†−k1

|vac〉, (184)

where φ20 = N[1 − (8/3)

√na3

t /π], νk = 1 + ck −√

ck(ck + 2) with ck ≡ k2/(8πatn), and thesummation is performed over the entire range of ky and kz, but the summation over kx is restrictedto the positive side to avoid double counting. In contrast to the single-particle state in Eq. (183),the Bogoliubov ground state is a pair-correlated state. In momentum space, the Bogoliubovground state exhibits pair correlation having opposite momenta; in real space, it describes thepairwise repulsive interaction. To show this, let us write down the many-body wave function inthe coordinate representation:

Ψ(r1α1, · · · , rNαN) = 〈vac|ψα1 (r1) · · · ψα1 (r1)|Φ〉. (185)

Substituting Eq. (184) into this and ignoring the terms of the order of 1/N, we obtain [87]

Ψ(r11, · · · , rN1) 'exp

(− 4

9 (3π − 8)N√

na32/π

)(2πNV2N)1/4 exp

−∑i< j

a2

ri je−ri j/ξ

, (186)

where ri j ≡ |ri − r j| and ξ ≡ (8πa2n)−1/2 is the correlation length. Thus, the many-body wavefunction decays exponentially when any two particles approach closer than the scattering lengthor the correlation (or healing) length.

5.1.2. Antiferromagnetic phaseWhen c1 > 0, the ground state is antiferromagnetic, and we have

ζ0 = 0, |ζ±1|2 =12± γ

2− 1

2N

∑k,0m

nk,m, (187)

where γ ≡ p/(c1n) and the last term is retained in order to construct a number-conserving Bo-goliubov theory. Substituting this into Eq. (169), we obtain

HAF =12

c0n(N − 1) +12

c1n(γ2N − 1

)− pγN

+∑

k,0,m

(εk + c1nδm,0)nkm +1

2Ω

∑k,0

c0(DkD−k + D†kDk)

+c1(S kS −k + S †kS k + n|ζ1ζ−1|ak,0a−k,0) + H.c., (188)

48

where

Dk =∑

m

a†0makm '√

N2

(√1 +

γ

2ak1 +

√1 − γ

2ak,−1

)(189)

S k =∑

m

a†0makm '√

N2

(√1 +

γ

2ak1 −

√1 − γ

2ak,−1

). (190)

We note that when γ , 0, the density and spin excitations are coupled because [Dk, S†k] = γ/2.

The m = 0 mode is decoupled and its dispersion relation can be calculated in a manner similarto that for the m = 1 mode of the ferromagnetic phase, with the result

EAFk,0 =

√ε2

k + 2c1nεk + p2. (191)

The m = ±1 modes are coupled, and their dispersion relations can be obtained by solving theHeisenberg equations of motion for the m = ±1 modes [6]:

EAFk,± =

√√√ε2

k+ 2(c0+c1)nεk± εk

√n2(c0−c1)2+

4p2c0

c1,

(192)

In the absence of an external magnetic field (p = γ = 0), Eq. (192) reduces to the excitationspectra of the uncoupled density and spin waves:

EAFk,+ =

√εk(εk + 2c0n), EAF

k,− =√εk(εk + 2c1n). (193)

5.1.3. Domain formationThe MIT group observed the formation of metastable spin domains [88]. They first prepared

all atoms in the m = 1 state and transferred half of them to the m = 0 state by irradiating therf field. Then, letting the system evolve freely in time while applying a uniform magnetic fieldto prevent the m = −1 component from appearing due to the quadratic Zeeman effect, theyfound that spin domains develop with the two components being aligned alternatively. It wasdiscussed [88, 89, 90] that this phenomenon is due to the imaginary frequencies of the excitationmodes. Here, we use the Bogoliubov theory to confirm this hypothesis and to derive a generalexpression for the dispersion relation. The experimental conditions in effect amount to setting

|ζ1|2 = |ζ0|2 =12− 1

2N

∑k,0

(nk1 + nk0), ζ−1 = 0. (194)

Substituting this in Eq. (169), we obtain

H =c0 + c1

2n(N − 1) − c1

8nN

+∑k,0

[(εk +

c0n2+

3c1n4

)nk1 +

(εk +

c0n2− c1n

4

)nk1

+c0

8n(ak0a−k0 + h.c.)

+c0 + c1

8n(ak1a−k1 + ak1a−k0 + a†k1a−k0 + h.c.)

]. (195)

49

The eigenspectrum can be obtained by writing down the equations of motion for a±k,1 and a±k,0.Rewriting the resulting equations in terms of A = ak,1 + a†−k,1 and B = ak,0 + a†−k,0, we obtain

(i~)2 d2

dt2 A =[ε2

k1 − (p + q)2]

A + 2(p + q)(εk1 − p − q)B,

(i~)2 d2

dt2 B =[ε2

k0 − p2]

B + 2(p + q)(εk0 − p)A. (196)

Assuming that A ∝ e−i~ Ekt and B ∝ e−

i~ Ekt, we obtain

(Ek)2 = ε2k + (2u + v)εk +

34

v2 ±[4(u2 + 2uv + 2v2)ε2

k

+2v2(2u + v)εk − v3(u +

34

v)] 1

2

, (197)

where u ≡ c0n/2 and v ≡ c1n/2. For parameters of the MIT experiment, n ∼ 1014 cm−3,at ∼ 29Å, and as ∼ 26Å [91], we find that v/u 1. In Eq. (197), ignoring higher-order powersof v/u, we obtain

(Ek)2 = ε2k + (2u + v)εk ± 2(u + v)εk,

where the plus (minus) sign corresponds to the density (spin) wave. We see that the energy ofthe spin wave becomes pure imaginary for εk < v, implying the formation of spin domains. Thecorresponding wavelength defines the characteristic length scale of the spin domains as

λc =

√3π

(a2 − a0)n. (198)

This result agrees with that of Ref. [90], except for a numerical factor. Using the above pa-rameters, we obtain λ ∼ 18 µm; this is in reasonable agreement with the observed value ofapproximately about 40 µm [88].

5.2. Spin-2 BECs

We again assume that the total number of particles is conserved:

NBEC +∑k,0

∑m

nk,m = N. (199)

Then, the one-body part of the Hamiltonian becomes

H0 = −pN fz +∑k,0

∑m

[εk − p(m − fz)

]nk,m, (200)

where

fz ≡∑

m

m|ζm|2. (201)

50

The interaction Hamiltonian of a spin-2 BEC is given by Eq. (30) and its Fourier transformbecomes

V =1

2Ω

∑k

(c0 : ρ†kρk : +c1 : F†k Fk : +

c2

5A†kAk

). (202)

Using Eqs. (156), (161), (164), and (165), we obtain the effective Hamiltonian for the spin-2BEC:

V ' N(

c0n2+

c1n2| f |2 + 2c2n

5|s−|2

)−

(c1n| f |2 + 4c2n

5|s−|2

)∑k,0

∑m

nk,m

+c0n2

∑k,0

∑mn

(2ζ∗mζna†k,nak,m + ζ∗mζ∗n ak,ma−k,n + ζmζna†k,ma−k,n)

+c1n∑k,0

∑mn

f · fmna†k,mak,n

+c1n2

∑k,0

∑i jmn

fi j · fmn(2ζ∗i ζna†k,mak, j + ζ∗i ζ∗mak, ja−k,n + ζ jζna†k,ia

†−k,m)

+c2n5

s−∗∑k,0

∑m

(−1)mak,ma−k,−m + H.c.

+

2c2n5

∑k,0

∑mn

(−1)m+nζ∗mζna†k,−mak,−n, (203)

where

f ≡∑mn

fmnζ∗mζn, (204)

s− ≡12

∑m

(−1)mζmζ−m. (205)

5.2.1. Ferromagnetic phaseWe assume that only the m = 2 magnetic sublevel is macroscopically occupied. Then, we

have Fz = 2n02 = 2(N −∑k,0m nkm), and the one-body part becomes

H0 = −2pN +∑

k,0,m

[εk + (2 − m)p

]nk,m. (206)

Because the spinor that is normalized to unity is ζm = δm,2, only the z-component is nonzero in fand s− = 0 and Eq. (203) becomes

V =12

g4nN + n∑k,0

[c0nk,2 + 2c1(2nk,2 − 2nk,0 − 3nk,−1 − 4nk,−2)

+25

c2nk,−2 +g4

2(ak,2a−k,2 + a†k,2a†−k,2)

], (207)

51

where g4 ≡ c0 + 4c1. Combining Eqs. (206) and (207), the total Hamiltonian is given by

HF =12

g4nN − 2pN

+∑k,0

[(εk + g4n) nk,2 +

12

g4n(ak,2a−k,2 + a†k,2a†−k,2)]

+∑k,0

[(εk + p)nk,1 + (εk + 2p − 4c1n) nk,0 + (εk + 3p − 6c1n) nk,−1

+ (εk + 4p − 8c1n + 2c2n/5) nk,−2]. (208)

It follows from this Hamiltonian that the m = 2 mode gives the Bogoliubov spectrum

EFm=2,k =

√ε2

k + 2g4nεk. (209)

For the system to be stable, the Bogoliubov excitation energy (209) must be positive. Thisimplies that the s-wave scattering length for the total spin-4 channel must be positive:

g4 =4π~2

Ma4 > 0. (210)

This condition is the same as the one required for the ferromagnetic mean field to be stable, thatis, the first term on the rhs of Eq. (208) being positive. We also note that the Bogoliubov spectrum(209) is independent of the applied magnetic field and remains massless in its presence. ThisGoldstone mode is a consequence of the global U(1) gauge invariance due to the conservation ofthe total number of bosons.

On the other hand, all the other modes give single-particle spectra:

EFm=1,k = εk + p, (211)

EFm=0,k = εk + 2p − 4c1n, (212)

EFm=−1,k = εk + 3p − 6c1n, (213)

EFm=−2,k = εk + 4p − 8c1n + 2c2n/5. (214)

For the system to be stable, the single-particle excitation energies must also be positive. Thisimplies that p > 2c1n and p > (2c1 − c2/10)n; these conditions are the same as the ones for theferromagnetic phase to be the lowest-energy mean field.

5.2.2. Antiferromagnetic phaseWe consider the situation in which only the m = ±2 modes are macroscopically occupied.

By requiring that the total magnetization be fz, the normalized spinor is given by

ζ2 = eiφ2

√12+

fz4, ζ−2 = eiφ−2

√12− fz

4, ζ0 = ζ±1 = 0. (215)

Because of the quantum depletion of the condensate, Fz has to be replaced by (N−∑k,0m nkm) fz,

and the one-body part of the Hamiltonian is therefore given by

H0 = −pFz +∑

k,0,m

(εk − pm)nkm

= −pN fz +∑

k,0,m

[εk − p(m − fz)

]nkm. (216)

52

The interaction Hamiltonian can be evaluated by noting that p ' fz(c1 − c2/20), and the totalHamiltonian is given by

HAF =12

(c0 + c2/5)nN − 12

pN fz

+∑k,0

(εk + g4n|ζ2|2)nk,2 + (εk + g4n|ζ−2|2)nk,−2

+

[g4n2

(ζ∗22 ak,2a−k,2 + ζ∗2−2ak,−2a−k,−2)

+(c0 − 4c1 + 2c2/5)n(ζ∗2ζ∗−2ak,2a−k,−2 + ζ2ζ

∗−2a†k,2ak,−2) + H.c.

]+

∑k,0

[εk + (c1 − c2/5)n +

12

(c1 + c2/10)n fz

]nk,1+[

εk + (c1 − c2/5)n − 12

(c1 + c2/10)n fz

]nk,−1

+2(c1 − c2/5)n(ζ∗2ζ∗−2ak,1a−k,−1 + H.c.)

+

∑k,0

(εk − c2n/5)nk,0 +

c2n5

(ζ∗2ζ∗−2ak,0a−k,0 + H.c.)

. (217)

We thus find that the spectrum is classified into three cases: the coupled m = ±2 modes, coupledm = ±1 modes, and single m = 0 mode.

We first investigate the eigenspectra of the m = ±2 modes. The relevant part of the Hamilto-nian is

h2 =∑k,0

εk,2nk,2 + εk,−2nk,−2

+g4n2

(ζ∗22 ak,2a−k,2 + ζ∗2−2ak,−2a−k,−2 + H.c.)

+

(c0 − 4c1 +

25

c2

)n(ζ∗2ζ

∗−2ak,2a−k,−2 + ζ2ζ

∗−2a†k,2ak,−2 + H.c.)

,

(218)

where εk,±2 = εk + g4n|ζ±2|2. Here, the complex numbers ζ±2 can be made real by means ofunitary transformations a±k,±2 → a±k,±2eiφ±2 . The resulting equations can be diagonalized bywriting down the Heisenberg equations of motion for ak,±2 + a†−k,±2, and seeking solutions of theform ak,±2 + a†−k,±2 ∝ exp(−iEAF

k,±2t/~). The resultant eigenspectra read

(EAFk,±2)2 = εk

εk + g4n ± g4n

√f 2z

4+

[1 − 8

g4

(c1 −

c2

20

)]2(1 −

f 2z

4

) . (219)

For the ground state to be stable, these energies must be positive; this implies the conditionsc1 − c2/20 > 0 and c0 + c2/5 > 0. The former condition is met if the antiferromagnetic phase isthe lowest-energy state, whereas the latter condition ensures that the antiferromagnetic phase ismechanically stable, that is, the first term on the rhs of Eq. (217) is positive. The spectra (219) aremassless regardless of the presence of the magnetic field. This is because they are the Goldstone

53

modes associated with the U(1) gauge symmetry and the relative gauge symmetry, which is therotational symmetry about the direction of the applied magnetic field.

Next, we investigate the eigenspectra for the coupled m = ±1 modes. The relevant part of theHamiltonian is

hAF1 =

∑k,0

εk,1nk,1 + εk,−1n−k,−1 + (δak,1a−k,−1 + H.c.)

, (220)

where δ ≡ 2(c1 − c2/5)nζ∗2ζ∗−2 and

εk,±1 ≡ εk +

(c1 −

c2

5

)n ± 1

2

(c1 +

c2

10

)n fz. (221)

The eigenspectrum can be obtained by writing down the Heisenberg equations of motion for ak1and a−k,−1, and seeking solutions of the form exp(∓iEk,±1t/~). The eigen matrix is 2 × 2 and itcan be solved straightforwardly, giving

Ek,±1 = ±12

(c1 +

c2

10

)n fz

+

√ε2

k + 2(c1 −

c2

5

)nεk +

14

(c1 −

c2

5

)2n2 f 2

z . (222)

The spectra become massive in the presence of a magnetic field. For the ground state to be stable,the spectra must be positive definite. Hence, we obtain the conditions c1 − c2/20 > p/2n > 0and c2 < 0; these are in agreement with the conditions for the antiferromagnetic phase to be thelowest-lying one.

The spectrum for the m = 0 can be obtained by means of the Bogoliubov transformation:

Ek,0 =

√ε2

k +2|c2|n

5εk +

(c2n10

)2f 2z . (223)

This mode becomes massive in the presence of a magnetic field.At zero magnetic field, all five dispersion relations in Eqs. (219), (222), and (223) become

gapless and linear, reflecting the fact that in the absence of the magnetic field, the ground state isdegenerate with respect to five continuous variables. Note, however, that only the first four areGoldstone modes because the continuous symmetry concerning the last mode does not reflectthe symmetry of the Hamiltonian but the hidden SO(5) symmetry [92].

5.2.3. Cyclic phaseThere are several mean-field solutions for the cyclic phase. Here, we consider the Bogoliubov

excitations from the following mean field:

ζ±2 =

√12

(1 ± fz

2

)eiφ±2 , ζ±1 = 0, ζ0 =

√1 − f 2

z /42

eiφ0 , (224)

54

where φ2 + φ−2 − 2φ0 = ±π. Because s− = 0 and 〈 f±〉 = 0, the interaction Hamiltonian becomes

V ' c0n2+

12

pN fz − c1n f 2z

∑k,0

∑m

nk,m

+c0

2V

∑k,0

: (DkD−k + D†kDk + H.c.) : +c1n fz∑k,0

∑m

mnk,m

+c1

2V

∑k,0

∑i jmn

fi j fmn(a†0,ia0,na†k,mak, j + a†0,ia†0,mak, ja−k,n + H.c.)

+2c2

5V

∑k,0

∑mn

(−1)m+na†0,ma0,na†k,−mak,−n. (225)

The interaction Hamiltonian shows that the spectra are divided into two parts: the m = ±1coupled modes and the m = ±2 and m = 0 coupled modes. The eigenspectra for the formermodes can be found by writing down the Heisenberg equations of motion for a±k,±1 with theresult

(EC±1)2 = ε2

k + β

(1 −

f 2z

8

)εk +

β2

32f 2z ±

β fz2

√(εk +

β

4

)[(1 − 3

16f 2z

)εk +

β

64f 2z

]. (226)

These excitation energies are always positive semidefinite and massive in the presence of a mag-netic field.

The eigenspectra for the m = ±2 and m = 0 coupled modes can be obtained by writing downthe equations of motion for X± ≡ ak,2 ± a†−k,2, Y± ≡ ak,0 ± a†−k,0, and Z± ≡ ak,−2 ± a†−k,−2. Thedesired eigenspectra are given by

EC±2,0 = εk +

2c2n5

(227)

and

(EC±2,0)2 = εk

εk+ c0n +4 + f 2

z

2c1n ±

√(c0n)2 −

(4 − 3 f 2

z

)c0c1n2 +

(4 + f 2z )2

4(c1n)2

. (228)

The solutions in Eq. (228) are always positive semidefinite and massless even in the presenceof an external magnetic field. This is a consequence of the fact that the mean-field solution isdegenerate with respect to at least two continuous variables. The solution in Eq. (227) is massiveand independent of the external magnetic field.

5.3. Dipolar BEC

In the presence of a magnetic dipole-dipole interaction (DDI), the Hamiltonian is

H =∫

dr∑

m

ψ∗m(r)(− ~2

2M∇2 + U(r) − pm + qm2

)ψm(r) +

c0

2n(r)2 + c1|F(r)|2

+

cdd

2

∫drdr′

∑µ,ν=x,y,z

Fµ(r)Qµν(r − r′)Fν(r′), (229)

55

where cdd = µ0(gFµB)2/(4π) and

Qµν(r) =δµν − 3rµrν

r3 . (230)

In general, the long-range nature of the DDI yields rich physics in a trapped system, such asnontrivial equilibrium shape and spin textures. The Bogoliubov spectrum in the presence of theDDI has been studied using a Gaussian ansatz and numerical analysis [44, 52, 45, 53]. Here, wediscuss the Bogoliubov spectrum for a spinor dipolar BEC under an external field [93, 94, 82].We assume that the system is strongly confined in the z-direction and free in other directionsso that it is effectively two-dimensional. Consequently, the order parameter is decomposed asfollows:

ψ(x, y, z) = h(z)ψ(x, y), (231)

where h(z) is assumed to be normalized to unity:∫ ∞

−∞h2(z)dz = 1. (232)

Substituting Eq. (231) into Eq. (229), we obtain an effective 2D Hamiltonian:

H2D =

∫dρ

∑m

ψ∗m(ρ)(− ~2

2M∇2 − pm + qm2

)ψm(ρ) +

c0

2n(ρ)2 +

c1

2|F(ρ)|2

+ H2Ddd , (233)

where ρ ≡ (x, y),

cα = cα

∫ ∞

−∞h4(z)dz, (234)

and Hdd is the two-dimensional version of the time-averaged dipolar interaction (149):

H2Ddd =

cdd

2∫

h4(z)dz

∫dρ

∫dρ′

∑µν

Fµ(ρ)Fν(ρ′)∑k2D

eik2D·(ρ−ρ′) ˜Q(2D)µν (k2D). (235)

Here, the integration kernel ˜Q(2D)µν (k2D) is given by

˜Q(2D)µν (k2D) =

∫dkz

2π

∫dzh2(z)

∫dz′h2(z′)eikz(z−z′) ˜Qµν(k3D) (236)

=

∫dkz

2π

∫dzh2(z)

∫dz′h2(z′)eikz(z−z′) 2π

3

[1 − 3(B · k3D)2

](δµν − 3BµBν), (237)

where ˜Qµν(k(3D)) is the Fourier transform of Eq. (149) and B is a unit vector along the externalfield.

