ArnaudKoetsier

Floris van Liere

Henk Stoof

The imbalanced antiferromagnet in an optical lattice

2

Introduction

• Fermions in an optical lattice• Described by the Hubbard model• Realised experimentally [Esslinger ’05]• Fermionic Mott insulator recently seen [Esslinger ’08, Bloch ’08]• There is currently a race to create the Néel state

• Imbalanced Fermi gases• Experimentally realised [Ketterle ’06, Hulet ’06]• High relevance to other areas of physics (particle physics, neutron stars, etc.)

• Imbalanced Fermi gases in an optical lattice ??

3

Fermi-Hubbard Model

Sums depend on:Filling NDimensionality (d=3)

On-site interaction: U Tunneling: t

Consider nearest-neighbor tunneling only.

The positive-U (repulsive) Fermi-Hubbard Model, relevant to High-Tc SC

H = −tPσ

Phjj0i

c†j,σcj0,σ + UPjc†j,↑c

†j,↓cj,↓cj,↑

4

Quantum Phases of the Fermi-Hubbard ModelFi

lling

Frac

tion

0

0.5

1

Mott Insulator (need large U)

Band Insulator

Conductor

Conductor

Conductor

• Positive U (repulsive on-site interaction):

• Negative U: Pairing occurs — BEC/BCS superfluid at all fillings.

5

• At half filling, when and we are deep in the Mott phase.Hopping is energetically supressedModel simplifies: only spin degrees of freedom remain (no transport)

• Integrate out the hopping fluctuations, then the Hubbard model reduces to the Heisenberg model:

U À t

Mott insulator: Heisenberg Model (no imbalance yet)

kBT ¿ U

J =4t2

U

Szi =1

2

³c†i,↑ci,↑ − c†i,↓ci,↓

´S+i =c

†i,↑ci,↓

S−i =c†i,↓ci,↑

H =J

2

Xhjki

Sj · Sk

Spin ½ operators: S = 12σ

Superexchange constant (virtual hops):

6

Néel State (no imbalance yet)

• The Néel state is the antiferromagnetic ground state for

• Néel order parameter measures amount of “anti-alignment”:

• Below some critical temperature Tc, we enter the Néel state and becomes non-zero.

0 Tc0

0.5

T

⟨n⟩

0 ≤ h|n|i ≤ 0.5

h|n|i

h|n|i

nj = (−1)jhSji

J > 0

7

• Until now, • Now take — spin population imbalance. • This gives rise to an overal magnetization

N