To proceed with the calculations, we assume that

h(z) =1

(2πd2)14

e−z2

4d2 . (238)

56

After some algebraic manipulations, we obtain

˜Q(2D)(k2D) =2π3

(δµν − 3BµBν)Q(k2D), (239)

Q(k) =1 − 3 cos2 θB + 3√πerfc(kd)kdek2d2

×[cos2 θB − sin2 θB(kx cos φB + ky sin φB)2

], (240)

where θB and φB specify the direction of the magnetic field: B = (cos φB sin θB, sin φB sin θB, cos θB).With this kernel, the Gross-Pitaevskii equation in the presence of the magnetic DDI can be ex-pressed as follows:

i~∂

∂tψm =

[− ~2

2M∇2 + U(r) + qm2 + c0n

]ψm + c1

∑n

F · fmnψn + cdd

∑n

b · fmnψn, (241)

where the linear Zeeman term is eliminated by moving into the rotating frame of reference inspin space with the Larmor frequency, and b is an effective dipole field given by

bµ ≡∫

dρ′∑k2D

eik2D·(ρ−ρ′)Q(k2D)∑µν

(δµν − 3BµBν)Fν(ρ′). (242)

Note that because the spin and orbital parts are decoupled in the dipole kernel (237), it is notnecessary to choose the coordinate axes in real space that coincide with those in the spin space. Inwhat follows, we take the spin quantization axis along the magnetic field, which is not necessarilyparallel to the direction of the strongly confining potential, and their relative direction is givenby B = (cos φB sin θB, sin φB sin θB, cos θB).

In the following discussions, we discuss the ferromagnetic phase (c1 < 0) such as the f = 1hyperfine manifold of 87Rb. As in this case, we assume that the coefficient of the quadraticZeeman effect is positive: q > 0. When the total longitudinal magnetization is set to zero, themagnetization is transverse due to the quadratic Zeeman effect. In fact, it can be shown that

ψ(0)1

ψ(0)0

ψ(0)−1

=

√1 − q/2√

(1 + q)/2√1 − q/2

, (243)

gives a stationary solution of Eq. (241) with U(r) = 0, where

q =q

2n(|c1| + cddQ(0)). (244)

The expectation value of the spin vector for this spinor is nonvanishing only for the x-component:

Fµ =∑mn

(ψ(0)m )∗(fµ)mnψ

(0)n =

√n(1 − q2)δµ,x. (245)

To investigate the Bogoliubov excitation, we expand the order parameter from the mean-fieldstationary state (243):

ψm = ψ(0)m +

∑k

[ukmei(k·r−ωkt/~) + v∗kme−i(k·r−ωkt/~)

]. (246)

57

Substituting Eq. (246) into Eq. (241), we obtain the 6 × 6 eigenvalue matrix equation:

ωk

(ukvk

)=

(H0 H1−H∗1 −H∗0

) (ukvk

), (247)

where uk and vk are three-component vectors and H0,1 is the 3 × 3 matrix. The spin wave modeu, v ∝ (1, 0,−1)T decouples from the other two modes and the eigenfrequency is obtained as

ω2k =

εk + q − cddn(1 − q)[2Q(k) + Q(0)]

εk + cddn(1 + q)[Q(k) − Q(0)]

. (248)

Note here that due to the anisotropy of Q(k), the right-hand side of Eq. (248) becomes neg-ative depending on the direction of k. This result implies that the initial state given by (243)is dynamically unstable. Figures 15 (a)–(c) show the distributions of |Imωk| (red) and −Reωk(blue) in momentum space which correspond to the dynamical instability and Landau instability,respectively. For ω2

k > 0, the sign of ωk is determined so that the corresponding eigenmodesatisfies the normalization condition u†kuk − v†kvk = 1.

kx/2¼ [¹m-1]

−0.05 0.050.0

kz/2¼ [¹m

-1] 0.

05

0.0

−0.

05

(a)

−Re !k|Im !k| 0 1.4 s-1

0 0.56 s-1

80 mG, no helix

(b)

−Re !k|Im !k| 0 0.44 s-1

0 0.29 s-1

160 mG, no helix

(c)

0 0.54 s-1|Im !k|

200 mG, no helix

Figure 15: Real and imaginary parts of ωk in Eq. (248) for the case of d = 1 µm and n = 2.3×1014 cm−3. The region withnonzero imaginary part shows the dynamical instability, while the region with Reωk < 0 shows the Landau instability.The external magnetic field is (a) 80 mG, (b) 160 mG, and (c) 200 mG.

The anisotropic distribution of the instability suggests the existence of the periodic pattern inreal space. Recently, the Berkeley group observed the crystallization of transverse magnetizationin a quasi two-dimensional 87Rb BEC [25]. In one of the experiment, they started from theinitial condition in which the magnetization form a helix and let the system evolve with time.The Bogoliubov spectrum (248) obtained above does exhibit dynamical instabilities that exhibitvarious patterns such as a helix and stripes. However, it fails to explain the Berkeley experimentsin terms of the time and length scale.

6. Vortices in Spinor BECs

A scalar BEC can host only one type of vortex, that is, a U(1) vortex or a gauge vortex.However, a spinor BEC can host a much richer variety of vortices. We begin by discussing whythis is the case. The properties of a vortex can be characterized by looking at how the orderparameter changes along a loop that encircles the vortex. Because the order parameter must besingle-valued, it should satisfy

ψ(ϕ, ξ) = ψ(ϕ + 2π, ξ), (249)58

where ψ is a 2 f + 1-component order parameter; ϕ, the azimuthal angle which specifies thelocation of the loop; and ξ, a set of “coordinates” which, together with ϕ, completely specifythe order parameter. For the case of a scalar BEC whose order parameter is a single complexfunction ψ = |ψ|eiφ, the vortex solution to Eq. (249) is

ψ(ϕ, ξ) = |ψ|einwϕ, (250)

where nw = 0,±1,±2, · · · is the winding number. If nw , 0, the density at the vortex core mustvanish. The corresponding superfluid velocity is given by

vs =~

2Mni[ψ∗∇ψ − (∇ψ∗)ψ], (251)

where M is the atomic mass and n = |ψ|2. Substituting Eq. (250) in Eq. (251), we find that

vs = nw~M∇φ. (252)

Therefore, the circulation of the superfluid velocity is quantized in units of κ = ~/M.Due to the internal degrees of freedom, spinor BECs allow several solutions other than

Eq. (250), depending on the symmetry of the order-parameter manifold. A general order pa-rameter of a spinor BEC is described as

ψ ≡√

nζ =√

neiφU(α, β, γ)ζ0, (253)

where n is the particle density; φ, the global gauge; U(α, β, γ) = e−ifzαe−ifyβe−ifzγ, an SO(3)rotation with spin matrices fx,y,z and Euler angles α, β, and γ; and ζ0, a representative normalizedspinor for the order parameter. Hence, the spacial variation of α, β, and γ as well as φ characterizethe structure of a vortex. In the following, we investigate what types of vortices are allowed ineach phase. A general classification of topological excitations based on homotopy theory is givenin Sec. 7.2.

6.1. Mass and spin supercurrents in spinor BECs

The mass current density j(mass) for a spinor BEC is defined as the sum of currents in all spincomponents:

j(mass) =~

2Mi

∑m

[ψ∗m(∇ψm) − (∇ψ∗m)ψm

]. (254)

It can be shown from the GPEs that j(mass) satisfies the continuity equation:

∂n∂t+ ∇ · j(mass) = 0. (255)

In a similar manner, we define the spin current density j(spin)µ as

j(spin)µ =

~2Mi

∑mn

(fµ)mn[ψ∗m(∇ψn) − (∇ψ∗m)ψn

]. (256)

59

At zero magnetic field, j(spin)µ satisfies the continuity equation for the µ component of the spin:

∂Fµ

∂t+ ∇ · j(spin)

µ = 0. (257)

In the presence of an external magnetic field, the spin angular momentum perpendicular to it isnot conserved, and Eq. (257) is replaced with the equation of motion of spins given by

∂Fµ

∂t+ ∇ · j(spin)

µ =p~

(B × F

)µ+

2qn~

(B × N B

)µ, (258)

where B is a unit vector in the direction of the magnetic field and N is the 3 × 3 symmetricnematic tensor defined in Eq. (71). The dipole interaction induces an effective magnetic field andtherefore modifies the first term in Eq. (258)

The superfluid velocity field for the particle and spins, defined as v(mass) = j(mass)/n andv(spin)µ = j(spin)

µ /n, respectively, are described in terms of a normalized spinor ζ as

v(mass) =~

2Mi

∑m

[ζ∗m(∇ζm) − (∇ζ∗m)ζm

], (259)

v(spin)µ =

~2Mi

∑mn

(fµ)mn[ζ∗m(∇ζn) − (∇ζ∗m)ζn

]. (260)

In contrast to the case of a scalar BEC, the circulations of Eqs. (259) and (260) are not alwaysquantized, as shown below.

6.2. Spin-1 BECThe rotation matrix for the order parameter of a spin-1 BEC is given by

U(α, β, γ) =

e−i(α+γ) cos2 β

2 − e−iα√

2sin β e−i(α−γ) sin2 β

2

e−iγ√

2sin β cos β − eiγ

√2

sin β

ei(α−γ) sin2 β2

eiα√

2sin β ei(α+γ) cos2 β

2

. (261)

The spin-1 BEC in the absence of an external magnetic field has two phases: ferromagnetic andpolar. Below, we discuss vortices in each phase.

6.2.1. Ferromagnetic phaseA representative order parameter of a spin-1 ferromagnetic BEC is ζferro

0 = (1, 0, 0)T , whereT denotes the matrix transpose. Substituting this and Eq. (261) in Eq. (253), we obtain a generalorder parameter of a spin-1 ferromagnetic BEC:

ψferro =√

nei(φ−γ)

e−iα cos2 β

21√2

sin β

eiα sin2 β2

. (262)

The direction of the magnetization for this state is given by

s ≡ Ff n= (sin β cosα, sin β sinα, cos β). (263)

60

The linear combination φ − γ in Eq. (262) reflects the spin-gauge symmetry which implies thatthe rotation in spin space through angle γ is equivalent to the gauge transformation by φ = −γ.For simplicity of notation, we set φ′ = φ − γ. We substitute Eq. (262) in Eq. (259), obtaining [7]

v(mass,F) =~M

[∇φ′ − cos β ∇α]. (264)

Thus, the spatial variation of the spin gives rise to a supercurrent, and vice versa.In the case of a scalar BEC, the superfluid velocity is given by vs = (~/M)∇φ; it is therefore

irrotational, i.e., ∇ × vs = 0. However, this is not the case with the ferromagnetic BEC. In fact,using

∇ × ( f a) = (∇ f ) × a + f (∇ × a), (265)

we find that

∇ × v(mass,F) =~M

sin β (∇β × ∇α). (266)

Introducing unit vectors t0 = (cos β cosα, cos β sinα,− sin β) and u0 = (− sinα, cosα, 0) whichsatisfy s = t0 × u0, we obtain ∇β = ∑

µ t0µ∇sµ and sin β∇α = ∑µ u0µ∇sµ. Then, we can express

Eq. (266) in terms of s as

∇ × v(mass,F) =~M

∑µν

(t0µ∇sµ) × (u0ν∇sν)

=~M

∑µνµ′ν′

δµµ′δνν′ (t0µ∇sµ′) × (u0ν∇sν′ )

=~M

∑λµνµ′ν′

(ελµνελµ′ν′ + δµν′δνµ′)(t0µ∇sµ′ ) × (u0ν∇sν′ )

=~M

∑λµν

[ελµν sλ(∇sµ × ∇sν) − (t0µ∇sµ) × (u0ν∇sν)

]=

~2M

∑µνλ

εµνλ sµ(∇sν × ∇sλ). (267)

This relation is known as the Mermin-Ho relation [95]. ∇ × v(mass,F)s is nonvanishing because of

the contribution from the Berry phase. To see this, we rewrite Eq. (264) as

v(mass,F) − ~M

(1 − cos β)∇α = ~M∇(φ′ − α)

and integrate both sides along a closed contour C. Due to the single-valuedness of the orderparameter, the integral of ∇(φ′ − α) should be an integral multiple of 2π. Hence, we obtain∮

Cv(mass,F) · d` − ~

M

∮C

(1 − cos β)∇α · d` = hM

nw (nw : integer). (268)

The term∮

C(1−cos β)∇α ·d` gives the Berry phase enclosed by the contour C. We thus concludethat in the ferromagnetic BEC, the circulation alone is not quantized, but the difference betweenthe circulation and the Berry phase is quantized.

61

As a special case, let us consider the order parameter (262) with α = −nwϕ and φ′ = nwϕ;then, Eq. (262) changes from ζ = (ei2nwϕ, 0, 0)T to (0, 0, 1)T as β changes from 0 to π. Thisimplies that the 2nw vortex is topologically unstable. On the other hand, if we take α = −nwϕand φ′ = (nw + 1)ϕ, then Eq. (262) changes from ζ = (ei(2nw+1)ϕ, 0, 0)T to (0, 0, eiϕ)T as β changesfrom 0 to π. Thus, the 2nw + 1 vortex is unstable against the decay to the singly quantized vortexwhich is stable [7].

We can utilize these properties to create a doubly quantized vortex from a vortex-free stateby changing β from 0 to π as a function of time while keeping α = ϕ and φ′ = −ϕ [96, 97]. Sucha scheme is realized if a spin polarized BEC is prepared in the m = 1 state under a quadrupolefield B(r, ϕ, z) = (B⊥(r, z) cos(−ϕ), B⊥(r, z) sin(−ϕ), Bz(r, z))T , where (r, ϕ, z) are the cylindricalcoordinates and B⊥ > 0. At t = 0, Bz is set to be positive and much greater than B⊥. As Bz ischanged adiabatically from Bz B⊥ to Bz −B⊥, the atomic spins follow the direction of thelocal magnetic field as α = ϕ and β = arctan(B⊥/Bz), resulting in the transformation of the orderparameter from (1, 0, 0)T to (0, 0, e−2iϕ)T ; thus, a doubly quantized vortex is imprinted. Usingthis method, the MIT group have observed multiply quantized vortices [98].

Any axisymmetric configuration of ψ is homotopic to one of the following two types ofvortices: a coreless vortex [Fig. 16 (c)] and a polar-core vortex [Fig. 16 (b)]. When φ′ = ±α, thevortex is coreless; in particular, the order parameter for the case of φ′ = α = ϕ is given by

ψcoreless =√

n

cos2 β

2eiϕ√

2sin β

e2iϕ sin2 β2

, (269)

in which the singularity can be removed by choosing β = 0 at r = 0. Let us consider the con-figuration in which β(r = 0, ϕ, z) = 0 and β(r = r0, ϕ, z) = π, where r0 is the radius at the(cylindrical) boundary of the system. Then, the spinor (1, 0, 0)T at the origin with no vortex sin-gularity and (0, 0, e2iϕ)T at the boundary with a doubly quantized vortex attached to the invertedspin configuration. The corresponding superfluid velocity is nonsingular at the origin; in fact,

v(mass,F) =~

Mr(1 − cos β)ϕ, (270)

where ϕ is the unit vector in the azimuthal direction. The coreless vortex has been observed bythe MIT group [99], where a vortex was imprinted using a quadrupolar field.

On the other hand, when φ′ = 0 and α = ϕ, the vortex core has a singularity in the ferromag-netic order, where the order parameter at r → ∞ is given by

ψpolar−core =√

n

e−iϕ cos2 β

21√2

sin β

eiϕ sin2 β2

. (271)

Because the m = ±1 components have nonzero vorticities, the density in Eq. (271) must vanishat the center of the vortex. In fact, the free-energy analysis shows that the vortex core is filledby a polar state in which only the m = 0 component is occupied; hence, this vortex is called apolar-core vortex. As a consequence, only the spin density vanishes at the vortex core and thenumber density remains finite. A variational order parameter that describes both the vortex and

62

its core simultaneously is given by

ψpolar−core′ =√

n

e−iϕ f (r) cos2 β

2√1 − f 2(r) cos4 β

2 − g2(r) sin4 β2

eiϕg(r) sin2 β2

, (272)

where f (r) and g(r) are the variational functions subject to f (0) = g(0) = 0 and f (∞) = g(∞) = 1and it can be obtained so as to minimize the Gross-Pitaevskii energy functional. The spontaneousformation of the polar-core vortices has been observed by the Berkeley group [100].

(a)

(c)

(b)

Figure 16: (a) Order parameter for the spin-1 ferromagnetic phase, where the arrow indicates the direction of the localmagnetization. (b) Polar-core vortex given by Eq. (271), where the ferromagnetic order parameter has a singularity at thevortex core which is filled by a polar state. (c) Coreless vortex given by Eq. (269) with β(r = 0) = 0 and β(r = r0) = π/2.

To calculate the spin superfluid velocity (260), it is convenient to use the Cartesian coordinatein spin space ζ = (ζx, ζy, ζz)T rather than the spherical coordinate ζ = (ζ1, ζ0, ζ−1)T . They arerelated to each other by

ζ = Uζ ≡ 1√

2

−1 0 1−i 0 −i0√

2 0

ζ. (273)

63

In the Cartesian coordinate, spin matrices are given by (fµ)νλ = −iεµνλ, where µ, ν, λ = x, y, z.The ferromagnetic order parameter (262) is then given by

ζferro = − eiφ′

√2

( t0 + iu0) = − 1√

2( t + iu), (274)

where t = cos φ′ t0 − sin φ′u0, u = sin φ′ t0 + cos φ′u0, and t0 and u0 are defined below Eq. (266).Substituting Eq. (274) in Eqs. (259) and (260) and using some vector calculus formulae, weobtain

v(mass,F) =~M

∑µ

tµ∇uµ, (275)

v(spin,F)µ = sµv(mass,F) − ~

2M

∑ν,λ

εµνλ sν∇sλ. (276)

The first term on the right-hand side of Eq. (276) is the flow of spins carried by the particle, andthe second term results from the spin flow induced by the gradient of spins that is perpendicularto s. The spin superfluid velocity v(spin,F) satisfies the following relations:

∇ · v(spin,F)µ = sµ∇ · v(mass,F) + v(mass,F) · ∇sµ −

~2M

(s × ∇2 s)µ, (277)∑µ

sµ∇ · v(spin,F)µ = ∇ · v(mass,F), (278)

∇ × v(spin,F)µ = ∇sµ × v(mass,F) +

~2M

∑νλσ

(sµ sν − δµν)ενλσ∇sλ × ∇sσ, (279)∑µ

sµ[∇ × v(spin,F)

µ

]= 0, (280)

where we have used the Mermin-Ho relation (267) to derive Eq. (279).

6.2.2. Polar phaseWe next consider the order parameter of a spin-1 polar BEC, where ζ0 = (0, 1, 0)T . A general

order parameter of the polar state is given by

ψpolar =√

neiφ

− e−iα√

2sin β

cos βeiα√

2sin β

. (281)

The fact that ψpolar does not depend on γ reflects the SO(2) symmetry around the quantizationaxis d = (sin β cosα, sin β sinα, cos β); exp(−if · d)ψpolar = ψpolar. Substituting Eq. (281) intoEq. (259), we obtain

v(mass,P) =~M∇φ. (282)

Thus, unlike the case of the ferromagnetic phase, the circulation is quantized; however, the unitof quantization is half the usual h/M. To understand this, let us consider a loop that encircles a

64

vortex with a fixed radius. Then, each point on the loop is specified by the azimuthal angle ϕ.We note that the single-valuedness of the order parameter (281) is met if we take, for example,α = φ = nwϕ/2 and β = π/2. Then, the order parameter at r → ∞ is given by

ψhalf−vortex =

√n2

−10

einwϕ

, (283)

where nw is an integer. It follows then that the circulation is quantized in units of h/(2M) ratherthan the usual h/M: ∮

v(mass,P) · d` = h2M

nw. (284)

The underlying physics for the half quantum number is the Z2 symmetry of the order parameterof the polar phase; Eq. (281) is invariant under π gauge transformation (φ → φ + π) combinedwith a π spin rotation around an axis perpendicular to d; d → −d. More generally, the single-valuedness of the order parameter is satisfied if φ changes by nwπ along the closed loop, wherenw is even if d(ϕ = 2π) = d(ϕ = 0), and odd if d(ϕ = 2π) = −d(ϕ = 0). Thus, the polarphase of a spin-1 BEC can host a half-quantum vortex for the case of nw = 1 [101], which isalso referred to as an Alice vortex or Alice string [102]. The dynamic creation of half-quantumvortices was discussed in Ref. [103]. As in the case of the polar-core vortex in the ferromagneticphase, the atomic density can be finite at the vortex core of the half-quantum vortex by filling theferromagnetic state [(1, 0, 0)T for the case of Eq. (283)] at the core.

(a) (b)

A

B

d

C2

0

¼

Figure 17: (a) Order parameter for the polar state, which is invariant under π rotation about the C2 axis together withthe π gauge transformation. (b) Half-quantum vortex of the polar phase of a spin-1 BEC. The order parameter rotatesthrough an angle π about the C2 axis defined in (a) as one circumnavigates the vortex. To satisfy the single-valuednessof the order parameter, this round trip must be accompanied by a phase change in π (phase difference between A and B),resulting in a half-quantized vortex.

To calculate the spin current, we again use the Cartesian coordinate in which the polar orderparameter is given by ζpolar = eiφ d. Substituting ζpolar in Eq. (260), we obtain

v(spin,P)µ = − ~

M

∑νλ

εµνλdν∇dλ. (285)

65

If d is restricted in a two-dimensional plane perpendicular to a unit vector a (in spin space), thecirculation of

∑µ aµv(spin,P)

µ is quantized:∮ ∑µ

aµv(spin,P)µ · d` = − h

2Mnw (nw : integer). (286)

Here, the unit of the quantization is h/2M due to the spin-gauge Z2 symmetry. The rotation anddivergence of the v(spin,P)

µ are calculated to be

∇ × v(spin,P)µ = − ~

M

∑νλ

εµνλ(∇dν) × (∇dλ), (287)

∇ · v(spin,P)µ = − ~

M(d × ∇2 d)µ. (288)

Note that at zero magnetic field, the continuity equation (257) becomes

∂Fµ

∂t= −∇ ·

(nv(spin,P)

µ

)=

~M

∑i=x,y,z

∑νλ

εµνλ[(∂in)dν(∂idλ) + ndν(∂i∂idλ)

]=

~M

∑i=x,y,z

∑νλ

εµνλdν∂i(n∂idλ)

∂F∂t=

~M

∑i=x,y,z

d × ∂i(n∂i d). (289)

This result implies that∑

i ∂i(n∂i d) has to be either zero or parallel to d in a stationary d texture.Otherwise, the local magnetization would develop with time and the order parameter would nolonger belong to a manifold for the polar state. The d-field around the half-quantum vortex ofEq. (283) is d = (cos nwϕ/2, sin nwϕ/2, 0), which satisfies

∑i ∂i(n∂i d) = −n2

w d/(4r2) if n(r, ϕ) isaxisymmetric, i.e., eϕ · ∇n = 0, where eϕ is the unit vector in the azimuthal direction. Therefore,the half-quantum vortex can be a stationary state.

6.2.3. Stability of vortex statesThe stability of a vortex state can be studied by using the multicomponent GP equations.

An important feature of the spinor BEC is that each of the components can accommodate dif-ferent vortices (see, for example, Eq. (281)); however, this gives rise to relative phases betweenspinor components, and from the energetics point of view, certain conditions must be met. FromEq. (44), we find that only the term c1|F|2/2 depends explicitly on the relative phases. Expressingthe term in terms of components, we obtain

c1

2| f |2 =

c1

2

[(|ψ1|2 − |ψ−1|2)2 + 2|ψ0|2(|ψ1|2 + |ψ−1|2)

+4|ψ20ψ1ψ−1| cos (χ1 + χ−1 − 2χ0)

], (290)

where χm ≡ arg(ψm) is the phase of the m-th component. To minimize the energy, χ1 + χ−1 − 2χ0must be πn, where n must be even if c1 < 0 and odd if c1 > 0. Substituting χm = χ

′m+χ

′′mϕ, where

66

ϕ is the azimuthal angle, we must have χ′1 + χ′−1 − 2χ′0 = πn and χ′′1 + χ

′′−1 − 2χ′′0 = 0. Assuming

that the maximum vorticity is 1, we find that the following three vortex states are allowed [104]:

(χ′′1 , χ′′0 , χ

′′−1) = (1, 1, 1), (1, 0,−1), (1, 1/2, 0), (291)

where the last state is allowed only if ψ0 = 0, and therefore, it is described as (1, none, 0).The (1, 1, 1) state is the usual singly quantized vortex state with an empty vortex core, where

the order parameter is given by

eiϕ(|ψ1|eiχ′1 , |ψ0|eiχ′0 , |ψ−1|eiχ′−1 )T . (292)

The (1, 0,−1) state has a vortex in the m = 1 component and an anti-vortex in the m = −1component with the vortex core filled by the m = 0 state, where the order parameter of this stateis given by

(|ψ1|ei(χ′+ϕ), |ψ0|, |ψ−1|e−i(χ′+ϕ))T . (293)

This is the polar-core vortex in the ferromagnetic phase or an integer spin vortex in the polarphase. The (1, none, 0) state corresponds to the Alice vortex [Eq. (283)] whose order parameteris given by

(|ψ1|ei(χ′1+ϕ), 0, |ψ−1|eiχ′−1 )T , (294)

where the vortex core is filled by the m = −1 component. Except for the (1,1,1) state, the vortexcore is filled by a non-vortex component which plays the role of a pinning potential, stabilizingthe vortex state in the absence of an external potential [104].

6.3. Spin-2 BEC

The rotation matrix is

U(α, β, γ) =

e−2i(α+γ)C4 −2e−i(2α+γ)C3S√

6e−2iαC2S 2 −2e−i(2α−γ)CS 3 e−2i(α−γ)S 4

2e−i(α+2γ)C3S −2e−i(α+γ)C2S 2 −√

38 e−iα sin 2β 2e−i(α−γ)C2S 2 −2e−i(α−2γ)CS 3

√6e−2iγC2S 2

√38 e−iγ sin 2β 1

4 (1 + 3 cos 2β) −√

38 eiγ sin 2β

√6e2iγC2S 2

2ei(α−2γ)CS 3 2ei(α−γ)C2S 2√

38 eiα sin 2β −2ei(α+γ)C2S 2 −2ei(α+2γ)C3S

e2i(α−γ)S 4 2ei(2α−γ)CS 3√

6e2iαC2S 2 2ei(2α+γ)C3S e2i(α+γ)C4

,

(295)

where C ≡ cos β2 and S ≡ sin β

2 . The spin-2 BEC has four mean-field ground-state phases inthe absence of the magnetic field: ferromagnetic, uniaxial nematic, biaxial nematic, and cyclic.The uniaxial and biaxial nematic phases are degenerate at the mean-field level (75) [105]. Theuniaxial and biaxial nematic phases are also referred to as the antiferromagnetic phase in Refs. [4,5] and the polar phase in Ref. [11].

67

6.3.1. Ferromagnetic phaseThe representative state of the ferromagnetic phase is ζferro

0 = (1, 0, 0, 0, 0)T . Substituting thisand Eq. (295) in Eq. (253), we obtain

ψferro =√

nei(φ−2γ)

e−2iα cos4 β

22e−iα cos3 β

2 sin β2√

6 cos2 β2 sin2 β

22eiα cos β

2 sin3 β2

e2iα sin4 β2

. (296)

Similar to the spin-1 case [see Eq. (262)], the order parameter has the spin-gauge symmetrywith γ now replaced by 2γ because the magnitude of the spin is 2 rather than 1. The superfluidvelocity is similarly obtained as

v(mass,F) =~M

[∇(φ − 2γ) − 2 cos β∇α], (297)

whose circulation is not quantized as in the case of the spin-1 ferromagnetic BEC. Applying thediscussions about the stability of the vortex states for the ferromagnetic spin-1 BEC, it can beshown that the 4nw + mw vortex (mw = 0, 1, 2, 3) is unstable against the decay to the mw vortex.

The relationships for the spin superfluid velocity corresponding to Eqs. (277)–(280) changeto the following ones:

∇ · v(spin,F)µ = 2

[sµ∇ · v(mass,F) + v(mass,F) · ∇sµ −

~2M

(s × ∇2 s)µ

], (298)∑

µ

sµ∇ · v(spin,F)µ = 2∇ · v(mass,F), (299)

∇ × v(spin,F)µ = 2∇sµ × v(mass,F) +

~M

∑νλσ

(2sµ sν − δµν)ενλσ∇sλ × ∇sσ (300)

∑µ

sµ[∇ × v(spin,F)

µ

]= ∇ × v(mass,F) =

~M

∑µνλ

εµνλ sµ(∇sν × ∇sλ). (301)

6.3.2. Uniaxial nematic phaseThe representative state of the spin-2 uniaxial nematic phase is ζuniax

0 = (0, 0, 1, 0, 0)T andsubstituting this and Eq. (295) in Eq. (253), we obtain the general order parameter of this phase:

ψuniax =

√6n4

eiφ

e−2iα sin2 β−2e−iα sin β cos β√

23 (3 cos2 β − 1)

2eiα sin β cos βe2iα sin2 β

. (302)

Similar to the case of the spin-1 polar phase, the spin degrees of freedom do not contributeto the superfluid velocity that only depends on the spatial variation of the gauge angle:

v(mass,uniax) =~M∇φ. (303)

68

This is a general property for the non-magnetized phase in spinor BECs.The shape of the uniaxial order parameter in the spin space is shown in Fig. 18 (a). As one

can see from Fig. 18 (a), the uniaxial nematic phase has a SO(2) symmetry around the directionof d = (cosα sin β, sinα sin β, cos β). The uniaxial nematic phase also has Z2 symmetry as in thecase of the spin-1 polar phase. However, in the present case, Z2 symmetry is independent of thegauge angle φ: ζuniax is invariant under a π rotation around an axis perpendicular to d. Hence,although the uniaxial nematic phase can accommodate a vortex, as shown in Fig. 18 (b), the masscirculation for this vortex is zero. We call it a 0-1/2 vortex, where 0 and 1/2 indicate the gaugetransformation and rotation angle in spin space around the vortex, respectively. An example ofthe order parameter around the 1/2-spin vortex is given as

ψspin−vortex =

√6n4

eiϕ

0√2/30

e−iϕ

, (304)

where ϕ is an azimuthal angle around the vortex.The spin superfluid velocity for the uniaxial nematic phase is calculated to be

v(spin,uniax)µ = −3~

M

∑νλ

εµνλdν∇dλ. (305)

If d is restricted in a plane perpendicular to a, the circulation of the spin superfluid velocity∑µ aµv(spin,uniax)

µ is quantized in terms of a unit of 3h/M.Because the spin and gauge is completely decoupled in this phase, the mass circulation is

quantized in terms of a unit of ~/M. An example of such a vortex is (0, 0, eiϕ, 0, 0)T .

(a) (b)U(1)

C2

0

¼

0

Figure 18: (a) Order parameter for the spin-2 uniaxial nematic phase, which has SO(2) symmetry corresponding torotations about d. The uniaxial nematic phase is also invariant under a π rotation about the C2 axis, i.e., this phase has aZ2 symmetry. (b) The order parameter configuration for the 0-1/2 vortex, around which the order parameter rotates by πabout the C2 axis in (a).

6.3.3. Biaxial nematic phaseThe representative state of the spin-2 biaxial nematic phase is ζbiax

0 = (1/√

2, 0, 0, 0, 1/√

2)T

and substituting this and Eq. (295) in Eq. (253), we obtain the general order parameter of this69

phase:

ψbiax =

√n2

eiφ

e−2iα[(

1 − 12 sin2 β

)cos 2γ − i cos β sin 2γ

]e−iα sin β(cos β cos 2γ − i sin 2γ)√

32 sin2 β cos 2γ

−eiα sin β(cos β cos 2γ + i sin 2γ)e2iα

[(1 − 1

2 sin2 β)

cos 2γ + i cos β sin 2γ]

. (306)

The superfluid velocity only depends on the spatial variation of the gauge angle:

v(mass,biax) =~M∇φ. (307)

The profile of ζbiax is depicted in Fig. 19 (a).When we rotate the order parameter by π/2 about the C4 axis in Fig. 19 (a) as we circum-

navigate a vortex, the order parameter changes its sign, as shown in Fig. 19 (b) (see, the phasedifference between A and B). Therefore, this vortex has a mass circulation of h/(2M). We callthis vortex the 1/2-1/4 vortex, where 1/2 and 1/4 denote the gauge transformation and the spin-rotation angle, respectively, around the vortex. If we rotate the order parameter by π about theC2 axis around a vortex, then it goes back to the initial state, as shown in Fig. 19 (c). In thiscase, the vortex has no mass circulation and it is referred to as a 0-1/2 vortex. On the otherhand, if we rotate the order parameter by π about the C′2 axis, the sign of the order parameter isreversed [Fig. 19 (d)]. This vortex has a mass circulation of h/(2M), and it is called a 1/2-1/2vortex. Because the roations about C4, C2, and C′2 are non-commutable, these vortices obey thenon-Abelian algebra, which is discussed in Sec. 6.6.

The vortices shown in Fig. 19 are the simplest examples, and in general, the order parametercan vary in a more complex manner. However, in order to meet the single-valuedness, the masscirculation in the biaxial nematic phase has to be quantized in terms of a unit of h/(2M). Thespin superfluid velocity is very complicated, and its circulation is not quantized.

6.3.4. Cyclic phaseA representative state of the spin-2 cyclic phase is ζcyclic

0 = (1/2, 0, i/√

2, 0, 1/2). Figure 20(a) shows the profile of the order parameter of the cyclic phaseΨ =

∑m Ym

2 ζcyclicm . One can clearly

see that it has a three-fold symmetry about the (1, 1, 1) axis [C3 axis in Fig. 20 (a)]. This impliesthat the system has a one-third vortex [106]. The rotation matrix about the (1, 1, 1) axis throughangle θ is given by

eiφe−i fx+fy+fz√3

θζcyclic

0 . (308)

For θ = 2π/3, this reduces to e4πi/3eiφζcyclic0 ; the single-valuedness of the order parameter is met

by the gauge transformation of φ→ φ+2π/3. Because this is one-third of the usual 2π, the cyclicphase can possess a one-third vortex. Similarly, for θ = 4π/3, Eq. (308) reduces to e2πi/3eiφζcyclic

0 ;the single-valuedness of the order parameter is met by the choice of ∆φ = 4π/3. Thus, the cyclicphase also has a two-third vortex. The cyclic phase can also accommodate the 1/2-spin vortex,around which the order parameter rotates by π about the direction of one of the lobes [Fig. 20(c)]. The vortices in the cyclic phase also obey the non-Abelian algebra (see Sec. 6.6).

70

(a) (b)

(c) (d)

C2

0¼

C'2

C4

A

B

A

B

Figure 19: (a) Order parameter for the spin-2 biaxial nematic phase, which has one symmetry axis of the fourth order(C4) and two axes of the second order (C2 and C′2). (b)–(d) The configuration of the order parameter around (b) a 1/2-1/4vortex, (c) a 0-1/2 vortex, and (d) a 1/2-1/2 vortex.

6.4. Rotating spinor BECThe stationary state of a rotating BEC can be obtained by minimizing F = E − ΩLz, where

the rotation axis is assumed to be in the z-direction. In the case of a spin-1 BEC, we have

F =∫

dr∑

m

ψ∗m

(−~2∇2

2M+ U(r) −Ω`z

)ψm +

c0

2n2 +

c1

2|F|2

. (309)

The stationary state is obtained by requiring that

δFδψ∗m

= µmψm, (310)

where µm is the chemical potential for the m-th component that is determined so as to conservethe total number of particles N =

∫dr

∑m |ψm|2, total magnetization Mz =

∫dr(|ψ1|2 − |ψ−1|2),

and total projected angular momentum Lz =∫

dr∑

i ψ∗i `zψi, where `z = −i~∂/∂ϕ. The vortex

lattice formation in the presence of an external rotation is discussed in Ref. [107]. The vortexphases of spin-2 BECs under rotation is discussed in Ref. [108].

6.5. Spin-polarized dipolar BECThe dipole interaction is long-range and anisotropic, and it has significant effects on the

ground state of rotating BECs [109, 110, 111, 112, 113]. Let us assume that all the dipoles of theatoms are polarized in the same direction. Then, the interaction is described by

V(r) = c0δ(r) + cdd1 − 3 cos2 θ

r3 , (311)71

(b) (c)

A

B

C2

0

2¼/3 C3

z

x

y4¼/3

(a)

Figure 20: Order parameter of the cyclic phase (1, 0, i/√

2, 0, 1)T , where the blue, yellow, and green colors denotearg[

∑m Ym

2 ζcyclicm ] = 0, 2π/3, and 4π/3, respectively. The order parameter has a two-fold symmetry about the x, y, and z

axes and a three-fold symmetry about the (±1,±1,±1) axes. (b) The 1/3-1/3 vortex, where the order parameter is rotatedby 2π/3 about the C3 axis from A to B in a clockwise direction, results in the gauge transformation of −2π/3. (c) The0-1/2 vortex, around which the order parameter is rotated by π about the y-axis.

where the first and second terms on the right-hand side are the s-wave contact interaction and thedipole interaction, respectively, and θ is the angle between the dipole moment and the relativecoordinate r between two atoms. Thus, depending on the direction of the polarization, the dipoleinteraction can be either attractive or repulsive. This anisotropy of the interaction leads to thedeformation of the vortex core and the structure of the vortex lattice.

Yi and Pu [112] considered a situation in which a system in an axisymmetric harmonic trapis rotated about the z-direction, and the dipoles are polarized in the x-direction in the frame ofreference corotating with the system, and found that the vortex core is deformed into an ellipticshape. This is because the dipoles undergo repulsive and attractive interaction in the x- andy-directions, respectively.

Cooper et al. [109] and Zhang and Zhai [111] considered a situation in which the interparticleinteraction is so weak that the lowest Landau level approximation holds, and found that with anincrease in the strength of the dipole interaction, the vortex lattice undergoes structural changesfrom triangular to square and to stripe configurations. Such changes were not found in Ref. [112],and possible reasons for the discrepancy are discussed in Ref. [113].

In the aforementioned work, the dipole interaction is treated as a perturbation to the s-wavecontact interaction. Very recently, the Rice group [26] used a shallow zero crossing in the wingof a Feshbach resonance of 7Li in the |F = 1,mF = 1〉 state to decrease the scattering length toas less as 0.01aB. This opens up several new possibilities on the dipole-interaction-dominatedBEC. An interesting question is how the degeneracy of a fast rotating BEC is lifted by a weakbut genuine dipole interaction and what is the corrsponding many-body ground state.

72

6.6. Non-Abelian vortices

The topological charge of a vortex determines how the order parameter changes as one cir-cumnavigates the vortex. For example, if the phase of the order parameter of a U(1) vortexchanges as einφ, the topological charge of this vortex is said to be n. One important feature ofa spinor BEC is that the order parameter has a geometrical shape in spin space, as illustrated inFigs. 16–20. Take, for example, the polar phase. This phase has two generators ei f ·dθ and eiπC2,where the former describes the rotation about the symmetry axis and the latter describes the π-rotation about the perpendicular axis followed by the gauge transformation by π [see Fig. 17(a)].These two operations commute with each other; thus, vortices of the polar phase of a spin-1 BECare Abelian. Similarly, the vortices in the ferromagnetic phase are also Abelian because thereis only one symmetry axis set by the magnetization. Therefore, the spin-1 BEC can host onlyAbelian vortices.

It is predicted [114] that the cyclic phase of a spin-2 BEC can host non-Abelian vortices.To understand this, we note that the order parameter of the cyclic phase is invariant under thefollowing 12 elements of the tetrahedral group T [106]: ~1, Ix = eifxπ, Iy = eifyπ, Iz = eifzπ,C = e2πi/3e−2πi(fx+fy+fz)/3

√3, C2, IxC, IyC, IzC, IxC2, IyC2, and IzC2, where fµ is the µ component

of the spin-2 matrices (µ = x, y, z) [see, Fig. 20]. Furthermore, these elements are classified intofour conjugacy classes:

• (I) integer vortices: ~1;

• (II) 1/2 - spin vortices: Ix, Iy, Iz;

• (III) 1/3 vortices: C, IxC, IyC, IzC;

• (IV) 2/3 vortices: C2, IxC2, IyC2, IzC2.

Topological charges in the same conjugacy class transform into one another under the globalgauge and/or spin rotation. The order parameter of a 1/2-spin vortex can be constructed fromthat of a non-vortex state ψnon−vortex as

ψ1/2vortex = eifzϕψnon−vortex, (312)

where ϕ is an azimuthal angle around a vortex line.The non-Abelian characteristics of the vortices manifest themselves most dramatically in

the collision dynamics. In general, when two vortices collide, they reconnect themselves, passthrough [Fig. 21(d)], or form a rung that bridges the two vortices [Fig. 21(b),(c)]. When twoAbelian vortices collide, all these three cases are possible, and one of them occurs dependingon the kinematic parameters and initial conditions. However, when two non-Abelian vorticescollide, only a rung can be formed; however, reconnection and passing through are topologicallyforbidden because the corresponding generators do not commute with each other. In fact, thenonzero commutator of the two generators gives the generator of the rung vortex. Figure 22illustrates a typical rung formation. When a core of 1/3 vortex in the cyclic phase is filled withthe ferromagnetic state, which is possible for a certain parameter set, it is possible to observesuch dynamics of vortex lines by using a phase-contrast imaging technique that can detect localmagnetization.

73

(a) (b) (c) (d)

A

B

A

ABA-1 A

A

A

A

ABA-1 ABA-1 A

B B B

B

BAB-1

BA-1AB

Figure 21: Collision dynamics of two vortices. (a) Initial configuration, where A or B represents an operation thatcharacterizes the corresponding vortex (a set of spin rotations and gauge transformations for the case of a cyclic BEC).The vortex on the right end, which is connected to B, is identified as ABA−1. The configuration in (a) is topologicallyequivalent to (b) and (c), where a rung is formed. If the vortices are a pair of vortex and anti-vortex, i.e., A = B−1, therung in (b) disappears, giving rise to reconnection, whereas the rung in (c) corresponds to a doubly quantized vortex thatcosts a large kinetic energy. If A and B are commutative, passing through is also possible because the configurationsof (a) and (d) will then be topologically equivalent. However, when A and B are not commutative, the collision alwaysresults in the formation of a rung.

(a) (b) (c)

Figure 22: Numerical simulation for the collision dynamics of non-Abelian vortices.

7. Topological Excitations

When a system undergoes Bose-Einstein condensation, some symmetries of the originalHamiltonian are spontaneously broken. The classification of the symmetry breaking can becarried out systematically using a group-theoretic method. The symmetry of the order parame-ter determines the types of possible topological excitations. In this section, we present a briefoverview of some basic notions and theoretical tools for investigating symmetries of the orderparameter and topological excitations.

7.1. Symmetry classification of broken symmetry states

7.1.1. Order parameter manifoldThe order parameter for a spin- f BEC is described with a set of 2 f + 1 complex amplitudes

ψ = ψm, and the order parameter manifold M is denoted as M = C2 f+1. In this section, we treata normalized spinor ζ = ψ/ ‖ ψ ‖ with its order parameter manifold given by

M = U(2 f + 1) =U(1) × SU(2 f + 1)

Z2 f+1. (313)

Consider a group G of transformations that act on M while leaving the mean-field energyfunctional invariant. We assume that the total spin and number of particles of the system are

74

conserved. Then, the energy functional is invariant under SO(3) rotations in spin space and U(1)gauge transformations. The full symmetry group G for a spinor BEC is therefore given by

G = SO(3)F × U(1)φ, (314)

where subscripts F and φ denote the spin and gauge, respectively. Any element g ∈ G can berepresented as g = eiφe−ifzαe−ifyβe−ifzγ, where α, β, and γ are Euler angles [see Fig. 23 (a)] andfx,y,z are (2 f + 1)× (2 f + 1) spin matrices. When an external magnetic field is applied, e.g. in thez-direction, G reduces to

GB = U(1)Fz × U(1)φ (315)

because the system will remain invariant only under rotations about the z-axis apart from thegauge transformation. Any element g′ ∈ GB is represented as g′ = eiφe−ifzα.

For each ζ ∈ M, the isotropy group (or the little group) Gζ is defined as the subgroup of Gthat leaves ζ invariant: Gζ = g ∈ G | gζ = ζ. For example, spin-1 polar phase ζpolar = (0, 1, 0)T

is invariant under a spin rotation about the z-axis, e−ifzα′ζpolar = ζpolar, and therefore, Gζpolar =

U(1)Fz .Next, we define the orbit G(ζ) of ζ by G(ζ) = gζ | g ∈ G, i.e., G(ζ) is the set of all points

obtained as a result of letting all transformations in G act on ζ. Because elements of G do notchange the mean-field energy, the orbit G(ζ) constitutes a degenerate space in M. For the case ofa spin-1 polar BEC, a general element of the orbit G(ζpolar) is given by

gζpolar = eiφe−ifzαe−ifyβe−ifzγ

010 = eiφ

− e−iα√

2sin β

cos βeiα√

2sin β

. (316)

The orbit is thus parametrized by φ and d ≡ (cosα sin β, sinα sin β, cos β) that specifies thequantization axis. The manifold of d (a three-dimensional unit vector) is a unit sphere which isreferred to as S 2, the two-sphere [see Fig. 23 (b)], whereas the manifold of φ is U(1). Therefore,we conclude that the order parameter manifold for the spin-1 polar state is given by Rpolar S 2

F × U(1)φ.

®

γ

¯

x

z

y

(a)

d

(b)

Figure 23: (a) Euler rotation with Euler angles (α, β, γ). (b) Two-sphere S 2 which is the manifold of a topological spaceof a three-dimensional unit vector d.

Mathematically, such a manifold is defined as a left coset of Gζ . For a subgroup H ⊂ G, theleft cosets of H are defined as G/H = gH | g ∈ G. The coset space is identical to the orbit of

75

H, and it forms a group if H is a normal subgroup of G, i.e., gHg−1 = H for ∀g ∈ G. The cosetspace G/H is equivalent to G/H′ if and only if H and H′ are conjugate, i.e., H′ = gHg−1. On theother hand, one can prove that the isotropy groups of all points on the same orbit are conjugate,i.e., Ggζ = gGζg−1 for ∃g ∈ G. Therefore, the coset space does not depend on the location in theorbit. We use H for a representation of the conjugate class of the isotropy groups gGζg−1, anddefine the order parameter manifold as

R = G/H. (317)

Here, H represents the remaining symmetry of the ordered state, and R describes the brokensymmetry, or a manifold of the degenerate space.

We next discuss the discrete symmetry of the order parameter manifold. The spin-1 polarphase has spin-gauge coupled discrete symmetry. In fact, ζpolar = (0, 1, 0)T is invariant undera π-rotation about an axis perpendicular to the z-axis followed by the gauge transformation ofeiπ. This can also be understood from the fact that Eq. (316) is invariant under the simultaneoustransformations of (φ, d)→ (φ+π,−d). A group for such an operation 1, eiπe−ifxπ is isomorphicto Z2 = 0, 1 because (eiπe−ifxπ)2 = 1. Therefore, the correct isotropy group of ζpolar is thedihedral group (D∞)Fz,φ U(1)Fz

o (Z2)F,φ, and the order parameter manifold is given by

Rpolar =SO(3)F × U(1)φU(1)Fz

o (Z2)F,φ

S 2F × U(1)φ(Z2)F,φ

, (318)

where U(1)Fzo (Z2)F,φ implies that when the nontrivial element of Z2 acts on an element g ∈

U(1), g changes to g−1.In the case of a spin-1 ferromagnetic phase, the order parameter ζferro = (1, 0, 0)T is invariant

under a spin rotation about the z-axis followed by a gauge transformation of the same amount asthe rotation angle; eiφe−ifzφζferro = ζferro. This is a spin-gauge coupled U(1) symmetry denotedas H = U(1)Fz+φ, where the subscript Fz + φ is shown to indicate the nature of the coupling. Ageneral form of the spin-1 ferromagnetic order parameter is given by

gζferro = eiφe−ifzαe−ifyβe−ifzγ

100 = ei(φ−γ)

e−iα cos2 β

21√2

sin β

eiα sin2 β2

. (319)

Note that the linear combination φ − γ in Eq. (319) manifestly exhibits a spin-gauge symmetry,i.e., the equivalence between phase change and spin rotation. Individual configurations of ζferro

are completely specified by the entire range of Euler angles (α, β, γ − φ), and therefore, the orderparameter manifold is given by

Rferro =SO(3)F × U(1)φ

U(1)Fz+φ

SO(3)F,φ. (320)

7.1.2. Symmetry of order parameter and stationary statesThere exists a close relationship between the symmetry of an order parameter and a stationary

state of the free-energy functional. To illustrate this, let us consider a smooth real functionf (x2 + y2) on manifold R2. The set of operations that leave f invariant constitute the groupG = SO(2) of rotations about the z-axis, where the origin x = y = 0 is the only point that has

76

SO(2) symmetry as an isotropy group, whereas any other point traces a circle, which is calledan orbit, because every element of G acts on that point (see Fig. 24); evidently, the isotropygroup of each point on a circle consists only of the identity element of G. The union of all orbitsthat have mutually conjugate isotropy groups constitutes what is referred to as a stratum. In thepresent example, all orbits of the concentric circles constitute one stratum, and the origin formsthe other isolated stratum because the isotropy group of the origin, which is SO(2), is differentfrom those of other neighboring points. Now, the fact that the origin is an extremum (maximum,minimum, or saddle point) of the function f (x2 + y2) suggests that an orbit that constitutes anisolated stratum gives a stationary orbit of a real smooth function, as illustrated in Fig. 24.

(a) (b)

Figure 24: (a) Orbits of f (x2 + y2) on a manifold R2. The union of all concentric circles constitute one stratum, and theorigin forms an isolated stratum. (b) The origin x = y = 0 is isolated in its stratum, and it is an extremum of any realsmooth function f (x2 + y2).

Michel generalized this idea by substituting R2 with an arbitrary manifold M, and SO(2) withan arbitrary compact Lie group G [115]. Michel showed that if the orbit G(ζ) is isolated in itsstratum S (ζ), i.e., if there exists a neighborhood U of G(ζ) such that U ∩ S (ζ) = G(ζ), then G(ζ)is a stationary orbit of a smooth real function on M that is invariant under G. Here, stratum S (ζ)is the union of all orbits that share the same type of isotropy group, that is, ζ and ζ′ belong to thesame stratum if and only if Gζ and Gζ′ are conjugate.

Michel’s theorem has been utilized to find the ground states of superfluid He-3 [116], p-wave [117, 118] and d-wave [105] superconductors, and spinor BECs [119, 120]. In the caseof a spin-1 BEC, we can uniquely determine the orbits of polar and ferromagnetic phases fromtheir isotropy groups. For example, ζferro = (1, 0, 0)T and ζpolar = (0, 1, 0)T are the only solutionsof eiφe−ifzφζ = ζ and e−ifzφζ = ζ, respectively, indicating that stratum S (ζferro,polar) consists ofG(ζferro,polar) alone. Therefore, the ferromagnetic and polar states are always extrema of themean-field energy functional. The order parameter for such states does not depend on parametersthat appear in the energy functional, and these states are called inert states. An inert state is robustagainst a change in the interaction parameters, implying that it is a stationary point of each termin the energy functional. In superfluid He-3, A, A1 and B phases are inert states.

It is possible, however, that the energy functional has other orbits of extrema that dependexplicitly on the coefficients involved in the functional. The corresponding states are callednon-inert states. Michel showed that the gradient of a G-invariant function is tangential to thestratum, implying that if we choose one stratum and find an orbit in it that minimizes the energyfunctional, then the orbit is stationary in the entire manifold. In contrast to inert states, the orderparameters of non-inert states are determined by the competition among several terms in theenergy functional.

In the case of spin-1 spinor BEC, non-inert states appear in the presence of an external mag-netic field, where the entire symmetry is reduced to GB = U(1)Fz

×U(1)φ. The mean-field energy

77

per particle for a spin-1 BEC in a uniform system is given by

ε[ζ] =12

c0n +12

c1n| f |2 − p∑

m

m|ζm|2 + q∑

m

m2|ζm|2, (321)

where f =∑

mn ζ∗mfmnζn is the spin expectation value per particle. Consider an order parameter

ζAF = (cos χ, 0, sin χ)T , (322)

where 0 < χ < π and χ , π/2. Different values of χ lead to different orbits, whereas therelative phase between ζ±1 and the global phase may vary in each orbit. The rotational symmetryabout the magnetic field is broken, and the isotropy group of ζAF is given by HAF = (C2)F,φ =

1, eiπe−ifzπ ⊂ GB. Substituting ζAF in Eq. (321) and minimizing ε with respect to χ, we obtainthe following stationary state:

ζAF =1√

2

(√1 − p

c1n, 0,

√1 +

pc1n

)T

,

∣∣∣∣∣ pc1n

∣∣∣∣∣ < 1, (323)

and the corresponding energy functional

ε[ζAF] =12

c0n − p2

4c1n+ q. (324)

The AF state corresponds to the spin part of the A2 phase in superfluid He-3 where the magneticfield changes the population of up-spin and down-spin Cooper pairs.

The energies for polar and ferromagnetic states are given by

ε[ζpolar] =12

c0n, (325)

ε[ζferro] =12

(c0 + c1)n − p + q. (326)

Comparing Eqs. (324), (325), and (326), we can obtain the phase diagram shown in Fig. 1.While symmetry considerations are helpful to find stationary states, they elude those states

that have no remaining symmetry. As an example, let us consider a broken-axisymmetry state [10]whose C2 symmetry axis is not parallel to the external field. This state cannot be obtained fromsymmetry considerations because GB is completely broken. Nonetheless, we can obtain a statethat has the same symmetry (as a subgroup of G) as the real ground state from the symmetryconsideration.

7.1.3. Procedure to find ground statesAccording to the discussions in the preceding subsection, the ground state that has remaining

symmetries can be found by the following procedure [121]:

1. List all subgroups H of G;2. for each H, find an order parameter invariant under h ∈ H;3. if a non-inert state is obtained, then find the parameters that appear in the order parameter

so as to minimize the mean-field energy;4. find the lowest-energy one among the obtained states.

78

Here, we present a list of the subgroups of SO(3) and show how to carry out step 2 of theabove procedure. The subgroups of SO(3) are well known and they are classified into continuousand discrete groups, as listed below.

• Continuous symmetriesThe only continuous subgroup of SO(3) is U(1). Let us choose the z-axis as the symmetryaxis. An infinitesimal transformation h ∈ H can be expressed as a combination of spinrotation and gauge transformation:

h = eiδφe−ifzδz ' 1 + iδφ − ifzδz, (327)

where δz and δφ are infinitesimal real values. The order parameter that is invariant under his an eigenstate of fz. Conversely, the eigenstate of fz with eigenvalue m has U(1) symme-try and it is invariant under a spin rotation e−ifzφ accompanied with gauge transformationeimφ. For a spin- f system, there are 2 f + 1 states that have the U(1) symmetry. We denotethis symmetry as U(1)Fz+mφ, where the subscript is shown to explicitly indicate the sym-metry between the spin and the gauge degrees of freedom. The combined symmetry form , 0 states is called continuous spin-gauge symmetry, and it relates spin textures to thesuperfluid current, as discussed in Sec. 6.

• Discrete symmetriesThe elements of the point groups of SO(3) can be described with a combination of thefollowing generators:

Cn : the cyclic group of rotations about the z-axis through angle 2πk/n with k = 1, · · · , n−1. The group is isomorphic to Zn and it has generators Cnz.

Dn : the dihedral group generated by Cnz and an additional rotation through π about anorthogonal axis. The group is isomorphic to Zn o Z2 and it has generators Cnz,C2x.

T : the point group of the tetrahedron whose generators are given by C2z,C3,x+y+z.O : the point group of the octahedron whose generators are given by C4z,C3,x+y+z,C2,x+y.Y : the point group of the icosahedron composed of 12 fivefold axes, 20 threefold axes,

and 30 twofold axes. It is shown [119] that a state with icosahedron symmetry doesnot exist for f ≤ 4.

Here, we specify the symmetry axis for convenience. The generator CnΩ denotes a 2π/nrotation around the direction of Ω. In the present case, CnΩ can be expressed in terms ofthe spin operator f as

CnΩ ≡ exp[−i

2πn

f · Ω|Ω|

], (328)

which is described by a (2 f +1) × (2 f +1) matrix. Combined with a gauge transformation,the state invariant under the transformation h = CnΩeiφ can be obtained by solving theeigenvalue equation

CnΩζ = e−iφζ. (329)

If the eigenvalue is 1, the invariance of the eigenstate is satisfied by a spin rotation alone,and therefore, the gauge symmetry is completely broken. On the other hand, the eigenstate

79

with an eigenvalue e−iφ , 1 is invariant under a simultaneous discrete transformation inspin and gauge. Such a symmetry is called discrete spin-gauge symmetry. What we needto do is to find simultaneous eigenstates of a set of generators for all discrete symmetrygroups.

The results for the spin-1 BEC are summarized in Table 5 and the structure of each orderparameter is shown in Fig. 25 (a).

phase H generators ζ ε

F+ U(1)Fz+φ eiφe−ifzφ (1, 0, 0) 12 (c0 + c1)n − p + q

F− U(1)Fz−φ e−iφe−ifzφ (0, 0, 1) 12 (c0 + c1)n + p + q

P (D∞)Fz,φ e−ifzφ, eiπC2x (0, 1, 0) 12 c0n

AF (C2)Fz,φ eiπC2z 1√2

(√1 + p

c1n , 0,√

1 − pc1n

)12 c0n − p2

2c1n + q

Table 5: Symmetry and order parameter structure for the spin-1 spinor BEC, where H is the isotropy group, ζ is arepresentative order parameter, and ε is the corresponding energy of the system.

7.1.4. Symmetry and order parameter structure of spin-2 spinor BECsIt is straightforward to carry out the procedure for spin-2 spinor BEC. The mean-field energy

for a spin-2 spinor BEC per particle is given by

ε[ζ] =12

c0n +12

c1n| f |2 + 12

c2n|A|2 − p∑

m

m|ζm|2 + q∑

m

m2|ζm|2, (330)

where A = (2ζ2ζ−2 − 2ζ1ζ−1 + ζ10 )/√

5 is the spin-singlet pair amplitude. We first considerstationary states that have isotropy groups H ∈ G. Among them, the state with H ⊂ GB cansurvive as a stationary state in the presence of an external field (p , 0 and q , 0).

• Continuous group U(1):There are five states that have continuous symmetries:

F2+ : (1, 0, 0, 0, 0)T , HF2+ = U(1)Fz+2φ : ei2φe−ifzφ, εF2+ = −2p + 4q + 2c1n, (331)

F2− : (0, 0, 0, 0, 1)T , HF2− = U(1)Fz−2φ : e−i2φe−ifzφ, εF2− = 2p + 4q + 2c1n, (332)

F1+ : (0, 1, 0, 0, 0)T , HF1+ = U(1)Fz+φ : eiφe−ifzφ, εF1+ = −p + q +12

c1n, (333)

F1− : (0, 0, 0, 1, 0)T , HF1− = U(1)Fz−φ : e−iφe−ifzφ, εF1− = p + q +12

c1n, (334)

UN : (0, 0, 1, 0, 0)T , HUN = (D∞)Fz : e−ifzφ, eiπC2x, εUN =110

c2n, (335)

where we indicate the generators of the isotropy group in curly brackets and ε = ε −c0n/2. These states are stationary in the presence of an external magnetic field, where theisotropy group HUN is reduced to U(1)Fz

.

80

U(1)Sz−Á

U(1)Sz

ei¼ C2

ei¼ C2

(i) F (ii) P (iii) AF

U(1)Sz−2Á

U(1)Sz−Á

(i) F2 (ii) F1

C2 U(1)Sz

(iii) UN (iv) C

ei2¼/3 C3

C2

(v) BAei¼ C4

C2

ei¼ C4 (viii) C4 ei2¼/3 C3

(ix) C3 (x) C2 (xi) C2'

ei¼ C2

C2

C2

C2

(vi) D2 (vii) D2'

C2

2¼0 phase

(a) spin 1

(b) spin 2

C2

C2

C2

Figure 25: Order parameters for (a) f = 1 and (b) f = 2 spinor BECs. Shown are the surface plots of the amplitude|∑m ψmY f ,m(s)|, where Y f ,m(s) is a rank- f spherical harmonics in spin space. The gray scale on the surface representsthe phase. The continuous and discrete symmetry axes are indicated along with the gauge transformation factors.

• Tetrahedron group T:We solve the simultaneous eigenstate of C2z and C3,x+y+z, whose matrix forms are given by

C2z =

1 0 0 0 00 −1 0 0 00 0 1 0 00 0 0 −1 00 0 0 0 1

, C3,x+y+z =14

−1 2i

√6 −2i −1

−2 2i 0 2i 2−√

6 0 −2 0 −√

6−2 −2i 0 −2i 2−1 −2i

√6 2i −1

. (336)

The only solution is ζC =12 (1, 0, i

√2, 0, 1)T (up to the phase factor). Here, ζC satisfies

C2zζC = ζC and C3,x+y+zζC = ei2π/3ζC. Therefore, the generators for the tetrahedron groupas a subgroup of G are C2z and e−i2π/3C3,x+y+z:

C :12

(1, 0, i√

2, 0, 1)T , HC = TF,φ : C2z, e−i2π/3C3,x+y+z, εC = 2q. (337)

This state is called a cyclic state. Cyclic states need not be stationary in the presence of anexternal field because T 1 GB.

81

• Dihedral groups Dn:In a manner similar to the tetrahedron group, we find

BN :1√

2(1, 0, 0, 0, 1)T , HBN = (D4)Fz,φ : eiπC4z,C2x, εBN = 4q +

c2n10

, (338)

D2 :(

cos χ√

2, 0, sin χ, 0,

cos χ√

2

)T

, HD2 = (D2)F : C2z,C2x, εD2 = 4q cos 2χ +c2n10

,

(339)

D2′ :12

√

1 − 10qc2n

, 0, i

√2(1 +

10qc2n

), 0,

√1 − 10q

c2n

T

,

HD2′ = (D2)F : C2z,C2x, εD2′ = 2q − 10q2

c2n. (340)

There are two energy minima D2 and D2′ in the order parameter manifold with an isotropygroup D2. When p = q = 0, UN, BN, and D2 states are all degenerate, and χ in D2 statecan take any arbitrary real value, while D2′ phase goes to the cyclic state. In the presence ofan external field, however, BN, D2 and D2′ states need not be stationary because Dn 1 GB.

• Cyclic groups Cn: All states that have cyclic symmetries are non-inert states. Here, wehave chosen the order parameter to minimize the mean-field energy.

C4 :1√

2

√

1 +p/2

(c1 − c2/20)n, 0, 0, 0,

√1 − p/2

(c1 − c2/20)n

T

,

HC4 = (C4)Fz,φ : eiπC4z, εC4 = 4q +110

c2n − p2

2(c1 − c2/20)n, (341)

C3+ :1√

3

(√1 +

p − qc1n

, 0, 0,√

2 − p − qc1n

, 0)T

,

HC3+ = (C3)Fz,φ : e−i2π/3C3z, εC3+ = 2q − (p − q)2

2c1n(342)

C3− :1√

3

(0,

√2 +

p + qc1n

, 0, 0,√

1 − p + qc1n

)T

,

HC3− = (C3)Fz,φ : ei2π/3C3z, εC3− = 2q − (p + q)2

2c1n(343)

C2 :1√

2

(0,

√1 +

p(c1 − c2/5)n

, 0,√

1 − p(c1 − c2/5)n

, 0)T

,

HC2 = (C2)Fz,φ : eiπC2z, εC2 = q +110

c2n − p2

2(c1 − c2/5)n, (344)

C2′ :1√

2(a, 0, b, 0, c)T , HC2′ = (C2)Fz : C2z. (345)

The order parameter and energy for C2′ state are the same as those of C2′ state in Table 4.

The obtained results are the same as those listed in Table 4. The structure of each orderparameter is shown in Fig. 25 (b).

82

7.2. Homotopy theory

7.2.1. Classification of topological excitationsBose-Einstein condensates can accommodate topological excitations such as vortices, monopoles,

and Skyrmions. These topological excitations are diverse in their physical properties but haveone thing in common; they can move freely in space and time without changing their charac-teristics that are distinguished by topological charges. The topological charges take on discretevalues and have very distinct characteristics independently of the material properties. It is thesematerial-independent universal characteristics and robustness to external perturbations that maketopological excitations unique.

The classification of topological excitations is best made by homotopy theory [122, 123].This theory describes what types of topological excitations are allowed in what order-parametermanifold. For example, π2(U(1)) = 0 implies that monopoles, which are characterized by thesecond homotopy group π2, are not topologically stable in systems described by scalar orderparameters. This theory also describes what happens if two defects coexist and how they coalesceor disintegrate.

By way of introduction, we first consider a BEC described by a scalar order parameter ψ(r) =|ψ(r)|eiφ(r). To classify line defects such as vortices, we take a loop in the condensate and considera map from every point r on the loop to the phase φ(r) of the order parameter ψ:

r 7→ φ(r) ≡ argψ(r). (346)

If the loop encircles no vortex (e.g., loop A in Fig. 26), the image of the map covers only a partof the unit circle, as shown in the right-hand side of Fig. 26. If the loop encircles a vortex (e.g.,loop B), the image covers the entire unit circle at least once. If it covers the circle n times, it issaid that the winding number of the vortex is n. The crucial observation here is that the windingnumber does not change if the loop deforms or moves in space as long as it does not cross thevortex. This property can be used to classify loops according to their winding number.

A

B

vortex

Pr

φ

1

S

Figure 26: Mapping from a loop in real space onto the order-parameter manifold S 1 (unit circle) according to thecorrespondence φ(r) : r→ argψ(r). This mapping defines the first homotopy group π1(S 1).

If two loops `a and `b are continuously transformable without crossing singularities of the or-der parameter, they are said to be homotopic to each other and written as `a ∼ `b. The homotopicrelationship is an equivalent one in that it is symmetric (i.e., `a ∼ `a), reflexive (i.e., if `a ∼ `b,

83

then `b ∼ `a), and transitive (i.e., if `a ∼ `b and `b ∼ `c, then `a ∼ `c). By this equivalencerelationship, all loops are classified into equivalent classes called homotopy classes:

[`1], [`2], [`3], · · · , (347)

where `i (i = 1, 2, 3, · · · ) is an arbitrarily chosen loop from the homotopy class [`i] because allloops belonging to the same class are equivalent and continuously deformable to each other.

Mathematically, a loop ` is defined as a mapping from I = [0, 1] to a topological space suchthat `(0) = `(1) = x0, where x0 is called a base point. If two loops `a and `b share the samebase point x0 and they are continuously deformable to each other, they are said to be homotopicat x0. If the base point is not shared, they are said to be freely homotopic. The constant loopc is defined as the one such that c(t) = x0 for ∀t ∈ [0, 1]. The inverse of loop ` is defined as`−1(t) ≡ `(1 − t) for ∀t ∈ [0, 1].

The product of two homotopy classes [`1] and [`2] is defined as

[`1] · [`2] = [`1 · `2], (348)

where `1 ·`2 denotes the product of two loops in which `1 is first traversed and then `2 is traversed.Members of [`1 · `2] that are homotopic to `1 · `2 need not return to the base point x0 en route tothe terminal point (see the dashed curve in Fig. 27).

0x

1l

2l

Figure 27: The product of two loops `1 · `2 in which `1 is first traversed and then `2 is traversed. Loops homotopic to`1 · `2 need not return to x0 en route to the terminal point, as shown for the dashed curve.

With the definition of the product (348), a set of homotopy classes form a group. In fact, theysatisfy the associative law

([`1] · [`2]) · [`3] = [`1] · ([`2] · [`3]); (349)

those loops that are homotopic to the constant loop c constitute the identity element [c] such that

[c] · [`] = [`] · [c] for any [`]; (350)

finally, the inverse [`−1] of [`] is the homotopy class that consists of inverse loops of [`] so that

[`−1] · [`] = [`] · [`−1] = [c]. (351)84

The group defined above is called the fundamental group or the first homotopy group and it isdefined as π1(M, x0), where x0 is the base point of all loops. In many situations in physics, thechoice of the base point is not important, and we shall omit it and write π1(M, x0) simply asπ1(M).

The fundamental group is said to be Abelian if any two elements commute (i.e., [m] · [n] =[n] · [m] for ∀[m] and ∀[n]) and non-Abelian if some elements do not commute. For the case ofU(1) vortices, the fundamental group is Abelian and · is simply addition. For example, if twosingly quantized U(1) vortices coalesce, a doubly quantized vortex results. This can be describedas [1] · [1] = [2]. Thus, the fundamental group is the additive group of integers Z:

π1(S 1) Z. (352)

Point defects such as monopoles can be characterized by considering a sphere Σ that coversthe object (see Fig. 28). We consider a map from each point r on Σ to a point m in the orderparameter space M:

m : Σ→ M. (353)

We classify elements of such a map by regarding as equivalent any two elements that can trans-form into each other in a continuous manner. For the case of the monopole, we consider M tobe a two-dimensional sphere or 2-sphere S 2, as shown in the right-hand side of Fig. 28. In thiscase, if the map (353) wraps S 2 n times, we have π2(S 2) [n].

O

2S

r

Σ

m(r)

Figure 28: Mapping from a sphere in real space onto the order-parameter manifold M according to the correspondenceΨ(r) : r→ M. This mapping defines the second homotopy group π2(M). Here, we take M to be S 2—a two-dimensionalunit sphere (2-sphere).

Skyrmions and knots are classified by the third homotopy group. We assume that the orderparameter becomes uniform at spatial infinity. Then, the three-dimensional space is compactedinto a three-dimensional sphere S 3. To help envisage it, imagine a two-dimensional plane andidentify all infinite points. Then, the two-dimensional plane is compacted into a two-dimensionalsphere S 2. By considering the map from S 3 to the order parameter space M and by identifyingmaps that can be continuously transformed into each other without crossing any singularity ofthe order parameter space as belonging to the same equivalent class, we can introduce elementsof the third homotopy group π3(M).

Higher homotopy groups can be defined in a similar manner. Moreover, the zeroth homotopyπ0(M) classifies the domain walls and gives the number of disconnected domains; in particular,

85

π0(M) = 0 implies that the order-parameter manifold is connected. Using homotopy theory,one can elicit insights into the topology of complicated structures, which is difficult to obtainintuitively. Here, we list some useful formulas of homotopy theory:

πn(S m)

Z if m = n ≥ 1;0 if m ≥ n ≥ 1;0 if m = 1 and n ≥ 2;Z if n = 3 and n = 2 (Hopf charge);

Z2 if n = m + 1 ≥ 4 or n = m + 2 ≥ 4,

(354)

where 0 implies that trivial configurations (i.e., those contractible to a point) exist and Z2 = 0, 1is the two-element group.

Let G be a Lie group and H be its subgroup. If G is connected (i.e., π0(G) = 0) and simplyconnected (i.e., π1(G) = 0), the following isomorphism holds:

π1(G/H) π0(H). (355)

Similarly, if π2(G) = π1(G) = 0, then

π2(G/H) π1(H). (356)

As a corollary, when G = SU(2) and H = U(1), we obtain

π2(SU(2)/U(1)) π1(U(1)) Z. (357)

Some order-parameter spaces R and their first, second, and third homotopy groups are sum-marized in Table 6. The topological objects classified by homotopy groups are summarized inTable 7. For example, in the case of ferromagnetic BECs, π1(SO(3)) Z2 implies that there existtwo types of linear defects that are singular and nonsingular; π2(SO(3)) 0 implies the absenceof point defects and 2D Skyrmions; and π3(SO(3)) Z implies the presence of a nonsingularsoliton-like object such as a Shankar monopole (Skyrmion) [124, 125].

Here, we define a Skyrmion as a nonsingular point object characterized by a π2 charge in twodimensions and by a π3 charge in three dimensions. In the polar phase of a spin-1 condensate, aπ2 Skyrmion takes a configuration in which the director points in the positive z-direction at thecenter, flares out in the radial direction, and eventually points in the negative z-direction as onegoes away from the center. On the other hand, a π3 Skyrmion is exemplified by a knot soliton,as discussed in Sec. 7.3.4. In contrast, a monopole is a singular point object in three dimensions(a singular point object in two dimensions is called a vortex). The charge of the monopole ischaracterized by a π2 charge; examples include a hedgehog (’t Hooft-Polyakov monopole) ina polar condensate (see Sec. 7.3.2). Note, however, that the π2 monopole cannot be createdin a ferromagnetic condensate because its order-parameter manifold is SO(3), which does notgive a non-trivial π2 charge; in this case, the monopole must be attached by a Dirac string (seeSec. 7.3.2).

7.2.2. Relative homotopy groupsTo fully understand broken-symmetry systems, it is important to take into account not only

the global boundary conditions but also various constraints imposed on the order-parameterfield. Relative homotopy groups provide a useful tool for characterizing effects when the order-parameter manifold changes its character on a surface or when it is constrained by boundary

86

R π1 π2 π3

scalar BEC U(1) Z 0 0Heisenberg spin S 2 0 Z Znematics RP2 S 2/Z2 Z2 Z Z

biaxial nematics SU(2)/Q Q 0 Zferromagnetic BEC SO(3)F,φ RP3

F,φ Z2 0 Z

spin-1 polar BEC [S 2F × U(1)φ]/(Z2)F,φ Z Z Z

cyclic BEC [SO(3)F × U(1)φ]/TF,φ T ∗ 0 Z

Table 6: List of homotopy groups. Subscripts F and φ denote the manifolds for the spin and gauge degrees of freedom,respectively. Q and T ∗ respectively denote the quaternion group and the binary tetrahedral group, the latter being asubgroup of SU(2) × R.

πn defects solitonsπ0 domain walls dark solitonsπ1 vortices nonsingular domain wallsπ2 monopoles 2D Skyrmionsπ3 Skyrmions, knots

Table 7: Topological objects (defects and solitons) described by homotopy groups.

conditions. Relative homotopy groups can also be used to discuss properties of a defect that isitself regarded as a singularity in the order-parameter field in ordinary homotopy theory. Finally,the relative homotopy group is useful to calculate the absolute homotopy group via the exactsequence of homomorphisms.

As an illustrative example, let us again consider a ferromagnetic BEC. Suppose that thesystem is contained in a cylinder of radius r0 and impose a boundary condition that the spinpoints in the positive z-direction at r = 0 and outward or downward on the wall at r = r0, wherer is the radial coordinate. Then, the spin texture would look like

ψcoreless(r, ϕ, z) =√

n

cos2 β(r)

2eiϕ√

2sin β(r)

e2iϕ sin2 β(r)2

. (358)

Because the direction of spin is s(r, ϕ, z) = (ex cosϕ + ey sinϕ) sin β(r) + ez cos β(r), the aboveboundary conditions are met if we choose β(0) = 0 and β(r0) = π/2 or β(r0) = π. The topologicalobjects corresponding to β(r0) = π/2 and β(r0) = π are referred to as the Mermin-Ho vortex andAnderson-Toulouse vortex, respectively. It can be shown that the circulation at the wall is 1for the Mermin-Ho vortex and 2 for the Anderson-Toulouse vortex. They are good examples ofrelative homotopy because both of them are not stable in free space and they can be stabilizedonly if the boundary conditions on the wall are imposed by some means.

In general, the nth relative homotopy group πn(R, R) consists of the elements of πn(R) fromwhich those of πn(R) are subtracted. We consider an n-sphere in R and let it expand into R bycontinuous deformation. Such an expansion may be regarded as a one-to-one mapping of πn(R)

87

into a subgroup of πn(R). Let Im(πn(R) → πn(R)) be the image of this homomorphism. Then,by the definition given above, the relative homotopy πn(R, R) is a quotient group of Im(πn(R) →πn(R)) in πn(R):

πn(R, R) πn(R)

Im(πn(R)→ πn(R)). (359)

The relative first homotopy group π1(R, R) can cope with planar defects with nontrivialboundaries. A well-known example is a planar soliton (Maki domain wall) in the A phase ofsuperfluid helium-3, where the order-parameter manifold is R = (S 2 × SO(3))/Z2 within a dipolehealing length from the domain wall and R = SO(3) outside this layer due to dipole locking.The domain wall may be regarded as a linear defect as seen from the edge of the wall. Wetake the axis perpendicular to the domain wall as the z-axis. Then, the order-parameter fielddefines a mapping from the z-axis into the contour in R whose end points belong to R. Becauseπ1(SO(3)) Z2, such contours are also classified as Z2, of which the nontrivial one correspondsto the Maki domain wall.

The relative second homotopy group π1(R, R) allows the stabilization of a point object subjectto a boundary condition on a sphere enclosing it. Another example is the boojum–a point defectthat appears in the A phase of superfluid helium-3 when it is contained in a hard-wall sphere.Because π2(SO(3)) = 0, the bulk superfluid A phase does not support a topologically stable pointdefect. However, because the l vector must be perpendicular to the surface, a point singularityfor the l-vector appears on the surface that is connected to a nonsingular vortex texture in thebulk.

7.3. Examples

7.3.1. Line defectsThe line defects are characterized by the first homotopy group (or the fundamental group)

π1(M). For the case of s-wave superconductors, liquid helium-4, and spin-polarized or spinlessgaseous BECs, the order parameter is scalar and the order-parameter manifold is M = U(1). Thefundamental group is therefore the additive group of integers:

π1(U(1)) Z. (360)

For the case of a spin-1 BEC, the ground state can be ferromagnetic or polar. The order-parameter manifold of the ferromagnetic phase is SO(3) and the fundamental group is π1(SO(3)) =Z2 = 0, 1, the two-element group. The spin configuration that corresponds to class 0 is continu-ously transformable to a uniform configuration, whereas the spin configuration that correspondsto class 1 describes a singly quantized vortex that is topologically stable. Because 1 + 1 = 0 inZ2, the coalescence of two singly quantized vortices is homotopic to a uniform configuration.Conversely, there is a configuration called the Anderson-Toulouse vortex that is nonsingular atthe origin but has circulation of two away from the origin.

Next, let us consider the polar phase of a spin-1 BEC. The order-parameter manifold isRpolar = [S 2

F × U(1)φ]/(Z2)F,φ. As we have discussed in Sec. 7.1.1, the Z2 symmetry arisesfrom the fact that the order parameter given by Eq. (316) is invariant under the simultaneoustransformation of (φ, d) → (φ + π,−d). Because G = S 2 × U(1) is not simply connected, to cal-culate π1(Rpolar), we lift G to G′ = S 2 × R and use Eq. (355). Then, the points that are identical

88

to (φ, d) in G′ are (φ + nπ, d) for even n and (φ + nπ,−d) for odd n, and those points constitutean additive group of integers n. Hence, we have

π1(Rpolar) ZF,φ. (361)

As in the case of a scalar BEC, vortices in the polar phase are classified by integers. However,due to the Z2 discrete symmetry, the minimum unit of circulation is one-half of the usual valueof h/M, as discussed in Sec. 6.2.2.

In a similar manner, we may expect that the uniaxial, biaxial, and cyclic phases in spin-2spinor BECs can support various types of vortices due to the discrete symmetry, as discussed inSec. 6. Some of them have fractional circulations, and some others have no mass circulation.

The topological charge can be expressed algebraically as a functional of the order-parameterfield. The topological charge corresponding to π1(S 1) can be expressed as a line integral of thevelocity field.

7.3.2. Point defectsPoint defects are characterized by the second homotopy group. Because π2(U(1)) = 0, scalar

BECs have no topologically stable point defects. When the order-parameter manifold is S 2, itcan host monopoles because π2(S 2) Z.

Here, we consider the situation shown in Fig. 28, where m(r) is a mapping from S 2 to S 2. Thetopological charge N2 of the point defect can be calculated as follows. By definition, N2 givesthe number of times the vector m(r) shown in Fig. 28 wraps S 2. Expressing the components ofthis vector in spherical coordinate as mx = sinα cos β, my = sinα sin β, and mz = cosα, we cancalculate N2 as

N2 =1

4π

∫Σ

dΩ∣∣∣∣∣∂(α, β)∂(θ, φ)

∣∣∣∣∣ , (362)

where dΩ ≡ sin θdθdφ, with θ and φ are the spherical coordinates of r, and the last term onthe right-hand side is the Jacobian of the transformation of the coordinates. The fact that theintegrand is the Jacobian explicitly indicates that N2 gives the degree of mapping from S 2 in realspace to S 2 in order-parameter space. The right-hand side of Eq. (362) can be directly expressedin terms of m as

N2 =1

4π

∫Σ

dθdφm ·(∂m∂θ× ∂m∂φ

)=

14π

∫j · dS, (363)

where

j =12εi jkmi(∇m j × ∇mk). (364)

We can use Gauss’ law to rewrite the right-hand side of Eq. (363) as a volume integral:

N2 =

∫nmdr, (365)

where

nm(r) =1

4π∇ · j(r) (366)

89

gives the density distribution of point singularities.

’t-Hooft-Polyakov monopole (hedgehog)Next, we consider the polar phase. The second homotopy group is given by π2(Rpolar) = Z,

and therefore, it can accommodate point defects. To investigate their nature, we write the orderparameter in the following form:

ψpolar =

√n2

eiφ

−dx + idy√

2dz

dx + idy

. (367)

By setting φ = 0, we obtain the spherical monopole called ’t-Hooft-Polyakov monopole orhedgehog:

d(r) =r|r| . (368)

It follows from Eq. (363) that the topological charge of the hedgehog is N2 = 1. Moreover, jdefined in Eq. (364) is related to the circulation of spin currents given by Eq. (287) as∑

µ

dµ(∇ × v(spin,P)

µ

)= −2~

Mj, (369)

which implies that the surface integral of the spin circulation is quantized in terms of 4h/M;∫ ∑µ

dµ(∇ × v(spin,P)

µ

)· dS = −4h

MN2. (370)

Alice ringThe monopole in the spin-1 polar phase is energetically unstable against deformation into an

Alice ring [126]. The Alice ring is a combined object of two defects characterized by π1 and π2(see Fig. 29). Viewed far from the origin, it appears to be a point defect (N2 = 1); however, alongthe ring C on which the order parameter is singular, it appears to be a continuous distribution ofline defects (N1 = 1). This topological object is called the Alice ring, and it has been predictedto be realized in an optically trapped spin-1 23Na BEC [126].

Dirac monopoleThe magnetic monopole was originally envisaged as a magnetic analogue of the quantized

electric charge:

∇ · B = 4πgδ(r), (371)

where g denotes the strength of the magnetic monopole. The solution of this equation is easilyfound to be B = gr/r3 and the corresponding vector potential is given by

A = − gr(r − z)

(−y, x, 0) = −g(1 + cos θ)r sin θ

eϕ, (372)

90

C

O

Figure 29: Alice ring comprised of continuously distributed half-quantum vortices along a contour C. Far from theorigin, it appears to be a monopole.

where (r, θ, ϕ) are the polar coordinates and eϕ is the unit vector in the ϕ-direction. The vectorpotential (372) reproduces the magnetic field (371), except on the positive z-axis along which themagnetic field exhibits a singularity called the Dirac string:

rotA = grr3 − 4πgδ(x)δ(y)θ(z)ez, (373)

where ez is the unit vector along the z-direction.The Dirac monopole can be created in the ferromagnetic phase of a spin-1 BEC [127]. Sub-

stituting φ→ −ϕ, α→ ϕ, and β→ θ in Eq. (319), we obtain

ζDirac ≡ 1√

nψDirac =

e−2iϕ cos2 θ

2e−iϕ√

2sin θ

sin2 θ2 .

(374)

The corresponding superfluid velocity v(mass) = (~/M)Im(ζ†∇ζ) [defined by Eq. (259)] is thesame as the vector potential (372) of the Dirac monopole with the identification g = ~/M.

It is interesting to note that the order parameter (374) exhibits a doubly quantized vortex onlyalong the positive z-axis. One can say that because the first homotopy group of SO(3) is Z2,which allows only a singly quantized vortex called a polar-core vortex as a stable topological ob-ject, the doubly quantized vortex must terminate at the magnetic monopole. In fact, the singularDirac monopole can deform continuously to a nonsingular spin texture if we take the followingparametrization:

ζDirac =

e−2iϕ cos2 β

2e−iϕ√

2sin β

sin2 β2

, β(r, ϕ, z) = θ(1 − t) + πt. (375)

We can see that as t increases from 0 to 1, the order parameter continuously changes from theDirac monopole at t = 0 to a nonsingular texture ζ = (0, 0, 1)T at t = 1.

7.3.3. Action of one type of defect on anotherWhen two different types of topological objects coexist, there exists an action of one object

on the other as they move relative to each other. As an example, consider a spin-1 polar phasein which two monopoles with charge 1 merge in the presence of a half-quantum vortex. As a

91

monopole turns around the vortex, it changes sign. Thus, if they merge along line 1, the order-parameter field must deform, as shown in Fig. 30 (b), and the combined object has the topologicalcharge of two. On the other hand, when they merge along line 2, the order-parameter manifoldcan deform into the one shown in Fig. 30 (c), and the topological charge is zero, indicating thatthe two monopoles annihilate in pairs. This example illustrates that the charge of the monopolechanges its sign as it moves around the half-quantum vortex along curve 2.

1

2

(a) (c)(b)

BA

half-quantum vortex

Figure 30: (a) Coalescence of two monopoles with charge one in the presence of a half-quantum vortex. (b) If they mergealong the dashed curve 1 shown in Fig. (a), the topological charge of the combined object is two. (c) If they merge alongthe dashed curve 2, the topological charge of the combined object is zero.

7.3.4. Skyrmions

Shankar SkyrmionThe third homotopy group characterizes topological objects called Skyrmions that extend

over the entire three-dimensional space. A prime example of this is the so-called Shankarmonopole [124, 125], although it is actually not a monopole but a Skyrmion. The topologyof the Shankar Skyrmion is π3(SO(3)) Z, which is supported by the ferromagnetic phase of aspin-1 BEC. The order parameter of the ferromagnetic BEC is characterized by the direction ofthe local spin s and the superfluid phase that can be parametrized in terms of two unit vectorssuch as t and u that form a triad with s: s = t × u. Concrete forms of s and t are introduced inSec. 6.2.1.

The class one element of π3(SO(3)) Z may be realized by rotating the triad at position rabout the direction [125, 124, 128, 129]

Ω =rr

f (r)N3, N3 ∈ Z (376)

through an angle f (r)N3, where f (0) = 2π and f (∞) = 0. It follows from the last condition thatthe triad is uniform in spatial infinity, and thus, the three-dimensional space is compacted to S 3.The case of n = 1 is shown in Fig. 31.

The topological charge of the third homotopy group can be introduced in a manner similar tothat of the second homotopy group. Now, the mapped vector has a four-component vector that isnormalized to unity. One such representation is

n =rr

sinf (r)2, n4 = cos

f (r)2. (377)

The invariant of such a mapping is again given by the Jacobian of the transformations:

N3 =1

12π2

∫drεαβγδεi jknα∂inβ∂ jnγ∂knδ, (378)

92

x

y

z

Figure 31: Shankar monopole with charge one.

where εi jk and εαβγδ are the completely antisymmetric tensors of the third and fourth orders, re-spectively, and the Roman letters run over x, y, z and the Greek letters run over 1, 2, 3, 4.

Knot solitonThe third homotopy group also classifies knot solitons. Knots differ from other topological

excitations such as vortices, monopoles, and Shankar monopoles in that they are classified bya linking number while others are classified by winding numbers. Knots are characterized bymappings from a three-dimensional sphere S 3 to S 2. As in the case of Shankar monopoles, theS 3 domain is prepared by imposing a boundary condition that the order parameter takes on thesame value in every direction at spatial infinity. Here, we consider the spin-1 polar phase [130].The order parameter manifold of the polar phase is given by Rpolar (S 2

S × U(1)φ)/(Z2)S,φ, ofwhich neither U(1) nor Z2 symmetry contributes to the homotopy groups in spaces higher thanone dimension; hence, we find that π3(Rpolar) π3(S 2) Z. The associated integer topologicalcharge Q is known as the Hopf charge, and it is given by

Q =1

4π2

∫d3x εi jkFi jAk, (379)

where Fi j = ∂iA j − ∂ jAi = d · (∂i d × ∂ j d) [131]. Note that the domain (r) is three-dimensional,while the target space (d) is two-dimensional. Consequently, the preimage of a point on the targetS 2 constitutes a closed loop in S 3, and the Hopf charge is interpreted as the linking number ofthese loops: if the d field has Hopf charge Q, two loops corresponding to the preimages of anytwo distinct points on the target S 2 will be linked Q times [see Fig. 32 (a)]. Figure 32 (b) shows

93

an example of the d field of a polar BEC with Hopf charge 11.Knots can be created by manipulating an external magnetic field. In the presence of an

external magnetic field, the linear Zeeman effect causes the Larmor precession of d, whereas dtends to become parallel to the magnetic field because of the quadratic Zeeman effect. Supposethat we prepare an optically trapped BEC in the m = 0 state, i.e., d = (0, 0, 1)T , by applyinga uniform magnetic field in the z-direction. Then, we suddenly turn off the uniform field andswitch on a quadrupole field. Because of the linear Zeeman effect, d starts rotating around thelocal magnetic field, and therefore, the d field winds as a function of t, resulting in the formationof knots.

Figure 33 shows the creation dynamics of knots in an optical trap subject to the quadrupolefield, where the upper panels show the snapshots of the preimages of d = −z and d = z andthe bottom panels show cross sections of the density for m = −1 (bottom) components on the xyplane. The density pattern in m = −1 components is characteristic of knots; a double-ring patternappears corresponding to one knot. As the d field winds in the dynamics, the number of ringsincreases. This prediction can be tested by the Stern-Gerlach experiment.

Topological objects are conjectured to play important roles in the formation of our universe.Among them, knot solitons have recently attracted the interest of cosmologists since Faddeevand Niemi suggested that knots might exist as stable solitons in a three-dimensional classicalfield theory [131].

S 2

r d

preimage

(a) (b)

Figure 32: (a) Preimages of two distinct points on S 2 form a link. (b) Spin configuration of a knot with Hopf charge 1in a polar BEC, where arrows inicate the d field of the polar phase. The solid and dashed lines trace the point where dpoints to x and −z, respectively, and form a link.

8. Many-Body Theory

In spinor BECs, the mean-field energy is much larger than the spin-exchange interaction.Therefore, we first determine the spatial dependence of the condensate wave function indepen-dently of the spin degrees of freedom, and then, we consider the spin state by assuming thatthe obtained spatial wave function is shared by all spin states. This approximation is called the

1Technically speaking, the configuration in Fig. 32 (b) is an unknot of a pair of rings with linking number 1, becausethe preimage of one point on S 2 forms a simple unknotted ring.

94

(c) 2.2 TL(b) 1.1 TL (d) 3.3 TL

d = (0, 0, ¡1)T

d = (1, 0, 0)T

10

(a) 0.5 TL

Figure 33: Dynamics of the creation of knots in a spherical optical trap under a quadrupole magnetic field. Snapshots ofthe preimages of d = (0, 0,−1)T and d = (1, 0, 0)T (top), and cross sections of the density for m = −1 components on thexy plane (bottom).

single-mode approximation. In this section, we examine the many-body spin states by using thisapproximation.

Suppose that the coordinate part of the ground-state wave function φ0(r) is independent ofthe spin state. Then, φ0(r) is the lowest-energy solution of the GP equation[

−~2∇2

2M+ Utrap + c0(N − 1)|φ0|2

]φ0 = µφ0, (380)

where Utrap is the trapping potential. Let µ0 and µ1 be the first- and second-lowest eigenvalues ofEq. (380), respectively. The criterion for the validity of the single-mode assumption is that thefirst excited energy µ1 − µ0 is much larger than other energy scales:

µ1 − µ0 |p|, |c1|N/Veff , |c2|N/Veff , (381)

where Veff ≡ (∫

dr|φ0|4)−1 is the effective volume of the system. When this condition (381) issatisfied, the field operator ψm (m = f , f − 1, · · · ,− f ) may be approximated as ψm(r) ' amφ0(r),where am is the annihilation operator of the bosons that have magnetic quantum number m andcoordinate wave function φ0.

8.1. Many-body states of spin-1 BECs

8.1.1. Eigenspectrum and eigenstatesAs explained above, the field operator of a spin-1 BEC in the single-mode approximation can

be expressed as [85, 4, 86]

ψm(r) = amφ0(r) (m = 1, 0,−1). (382)

95

For the sake of simplicity, let us consider the case of zero magnetic field (i.e., p = q = 0).Substituting Eq. (382) in Eq. (24) and replacing

∑m a†mam and

∑m,n a†ma†nanam with N and N(N −

1), respectively, we obtain

H = N∫

dr φ∗0

(−~2∇2

2M+ U(r) +

c0

2(N − 1)|φ0|2

)φ0 + c′1 : F2 :, (383)

where

F =1∑

m,n=−1

fmna†man (384)

and c′1 ≡ (c1/2)∫

dr|φ0|4; φ0 is determined by Eq. (381) subject to the normalization condition∫|φ0|2dr = 1. (385)

Substituting the solution of Eq. (381) back into (383), we obtain

H = µN − c′0N(N − 1) + c′1 : F2 :, (386)

where c′0 ≡ (c0/2)∫

dr|φ0|4. From Eq. (16), we find that the total spin operator F satisfies

: F2 := N(N − 1) − 3 S †S , (387)

where

S ≡ 1√

3(a2

0 − 2a1a−1) (388)

is the spin-singlet pair operator. We use Eq. (387) to rewrite Eq. (386) as

H = µN − c′0N(N − 1) + c′1[N(N − 1) − 3S †S

]. (389)

Because N and S †S commute, the eigenvalue problem reduces to finding their simultaneouseigenstates.

Let |0〉 be the state in which there is no spin-singlet pair, i.e., S |0〉 = 0. Using the commuta-tion relation [

S , (S †)k]=

23

k (2N + 2k + 1)(S †)k−1, (390)

we obtain

S †S (S †)k |0〉 = 23

k (2N − 2k + 1)(S †)k |0〉. (391)

We then define the k-pair state |k〉:

|k〉 ≡ (S †)k |0〉√〈0|S k(S †)k |0〉

. (392)

It follows from the commutation relation (390) that

S †S |k〉 = 23

k (2N − 2k + 1)|k〉. (393)

96

8.1.2. FragmentationNext, we consider a state |k, d〉 in which 2k atoms form spin-singlet pairs and the remaining

N − 2k atoms reside in the m = 0 state along the d direction (|d| = 1). Defining

ad ≡ dxax + dyay + dz az

= −dx + idy√

2a1 + dza0 +

dx − idy√2

a−1, (394)

the corresponding eigenstate and eigenenergy are given by

|k, d〉 = Z−12

m (a†d)N−2k(S †)k |0〉, (395)Ek = µN − c′0N(N−1) + c′1 [N(N−1)−2k(2N−2k + 1)], (396)

where Zm is the normalization constant. In the absence of an external magnetic field, all magneticsublevels are degenerate. If we take the quantization axis along the z-direction, we can specify theeigenstate in terms of the total spin F, magnetic quantum number Fz, and number of spin-singletpairs k:

|k, F, Fz〉 = Z−12 (S †)k(F−)F−Fz (a†1)F |vac〉, (397)

where F = N − 2k, and

F− ≡∑m,n

(f−)mn a†man =√

2(a†0a1 + a†−1a0) (398)

is the lowering operator of the magnetic quantum number.

Next, we consider a situation in which all F atoms points in the z-direction in addition to thek spin-singlet pairs, i.e., Fz = F in Eq. (397), and seek the number of the atoms that are presentin the m = 0 state. We introduce the basis state |n1, n0, n−1〉 in which the magnetic sublevelsm = 1, 0,−1 are occupied by n1, n0, and n−1 atoms, respectively. Then, |vac〉 = |0, 0, 0〉 and

〈a†0a0〉 =〈vac|aF

1 S ka†0a0(S †)k(a†1)F |vac〉〈vac|aF

1 S k(S †)k(a†1)F |vac〉

=〈F, 0, 0|S ka†0a0(S †)k |F, 0, 0〉〈F, 0, 0|S k(S †)k |F, 0, 0〉

. (399)

Substituting a†0a0 = a0a†0 − 1 and using the fact that a†0 commutes with S †, we have

〈a†0a0〉 =〈F, 1, 0|S k(S †)k |F, 1, 0〉〈F, 0, 0|S k(S †)k |F, 0, 0〉

− 1. (400)

We use Eq. (390) to rewrite

S (S †)k = [S , (S †)k] + (S †)kS

=23

k (2N + 2k + 1)(S †)k−1 + (S †)kS . (401)

97

Noting that S |F, 1, 0〉 = S |F, 0, 0〉 = 0 and N|F, 1, 0〉 = (F + 1)|F, 1, 0〉, we have

〈a†0a0〉 =(2F + 2k + 3)〈F, 1, 0|S k−1(S †)k−1|F, 1, 0〉(2F + 2k + 1)〈F, 0, 0|S k−1(S †)k−1|F, 0, 0〉

− 1

=(2F + 2k + 3)(2F + 2k + 1) · · · · (2F + 5)(2F + 2k + 1)(2F + 2k − 1) · · · · (2F + 3)

− 1

=2k

2F + 3=

N − F2F + 3

.

Because 〈a†1a1〉 − 〈a†−1a−1〉 = F and∑1

m=−1〈a†mam〉 = N, we obtain

n1 ≡ 〈a†1a1〉 =NF + F2 + N + 2F

2F + 3, (402)

n0 ≡ 〈a†0a0〉 =N − F2F + 3

, (403)

n−1 ≡ 〈a†−1a−1〉 =(N − F)(F + 1)

2F + 3. (404)

When all atoms form spin-singlet pairs, i.e., F = 0, Eqs. (402)–(404) show that all magneticsublevels are equally populated:

〈a†1a1〉 = 〈a†0a0〉 = 〈a†−1a−1〉 =N3. (405)

That is, the condensate is fragmented in the sense that more than one single-particle state ismacroscopically occupied [132, 4, 86, 133]. This is in sharp contrast with the mean-field re-sult that predicts that for c1 > 0, the ground state is either polar (n0 = N) or antiferromagnetic(n1 = n−1 =

N2 ). This discrepancy originates from the mean-field assumption that there exists one

and only one BEC. However, such an assumption cannot be justified when the system possessescertain exact symmetries such as rotational symmetry in the present case. Equation (403) showsthat n0 decreases rapidly with an increase in F. In fact, when F = O(

√N), we have n0 = O(

√N).

This implies that although mean-field results break down at zero magnetic field, the validity ofmean-field theory quickly recovers as the magnetic field increases. This affords an example ofwhy fragmented BECs are fragile against symmetry-breaking perturbations.

Finally, we note an interesting relationship between a mean-field state and the many-bodyspin-singlet state. A mean-field state is one in which all particles occupy the same single-particlestate:

(a†d)N

√N!|vac〉, (406)

where ad is defined in Eq. (394). Substituting dx = sin θ cos φ, dy = sin θ sin φ, and dz = cos θ inEq. (394), we obtain

|θ, φ〉 = 1√

N!

− a†1√2

sin θeiφ + a†0 cos θ +a†−1√

2sin θe−iφ

N

|vac〉. (407)

98

We calculate the equal-weighted superposition state of |θ, φ〉 over all solid angles dΩ = sin θdθdφ.

|sym〉 = 14π

∫ π

0sin θ dθ

∫ 2π

0dφ |θ, φ〉

=

0 if N is odd;1

(N+1)√

N!(a†20 − 2a†1a†−1)

N2 |vac〉 if N is even. (408)

This result shows that the spin-singlet state is the superposition state of a ferromagnetic state|θ, φ〉 over all angles of magnetization. Conversely, the mean-field state |θ, φ〉may be interpretedas a broken symmetry state of the spin-singlet state with respect to the direction of quantization(θ, φ).

8.2. Many-body states of spin-2 BECsNext, we discuss the many-body states of spin-2 BECs in the single-mode approximation.

Substituting ψm(r) = amφ(r) in Eq. (30), we obtain the spin-dependent part of the Hamiltonianas

H =c1

2Veff : F 2 : +2c2

5Veff S+S− − pFz, (409)

where

F ≡∑mn

fmna†man (410)

is the spin vector operator,

Fz ≡∑

m

ma†mam (411)

is its z-component, and

S− = (S+)† ≡ 12

∑m

(−1)mama−m (412)

are the creation (S+) and annihilation (S−) operators of a spin-singlet pair.While the operator S+, when applied to a vacuum, creates a pair of bosons in the spin-singlet

state, the pair should not be regarded as a single composite boson because S+ does not satisfythe commutation relations of bosons. In fact, the operators S± together with Sz ≡ (2N + 5)/4satisfy the S U(1, 1) commutation relations:

[Sz, S±] = ±S±, [S+, S−] = −2Sz. (413)

Here, the minus sign in the last equation is the only difference from the usual spin commutationrelations. This difference, however, leads to some important consequences. In particular, theCasimir operator S2, which commutes with S± and Sz, does not take the usual form of thesquared sum of spin components but instead takes the form

S2 ≡ −S+S− + S2z − Sz. (414)

99

8.2.1. Eigenspectrum and eigenstatesThe eigenstates are classified according to the eigenvalues of the Casimir operator (414).

Because S+S− = S2z − Sz − S2 is positive semidefinite, Sz must have a minimum value Smin

z .Recalling that Sz ≡ (2N + 5)/4, Smin

z can be expressed in terms of a non-negative integer N0 asSmin

z = (2N0 + 5)/4. Let |φ〉 be the eigenvector corresponding to this minimum eigenvalue; then,S−|φ〉 = 0. It follows that

S2|φ〉 = Sminz (Smin

z − 1)|φ〉; (415)

hence, the eigenvalue of S2 is given by ν = S (S − 1) with S = (2N0 + 5)/4.The operation of S+ on |φ〉 increases the eigenvalue of Sz by 1 and that of N increases by

2, as observed from the commutation relations (413). Therefore, the allowed combinations ofeigenvalues S (S − 1) and S z for S2 and Sz, respectively, are given by

S = (2N0 + 5)/4 (N0 = 0, 1, 2, . . .) (416)

andS z = S + NS (NS = 0, 1, 2, . . .), (417)

where N0 and NS satisfyN = 2NS + N0, (418)

and we may therefore interpret NS as the number of spin-singlet “pairs” and N0 the number ofremaining bosons.

To find the exact energy eigenvalues of Hamiltonian (409), we first note that S± and Sz

commute with F :

[S±, F ] = 0, [Sz, F ] = 0. (419)

The energy eigenstates can be classified according to quantum numbers N0 and NS, total spinF, and magnetic quantum number Fz. We denote the eigenstates as |N0,NS, F, Fz; λ〉, whereλ = 1, 2, . . . , gN0,F labels orthonormal degenerate states with gN0,F provided in Ref. [5].

The first commutator in Eq. (413) implies that S± plays the role of changing the eigenvalueof Sz by ±1 by creating or annihilating a spin-singlet pair. The eigensystem can be constructedby first preparing the eigenstates that do not involve spin-singlet pairs,

|N0, 0, F, Fz; λ〉, (420)

and then, by generating new eigenstates through successive operations of S NS+ on it:

|N0,NS, F, Fz; λ〉 =S NS+ |N0, 0, F, Fz; λ〉

‖ S NS+ |N0, 0, F, Fz; λ〉 ‖

, (421)

where ‖ |ψ〉 ‖≡√〈ψ|ψ〉, and the orthnormality condition is

〈N0,NS, F, Fz; λ′|N0,NS, F, Fz; λ〉 = δλ′,λ. (422)

To write down the eigenstates (421) explicitly, we need to find the matrix elements of S +, ofwhich only the nonvanishing ones are

〈N0,NS + 1, F, Fz; λ|S +|N0,NS, F, Fz; λ〉 =√

(NS + 1)(NS + N0 + 5/2). (423)100

Finally, the energy eigenvalue for the state |N0,NS, F, Fz; λ〉 is given by

E =c1

2Veff [F(F + 1) − 6N] +c2

5Veff NS(N + N0 + 3) − pFz, (424)

where the relationship 2NS + N0 = N is used.The total spin F can, in general, take integer values in the range 0 ≤ F ≤ 2N0. However, there

are some forbidden values [5]. That is, F = 1, 2, 5, 2N0−1 are not allowed when N0 = 3k (k ∈ Z),and F = 0, 1, 3, 2N0 − 1 are forbidden when N0 = 3k ± 1. We prove this at the end of thissubsection.

To gain a physical insight into the nature of the energy eigenstates |N0,NS, F, Fz; λ〉, it isuseful to express them in terms of some building blocks. In fact, we can express them in terms ofone-, two-, and three-boson creation operators. Let A(n)

f† be the operator such that when applied

to the vacuum state, it creates n bosons that have a total spin F = f and magnetic quantumnumber Fz = f . While the choice of a complete set of operators for the building blocks is notunique, we choose the following five operators for constructing the eigenstates:

A(1)†2 = a2

† (425)

A(2)†0 =

1√

10[(a†0)2 − 2a†1a†−1 + 2a†2a†−2] =

√25S+ (426)

A(2)†2 =

1√

14[2√

2a†2a†0 −√

3(a†1)2] (427)

A(3)†0 =

1√

210[√

2(a†0)3 − 3√

2a1 † a†0a†−1 + 3√

3(a†1)2a†−2

+3√

3a†2(a†−1)2 − 6√

2a†2a†0a†−2] (428)

A(3)†3 =

1√

20[(a†1)3 −

√6a†2a†1a†0 + 2(a†2)2a†−1]. (429)

Note that A(2)†1 and A(3)†

1 vanish identically because of the Bose symmetry, and that the operatorsA(n)†

f commute with F+.We next consider a set B of unnormalized states:

|n12, n20, n22, n30, n33〉 ≡ (a†2)n12 (A(2)†0 )n20 (A(2)†

2 )n22 (A(3)†0 )n30 (A(3)†

3 )n33 |vac〉,(430)

with n12, n20, n22, n30 = 0, 1, 2, . . . ,∞ and n33 = 0, 1. The total number of bosons and the totalspin of the state (430) are N = n12 + 2(n20 + n22)+ 3(n30 + n33) and F = Fz = 2(n12 + n22)+ 3n33,respectively. For given N and F, n33 is uniquely determined by the parity of F:

n33 = F mod 2. (431)

It can be shown [5] that B forms a nonorthogonal complete basis set of the subspace H(Fz=F) inwhich the magnetic quantum number Fz is equal to the total spin F. It can also be shown [5] thatthe energy eigenstates can be represented as

(F−)∆F(A(2)†0 )n20 P(NS=0)(a

†2)n12 (A(2)†

2 )n22 (A(3)†0 )n30 (A(3)†

3 )n33 |vac〉, (432)

101

where n12, n20, n22, n30 = 0, 1, 2, . . . ,∞, n33 = 0, 1, and ∆F = 0, 1, . . . , 2F. These parameters arerelated to N0,NS, F, Fz as

N0 = n12 + 2n22 + 3n30 + 3n33, (433)NS = n20, (434)F = 2n12 + 2n22 + 3n33, (435)

Fz = F − ∆F, (436)

and the corresponding eigenenergy is given by Eq. (424). Note that the states defined in (432)are unnormalized, and the states having the same energy (i.e., those belonging to the same set ofparameter values N0,NS, F, Fz) are nonorthogonal.

It might be tempting to envisage a physical picture that the system, as in the case of 4He, tobe made up of nn f composite bosons whose creation operator is given by A(n)†

f . However, this is

an oversimplification because the operator A(n)f does not obey the boson commutation relation.

Moreover, the projection operator P(NS=0) in (432) imposes many-body spin correlations suchthat the spin correlation between any two bosons must have a vanishing spin-singlet component.Note that two bosons with independently fluctuating spins generally have a nonzero overlap withthe spin-singlet state. The many-body spin correlations of the energy eigenstates are thus farmore complicated than what a naive picture of composite bosons suggests. On the other hand, aslong as quantities such as the number of bosons, magnetization, and energy are concerned, theabove simplified picture is quite helpful. As an illustration, we provide an explanation for theexistence of forbidden values for the total spin F. For example, to construct a state with F = 0or F = 3, composite particles with total spin 2 must be avoided, namely, n12 = n22 = 0. Then,we have N0 = 3(n30 + n33), implying that F = 0 or F = 3 is only possible when N0 = 3k(k ∈ Z).For a state with F = 2 or F = 5, we have n12 + n22 = 1 and N0 = 1+ n22 + 3(n30 + n33), implyingthat N0 , 3k(k ∈ Z). The above simplified picture is also helpful when we consider the magneticresponse discussed below.

8.2.2. Magnetic responseHere, we consider how the ground state and the magnetization Fz respond to the applied

magnetic field p. From Eq. (424), we see that the minimum energy states always satisfy Fz = Fwhen p > 0. The problem thus reduces to minimizing the function

E(Fz,Ns) =c1

2Veff [Fz(Fz + 1) − 6N] +c2

5Veff NS(2N − 2NS + 3) − pFz. (437)

Here, we only consider the case of c2 < 0. Detailed investigations of this problem can befound in Ref. [5]. In this case, it is convenient to introduce a new parameter

c′1 ≡ c1 −c2

20, (438)

and consider the energy as a function of Fz and l ≡ 2N0 − Fz:

E(Fz, l) =c′1

2Veff

[Fz −

Veff

c′1

(p +

c2

8Veff

)+

12

]2

− c2

40Veff l(l + 2F + 6) + const. (439)102

While the averaged slope∆Fz/∆p ∼ Veff/c′1 coincides with that in MFT, the offset term c2/(8Veff)in Eq. (439) makes a qualitative distinction from MFT, that is, the onset of the magnetizationdisplaces from p = 0 to p = |c2|/(8Veff). Note that the slope Veff/c′1 and the offset |c2|/(8Veff)are determined by independent parameters. A typical behavior of the magnetic response when|c2| c′1 is shown in Fig. 34.

Figure 34: Typical dependence of the ground-state magnetization on the applied magnetic-field strength, for c2 < 0 andc′1 > 0.

We now calculate the Zeeman-level populations of the ground states for c2 < 0. In MFT, thelowest-energy states for c2 < 0 have vanishing population in the m = 0,±1 levels. In contrast,the exact ground states derived in the preceding subsection, (A(2)†

0 )NS (a†2)n12 (A(2)†2 )n22 (A(3)†

3 )n33 |vac〉with n22 = 0, 1 and n33 = 0, 1, have nonzero populations in the m = 0,±1 levels. The exact formsfor the averaged population 〈a†mam〉 are calculated as follows. The above ground states have theform (A(2)†

0 )NS |φ〉 ∝ (S+)NS |φ〉, with |φ〉 being a state with a fixed number (s ≡ n12 + 2n22 + 3n33)of bosons satisfying S−|φ〉 = 0. The average Zeeman population for the ground states,

〈a†mam〉 ≡〈φ|(S−)NS a†mam(S+)NS |φ〉〈φ|(S−)NS (S+)NS |φ〉

, (440)

is then simply related to the average Zeeman populations for the state |φ〉 as

〈a†mam〉 = 〈a†mam〉0 +NS

s + 5/2(〈a†mam〉0 + 〈a†−ma−m〉0 + 1), (441)

where 〈a†mam〉0 ≡ 〈φ|a†mam|φ〉/〈φ|φ〉. This formula implies that when NS s, the Zeeman popu-lations of the ground states are sensitive to the form of |φ〉.

With this formula, it is a straightforward task to calculate the average Zeeman-level popula-tions for four types of ground states,

(A(2)†0 )NS (a†2)n12 (A(2)†

2 )n22 (A(3)†3 )n33 |vac〉 (442)

103

with n22 = 0, 1 and n33 = 0, 1. A striking feature appears in the leading terms under the condition1 n12 NS. The results are summarized as

〈a†1a1〉 ∼ 〈a†−1a−1〉 ∼ NS(1 + n33)/n12 (443)

and〈a†0a0〉 ∼ NS(1 + 2n22)/n12. (444)

These results show that the population of each magnetic sublevel depends very sensitively on thespin correlations. As a consequence, a minor change in the magnetization might lead to a majorchange in the population. Such dramatic changes originate from bosonic stimulations caused bythe term a†2a†0 in A(2)†

2 or the term (a†2)2a†−1 in A(3)†3 .

8.2.3. Symmetry considerations on possible phasesThe possible ground states of spinor BECs can be inferred based on symmetry arguments

without solving the GPEs. The spin-2 BEC has five internal degrees of freedom correspondingto magnetic quantum numbers m = 2, 1, 0,−1,−2. Let am’s be their amplitudes. Under rotationof the coordinate system, they must transform so as to guarantee that the linear combination

ψ =

2∑m=−2

amYm2 (θ, φ) (445)

is scalar, where Ym2 (θ, φ) is the spherical harmonic function of rank 2 with θ and φ being the polar

and azimuthal angles, respectively, in the spherical coordinate system. Let

k = (kx, ky, kz) = (sin θ cos φ, sin θ sin φ, cos θ) (446)

be the unit vector in the (θ, φ) direction. In terms of the unit vector k, we can rewrite Eq. (445)as

ψ =

√158π

kT Mk, (447)

where superscript T denotes the transpose, and

M =12

a2 + a−2 −

√23 a0 i(a2 − a−2) −a1 + a−1

i(a2 − a−2) −a2 − a−2 −√

23 a0 −i(a1 + a−1)

−a1 + a−1 −i(a1 + a−1) 2√

23 a0

≡ 1√

3

ψxx ψxy ψxz

ψxy ψyy ψyz

ψxz ψyz ψzz

. (448)

The order parameter is thus characterized by a 3 × 3 traceless symmetric matrix TrM = 0 withunit normalization

Tr(M∗M) = 1. (449)

104

Let us now consider the eigenvalues of this order parameter matrix:

det(E1 −

√3M

)= E3 − 3S E +

A

3√

3= 0, (450)

where√

3 is added in front of the matrix for the sake of convenience, and S and T are theamplitudes of the spin-singlet pair and spin-singlet trio, respectively:

S =12

∑i, j

ψ2i j =

12ψ2

0 − ψ1ψ−1 + ψ2ψ−2, (451)

T =∑i, j,k

εi jkψxiψy jψzk

= −9(ψ2ψ2−1 + ψ

21ψ−2) +

√6ψ0(−ψ2

0 + 3ψ1ψ−1 + 6ψ2ψ−2). (452)

Substituting S = xy and A = 2√

3(x3 + y3), Eq. (450) reduces to

E3 + x3 + y3 − 3Exy = 0, (453)

and hence, the solutions are

E = −(x + y),−(ωx + ω2y),−(ω2x + ωy), (454)

where ω = e2πi/3 and

x =Sy=

1

4816

(A +√

A2 − 48S 3) 1

3. (455)

We can infer from this that the spin-singlet pair and the spin-singlet trio of atoms form buildingblocks of the ground state of the spin-2 BEC. It is noteworthy that this fact arises solely from thefact that the order parameter matrix of the spin-2 BEC is traceless symmetric.

9. Special Topics

9.1. Quenched BEC: Kibble-Zurek mechanismTopological excitations can be spontaneously created when a system undergoes a phase tran-

sition in a finite time. The basic concept is that after the phase transition domains of the newphase emerge at causally disconnected places, and therefore, the order parameters of the newphase at different locations are not correlated. When the order parameters grow to overlap, theymay be able to adjust dynamically so that they are connected smoothly; if they cannot, singu-larities should develop in the order parameter space, giving rise to topological excitations. Sucha scenario of topological defect formation was first discussed by Kibble [134] in the context ofcosmic-string and monopole formations in the early Universe, and an experimental test for theKibble scenario in condensed matter systems was proposed by Zurek [135] (see Ref. [136] for areview).

Bose-Einstein condensates of dilute atomic vapor offer an ideal opportunity for studying theKibble-Zurek mechanism because the temperature, strength of interaction, and external param-eters such as magnetic field and trapping potentials can be changed in a time shorter than thecharacteristic time scale of topological defect formation. In fact, scalar vortices [137, 138] and

105

spin vortices [100, 25] have been observed to emerge spontaneously upon both temperature andmagnetic-field quenches.

Topological defect formation via the Kibble-Zurek mechanism can occur in a ferromagneticphase of a spinor BEC, where the phase transition is triggered by a sudden change in a magneticfield (magnetic field quenching) [100, 139, 140, 141]. The ground-state phase of a spin-1 ferro-magnetic BEC is shown in Fig. 1, where p and q are the coefficients of the linear and quadraticZeeman effects, respectively. When we consider a type of magnetic-field quench for which thetotal magnetization of the system is conserved, the quench occurs along a constant-p line of thephase diagram. As in the experiment described in Ref. [100], we consider the case of p = 0 thatimplies that the total magnetization of the system is kept to be zero. Then, the state of the systemcan be in one of the following two phases.

When q is larger than the critical value qc = 2|c1|n, with n being the number density, theground state is the polar phase ψpolar = (0,

√n, 0) which is nonmagnetic. When q < qc, the

ground state is the broken-axisymmetry (BA) phase in which the system develops transversemagnetization. The ground state for q < qc is the broken-antisymmetry (BA) phase with theorder parameter given by

ψBA =√

n

e−iα

√1 − q/2√

(1 + q)/2eiα

√1 − q/2

, (456)

where q ≡ q/qc. The BA phase has transverse magnetization F+ ≡ Fx + iFy = eiα√

1 − q2

which breaks the U(1) symmetry corresponding to the direction of the transverse magnetization.By rapidly quenching the magnetic field from above to below qc, local transverse magnetizationdevelops in random directions, and thus, spin vortices are spontaneously created.

To analyze such a situation, let us assume that the initial state is the polar phase (0,√

n, 0)and expand the field operator as

ψm =1√Ω

∑k

(√nδm,0 + ak,m

)ei(k·r− 1

~ c0nt). (457)

Substituting this into the Hamiltonian and keeping the terms up to the second order in ak,m withk , 0, we obtain the Bogoliubov Hamiltonian for the polar phase. We are interested in how themodes m = ±1 grow with time. The Heisenberg equations of motion for the modes are [139]

i~ddt

ak,±1 = (εk + q + c1n)ak,±1 + c1na−k,∓1, (458)

with the solutions given by

ak,±1(t) =

(cos

Ekt~− i

εk + q + c1nEk

sinEkt~

)ak,±1(0)

−ic1nEk

sinEkt~

a†−k,∓1(0), (459)

where

Ek =

√(~2k2

2M+ q

) (~2k2

2M+ q + 2c1n

)(460)

106

gives the Bogoliubov spectrum. If Ek has an imaginary part, the corresponding mode is dynam-ically unstable and it grows exponentially. Because c1 < 0 and q > 0 for spin-1 87Rb atoms,exponential growth occurs for q < qc, in agreement with the phase boundary.

9.1.1. Instantaneous quenchWe can use the solution (459) to investigate how the transverse magnetization develops with

time. Because ψ0 '√

n, the magnetization operator F+ = F†− = Fx + iFy is written as

F+ =√

2n(ψ†1(r) + ψ−1(r)). (461)

It follows that the correlation function of the transverse magnetization is given by

〈F+(r, t)F−(r′, t)〉 ' n2Ω

∑k<kc

qc

qc − q − εke

2~ |Ek |t+ik·(r−r′). (462)

The exponential terms describe dynamical instabilities of which the dominant term is the one forwhich the imaginary part of Ek is maximal. Let kmu be the “most unstable” wave number forwhich |ImEk| is maximal. It follows from Eq. (460) that kmu = 0 for qc/2 < q < qc and

kmu = ±√

2M~2

(qc

2− q

)(463)

for q < qc/2. Thus, for q < qc/2, the typical size of magnetic domains is ∼ k−1mu; for qc/2 < q <

qc, it is determined by the width of the distribution of ImEk. In any case, the size of the spindomains grows as the final value of q increases, as experimentally observed in Ref. [3]. The timescale τ for the magnetization is given by

τ =~

2|Ekmu |, (464)

which is ~/√

4q(qc − q) for qc/2 < q < qc and ~/qc for q < qc/2.

9.1.2. Finite-time quenchNext, we consider the case in which the magnetic field is quenched linearly in a finite time

τQ:

q(t) = qc

(1 − t

τQ

). (465)

In this case, the Bogoliubov excitation energy (460) also varies with time and the spin correlationfunction is estimated to be

〈F+(r, t)F−(r′, t)〉 ∝∫

dk exp[

2~

∫ t

0|Ek(t′)|t′dt′ + ik · (r − r′)

]∝ exp

[f (t) − |r − r′|2/ξ2

Q

], (466)

where ξQ is the correlation length and f (t) determines the growth rate of magnetization. In twodimensions, we have [139]

ξQ =

[16~2

M2 t(τQ − t)] 1

4

(467)

107

and

f (t) =τQqc

2~

tan−1√

tτQ − t

−

√tτQ

(1 − t

τQ

) (1 − 2t

τQ

) . (468)

The time scale for magnetization is determined by f (t) = 1. Solving this equation by assumingthat t τQ, we obtain

tQ ∼(~2τQ

q2c

) 13

. (469)

Substituting this into Eq. (467), we obtain

ξQ ∼(

~4

M3qc

) 16

τ13Q. (470)

The same power law was obtained in Ref. [140].

9.1.3. Numerical simulationsThe topological defect formation caused by a quantum phase transition can be best illustrated

by numerical simulations. Figure 35 shows the time evolution following an instantaneous quenchof the magnetic field from q > qc to q = 0 (Fig. 35(a)) and q = qc/2. Here, we assume that thesystem is a two-dimensional disk with a hard wall at radius 100 µm and the potential is assumedto be flat inside the wall. The initial state is considered to be a stationary state of the GP equationsand the following quench dynamics is obtained by using the time-dependent GP equations. Totrigger the dynamical instabilities that cause defect formation, we assume small fluctuations inthe initial amplitudes of the m = ±1 components according to

ak,±1(0) = α + iβ, (471)

where α and β are random variables whose amplitudes x follow the normal distribution p(x) =√2/π exp(−2x2).

Figure 35 shows snapshots of the amplitude |F+(r)| and the phase argF+(r) of the transversemagnetization after the quench. We observe that many holes emerge spontaneously after 100 ms.These holes represent topological excitations called polar-core vortices in which the m = ±1components have vortices with the cores filled by the m = 0 components.

By counting the number of vortices in the final state, we can determine the number of spinvortices. Alternatively, we can calculate the same number—the spin winding number w—fromthe algebraic relation

w =1

2π

∮C(R)

F−∇F+ − F+∇F−2i|F+|2

· d` = 12π

∮∇α · d`, (472)

where C(R) is a circle of radius R and w is a count of the number of rotations of the spin vectorin the xy plane along C(R). Figure 36 shows the R dependence of the ensemble average of w2(R).We note that 〈w2(R)〉avg is proportional to R for large R in agreement with the KZ theory, whereasit is proportional to R2 for small R.

108

0

1

0

(b) q = q / 2

(a) q = 0

c

t = 50 ms t = 100 mst = 75 ms t = 200 ms

t = 50 ms t = 100 mst = 75 ms t = 200 ms

t = 0 ms

t = 0 ms

|F+|

arg|F+|

|F+|

arg|F+|

Figure 35: Spontaneous magnetization following the quench from q > qc to q < qc. Shown are time developments ofthe magnetization |F+ | (upper) and its direction arg |F+ | (lower) for (a) q = 0 and (b) q = qc/2. The black circles in (b)indicate a topological defect. Reprinted from Ref. [139].

If spin vortices can be generated randomly, 〈w2(R)〉avg should be proportional to the area ofthe system and hence to R2. The KZ scaling law ∝ R that we observe for large R arises fromthe fact that the number of vortices matches that of anti-vortices inside the disk due to spinconservation. In other words, for small q, magnetic domains are aligned in such a manner as tocancel the local spin when averaged over the spin correlation length. Thus, over a greater lengthscale, the magnetic domains can be created independently. This is the underlying physics thatmakes the spin conservation compatible with the KZ postulate of independent defect creation ata long distance and the KZ scaling law valid for 2D systems.

9.2. Low-dimensional systems

It had been conjectured from the work of Peierls and Landau in the 1930s [142, 143] thattwo-dimensional systems with continuous order parameters could not have a conventional long-range order, and this was confirmed by Mermin and Wagner [144] and by Hohenberg [145].Long wavelength fluctuations are more important in one and two dimensions than in three di-mensions and deviations from the mean-field theory are much greater. However, there still existsa type of quasi-long-range order in the low-temperature phase in two-dimensional systems thatis characterized by a power-law decay of the order parameter correlation function. Such a phase

109

0.01

0.1

1

10

1 10 100

w2

avg

R [µm]

q=0

q=qc/2

slope = 1

slope

= 2

Figure 36: R dependence of the variance of the spin winding number w(R) for instantaneous quench of the magneticfield to q = 0 and to q = qc/2. The dashed and dotted lines are respectively proportional to R and R2. Reprinted fromRef. [139].

transition is known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. In this subsection,we first explain the BKT transition for a single-component Bose gas, and then, we extend it to aspinor gas.

9.2.1. Berezinskii-Kosterlitz-Thouless transition in a single-component Bose gasWe start from a Ginzburg-Landau energy functional

H[ψ] =∫

d2r[ J2|∇ψ|2 + α

2|ψ|2 + β

4|ψ|4

], (473)

which gives an equilibrium probability for ψ as Peq[ψ] ∝ exp[− HkBT ]. Here, β > 0 and α changes

its sign from positive to negative as the temperature decreases from above to below the criticaltemperature Tc, so that the equilibrium value of ψ changes from ψ = 0 at T > Tc to |ψ| = ψ0 ≡√|α|/β at T < Tc.

In three dimensions, the one-body correlation function 〈ψ∗(r)ψ(0)〉 remains finite at largedistance below Tc, exhibiting a long-range order. In constrast, above Tc, the correlation functiondecays exponentially:

〈ψ∗(r)ψ(0)〉 ∝ e−r/ξ+

rfor r → ∞, (474)

where ξ+ =√

J/|α| ∝ (T − Tc)−1/2 except in the immediate vicinity of Tc (see e.g. Ref. [146]).In this case, the order is short-ranged.

In two dimensions, the effect of fluctuations is much more drastic. We first assume that|ψ| takes its equilibrium value everywhere and consider only phase fluctuations. Substitutingψ(r) = ψ0eiφ(r) in Eq. (473), we have

H =12

∫K0|∇φ|2d2r =

12

K0

∑k

k2φ∗kφk, (475)

110

where K0 ≡ J0|ψ0|2 and φk is the Fourier component of φ(r) satisfying φ−k = φ∗k. Because the

gradient of φ induces a superfluid current as vs = (~/M)∇φ, the coefficient K0 is related to thesuperfluid density ρ0

s as

ρ0s =

M2K0

~2 , (476)

so that H is rewritten as

H =12

∫d2r ρ0

s |vs|2. (477)

Because Eq. (477) is quadratic in φk, we can calculate the correlation function as

〈ψ∗(r)ψ(0)〉 = |ψ0|2〈e−iφ(r)−φ(0)〉 (478)

= |ψ0|2 exp

−∑k

kBTK0k2 1 − cos(k · r)

. (479)

In a two-dimensional system, the summation of k is replaced by an integral as∑k

kBTK0k2 1 − cos(k · r) = 1

(2π)2

kBTK0

∫dkk

∫ 2π

0dθ[1 − cos(kr cos θ)] (480)

=kBT2πK0

∫dk

1 − J0(kr)k

, (481)

where J0(z) is the zeroth Bessel function. For k ≥ 1/r, we can ignore the rapidly oscillatingfunction J0(kr), whereas the contribution from k < 1/r is small. We also introduce a cutoffwavelength ξ corresponding to the coherence length, and neglect spatial structures smaller thanξ. Then, the correlation function is calculated as

〈ψ∗(r)ψ(0)〉 = |ψ0|2 exp

− kBT2πK0

∫ ξ−1

r−1

dkk

∝ (rξ

)−η, (482)

where

η =kBT2πK0

. (483)

Thus, phase fluctuations destroy the true long-range order; however, the quasi long-range ordercharacterized by a power-law behavior remains.

Next, we consider the effect of fluctuations on the amplitude |ψ|. A crucial effect arises when|ψ| passes through zero. If |ψ(r)| is finite everywhere in a singly connected region, the phase φ ofthe order parameter is defined as a single-valued function. On the other hand, if ψ = 0 at r = ri,φ can be a multi-valued function. The points ri correspond to vortices around which φ variesby an integral multiple of 2π.

The phase transition for the appearance of a quasi-long-range order can be captured by asimple discussion of vortex nucleation. When a single vortex passes between two distinct points,the relative phase between these two points changes by ±2π, implying that free vortices destroythe phase coherence. Hence, the quasi-long-range order is destroyed when free vortices arethermally excited.

111

The free energy for a single vortex can be calculated as follows. The circulation of thesuperfluid velocity around the vortex is quantized as

∮vs · d` = h/M. Assuming axisymmetry

around the vortex, we have vs = ~eϕ/(Mr), where (r, ϕ) is the polar coordinate around the vortexand eϕ is a unit vector in the direction of ϕ. Then, the energy cost required to create a singlevortex is given by

Ev = Ec +12

∫ R

ξ

rdr∫ 2π

0dϕ ρ0

s |vs|2 (484)

= Ec + πK0 ln(

Rξ

), (485)

where R is the radius of the system; ξ, the radius of the vortex core; and Ec, the vortex coreenergy arising from the region of r < ξ. On the other hand, because the vortex can freely movein the area of R2, the entropy is given by S = kB ln(R2/ξ2). Then, we obtain the free energy of afree vortex as

F = Ev − TS = (πK0 − 2kBT ) ln(

Rξ

)+ Ec. (486)

One can immediately see from this free energy form that the phase transition occurs at

Tc0 =πK0

2kB, (487)

above which free vortices are generated and destroy the phase coherence. From Eqs. (483) and(487), the exponent η must satisfy η ≤ 1/4.

A more careful argument requires the interaction between vortices to be considered. If thereexists a pair of a vortex (charge 1) and an anti-vortex (charge −1) at distance d, the velocity fieldsgenerated by these vortices at a distance larger than d cancel each other out. As a result, thevortex pair has a finite energy of

Epair(d) = 2Ec + 2πK0 lndξ, (488)

which indicates that a vortex and an anti-vortex attract each other and form a pair at low temper-ature. Because vortex pairs do not destroy phase coherence (see Fig. 37), the superfluid phasetransition can be interpreted as a dissociation of vortex pairs.

If there exist many vortex pairs, the attractive interaction between a vortex and an anti-vortexis weakened by the screening effect. Kosterlitz and Thouless [147] analyzed this phase transitionby mapping the system to a two-dimensional Coulomb gas of unit charge q =

√πK0. They

introduced a “dielectric constant” ε that renormalizes K0 to K = K0/ε, and constructed therenormalization-group flow equations. Their main result is that the superfluid density jumps atthe transition temperature that satisfies the universal relation:

K(Tc)kBTc

=~2ρs(Tc)M2kBTc

=2π. (489)

This universal behavior in superfluid helium was observed by Bishop and Reppy [148]. Withregard to cold atom systems, the BKT transition of a single-component Bose gas was observedin Ref. [149]

112

(a) (b)

Figure 37: Local phase of the order parameter around (a) a single vortex and (b) a vortex pair, where the direction ofeach arrow indicates a phase that varies between 0 and 2π. The phase around a single vortex varies, causing a velocityfield proportional to 1/r, whereas that away from the vortex pair can be uniform.

9.2.2. 2D spinor gasesWe now consider the case of a spinor. As discussed above, vortices play a crucial role in the

superfluid phase transition in two-dimensional systems. When a system acquires spin degrees offreedom, the BKT phase transition becomes highly nontrivial because the nature of the vorticesdepends on the order parameter manifold, as discussed in Sec. 7. It is well known that thoughthe XY spin model exhibits a BKT phase transition where the order parameter manifold U(1)is the same as that of a single-component Bose gas, the BKT transition does not occur for theHeisenberg spin model at finite temperature [147]. This is because the order parameter manifoldfor the Hisenberg model, which is isomorphic to S 2, does not accommodate a topologicallystable vortex because π1(S 2) = 0. As discussed in Sec. 7, the order parameter manifold for thepolar phase of a spin-1 BEC can accommodate half-quantum vortices, and therefore, the BKTtransition occurs due to the binding of half-quantum vortex pairs. On the other hand, if thecondensate accommodates several types of vortices, the BKT transition is expected to occur inseveral stages. Here, we discuss two BKT transitions of ferromagnetic spinor gases that occurunder an external magnetic field.

For simplicity, we neglect the trapping potential and assume an infinite system. The energyfunctional for a spin-1 spinor gas under an external magnetic field B in the z-direction is givenby

H =∫

d2r

~2

2M

∑m=0,±1

|∇ψm|2 + c0n2 + c1|F|2 + q(|ψ1|2 + |ψ−1|2

) , (490)

where n =∑

m |ψm|2 and F =∑

mn ψ∗m(f)mnψn are the number density and spin density, re-

spectively, and q is the quadratic Zeeman energy proportional to B2. For spin-1 BECs, q =(µBB)2/(4∆hf). The linear Zeeman term simply causes Larmor precession due to the spin con-servation, and therefore, it is omitted by going onto a rotating frame of reference in the spinspace.

We consider the ferromagnetic interaction, c1 < 0. At zero temperature, the ground statephase is the same as that we have obtained for a three-dimensional system in Sec. 3. Here,we assume that the total longitudinal magnetization is conserved to be zero. Then, the order

113

parameter for q > qc ≡ 2|c1|n0 is the polar phase whose order parameter is given by

ψpolar =√

n0eiφ

010 , (491)

whereas for q < qc, the broken-axisymmetry phase appears:

ψBA =

√n0eiφ

2

e−iα

√1 − q√

2(1 + q)eiα

√1 − q

, (492)

where n0 is the absolute square of the order parameter at zero temperature and q ≡ q/qc.For q > qc, there exists only one type of vortex; a conventional phase vortex. Therefore, the

BKT transition for q > qc is the same as that in single-component gases. Unbinding of phasevortex pairs destroys the coherence of the U(1) phase that appears in the correlation function ofthe m = 0 component:

〈ψ∗0(r)ψ0(0)〉 ∝ r−η, (493)

where η is given by Eq. (483) with renormalized K instead of K0 = ~2n0/M. The universal jumpis given by Eq. (489).

On the other hand, for q < qc, there exist two types of vortices, i.e., a phase vortex anda spin vortex. The phase vortex is the same as that in the polar state, around which the U(1)phase φ changes by integer multiples of 2π. The spin vortex is defined as a vortex around whichthe direction of transverse magnetization (i.e., α) varies by integer multiple of 2π. Here, thetransverse magnetization appears at q < qc and it is given by

F+ = Fx + iFy =√

2(ψ∗1ψ0 + ψ

∗0ψ−1

)= n0

√1 − q2eiα, (494)

Fz = |ψ1|2 − |ψ−1|2 = 0. (495)

The effects of these vortices are independent because the energy for these fluctuations are decou-pled as

H =12

∫d2r

[Kp

0 |∇φ|2 + Ksp

0 |∇α|2], (496)

where

Kp0 ≡

~2n0

M, (497)

Ksp0 ≡

~2n0(1 − q)2M

. (498)

Similar to the discussion for Eq. (486), a rough estimation for the transition temperature for theappearance of a free phase vortex and a free spin vortex are given by

T pc0 =

πKp0

2kB=π~2n0

2MkB, (499)

T spc0 =

πKsp0

2kB=π~2n0(1 − q)

4MkB=

1 − q2

T pc0, (500)

114

respectively. Hence, as the temperature decreases, phase vortex pairs first bind and then spin vor-tices form pairs at lower temperature. More proper treatment following the Kosterlitz-Thouless’srenormalization method gives the relation between the universal jump and the transition temper-ature:

Kp(T pc )

kBT pc=

Ksp(T spc )

kBT spc=

2π. (501)

The phase transitions related to the binding of the phase and spin vortices can be observedas an appearance of the quasi-long-range order in the correlation function of m = 0 and m = ±1components, respectively, which are given by

〈ψ∗0(r)ψ0(0)〉 ∝ 〈e−i(φ(r)−φ(0))〉 ∝ r−ηp , (502)

〈ψ∗±1(r)ψ±1(0)〉 ∝ 〈e−i(φ(r)−φ(0))〉〈e∓i(α(r)−α(0))〉 ∝ r−ηp−ηsp , (503)

with

ηp =kBT

2πKp , ηsp =kBT

2πKsp . (504)

The binding of spin vortices also contributes to the ferromagnetic long-range order

〈F+(r)F−(0)〉 ∝ 〈e−iα(r)−α(0)〉 ∝ r−ηsp . (505)

Figure 38 summarizes the above results. Depending on the temperature and the externalmagnetic field, there exist three ordered phases: (I) quasi-long-range order arises only in them = 0 component, where the gas has no local magnetization; (II) a quasi-long-range order arisesonly in the m = 0 component, where the gas is locally magnetized, but with no long-rangeorder of magnetization; (III) a quasi-long-range order appears in every m component, where thetransverse magnetization also exhibits the quasi-long-range order.

Here, we discuss the phase diagram at low magnetic fields. In the above system, spin vorticesare stable because the quadratic Zeeman effect suppresses the fluctuation of longitudinal magne-tization. As the external field decreases, however, the contribution of such fluctuations becomeslarger and induces coupling between the spin and the phase vortices. Then, both transition tem-peratures are expected to be suppressed. At zero magnetic field, the quasi-long-range order doesnot appear at finite temperature as in the case of the Hisenberg spin system. Understanding thedetailed behavior in the low-magnetic-field region requires further investigation.

10. Summary and Future Prospects

In the present paper, we have reviewed the basic knowledge concerning spinor Bose-Einsteincondensates (BECs) that has been accumulated thus far. The fundamental characteristics ofspinor BECs are rotational invariance, coupling between the spin and the gauge degrees of free-dom, and magnetism arising from the magnetic moment of the spin.

The rotational invariance and gauge invariance alone uniquely determine the microscopicHamiltonian of the spinor BEC, as discussed in Sec. 2. The mean-field theory described inSec. 3 discribes the ground-state phases and the dynamics of spinor BECs. In a scalar BEC,the measurements of the collective modes are found to agree with the theoretical values withan accuracy better than 1% (see Ref. [150] for a comprehensive review). In the near future,

115

T

q

normal

qc

Tcphase

Tcspin

III

III

Figure 38: Schematic phase diagram of a 2D ferromagnetic Bose gas. (I) a quasi-long-range order appears only in them = 0 component with no local magnetization; (II) a quasi-long-range order arises only in the m = 0 component, wherethe gas is locally magnetized, but without any long-range order of the magnetization; (III) a quasi-long-range orderappears in every m component, where the transverse magnetization also exhibits a quasi-long-range order. In the shadedregion, the fluctuation of the longitudinal magnetization is expected to destroy the quasi-long-range order.

experimental tasks to be carried out include investigating the extent to which the predictions ofthe Bogoliubov theory described in Sec. 5 can be verified experimentally.

The topological excitations described in Sec. 7 are of cross-disciplinary interest; in particu-lar, the non-Abelian nature of vortices in the nematic and cyclic phases of spin-2 BECs will havefundamental implications on quantum turbulence and other subfields of physics. The topologicalexcitations are usually considered within a given order parameter manifold. However, as exem-plified in Fig. 33 in Sec. 7.3.4, the order-parameter manifold can change dynamically from one toanother, and during a transition, the topological charge is no longer conserved. It is an interestingtheoretical question to investigate whether, if any, a more general topologically invariant quantitycan be defined in such a general situation. The relative homotopy theory described in Sec. 7.2.2may be used as a guiding theoretical framework.

A closely related subject is vortex nucleation dynamics and vortex lattice formation. Vorticesare singularities of the order parameter. In a scalar BEC, the only possible singularity is thedensity hole in the condensate. However, the spinor BEC has several different phases and thechoice of creating a density hole is energetically most costly. Thus, in spinor BECs, the vortexcore is filled by a state that does not belong to the order parameter manifold of the condensate.As discussed in Sec. 6, the spin-1 polar BEC has a ferromagnetic vortex core, and the spin-1ferromagnetic BEC can host a polar-core vortex. For spin-2 and higher-spin BECs, more than twopossible phases can fill the vortex core. Because each phase has its own internal spin structure,an interesting question arises as to how the order parameter of the BEC smoothly connects thespin structure of the vortex core [151].

The many-body quantum correlations of spinor BECs described in Sec. 8 are fragile againstan external magnetic field because the bosonic stimulation favors the mean field. The spinor-dipolar phenomena described in Sec. 4.3 are also vulnerable against a magnetic field because

116

the Larmor precession averages out the part of the dipole interaction that couples the differenttotal spin components. It is noteworthy that both many-body spin correlations and spinor-dipolareffects become significant as the external magnetic field is reduced down to approximately 10 µG.We have good reasons to expect that further symmetry breaking occurs in such an environmentof ultralow magnetic field.

From the viewpoint of fundamental physics, the topological defect formation based on theKibble-Zurek mechanism discussed in Sec. 9.1 merits further experimental and theoretical study.In particular, this effect combined with the spinor Berezinskii-Kosterlitz-Thouless transition intwo dimensions described in Sec. 9.2.1 raises many interesting questions. In fact, the gauge,spin, and dipole-dipole interactions set very different energy scales that, we can conceive, leadto topological phase transitions at different energy and temperature scales.

All the discussions presented in this review are restricted to absolute zero. However, experi-ments on spinor BECs have been carried out at temperatures much higher than the typical energyscale of the spin-exchange interaction. Nevertheless, once the system undergoes Bose-Einsteincondensation, all the condensed particles respond to an external perturbation in exactly the samemanner. When the number of condensate particles is of the order of 105, the collective energyfor the dipole interaction amounts to 1 µK for 87Rb and 100 µK for 52Cr. Therefore, for temper-atures below 0.1 µK, the dynamics of the system can be understood, at least qualitatively, basedon the zero-temperature theory. Nonetheless, it is likely that the precise measurements of collec-tive modes are sensitive to finite-temperature effects. Moreover, when the long-time dynamicsis considered, the thermalization process is expected to play a crucial role in determining themagnetic structure of the system. In fact, very recent experiments on the 87Rb f = 1 BEC showevidence of magnetic ordering [25] as a consequence of thermalization. To fully understandsuch phenomena and to explore new types of temperature-quenched symmetry-breaking phasetransitions, it is obvious that we need to develop theoretical tools to investigate the dynamics ofspinor-dipolar physics at finite temperatures.

Acknowledgements

We acknowledge the fruitful collaborations with Hiroki Saito and Michikazu Kobayashi. MUacknowledges the Aspen Center for Physics, where part of this work was carried out.

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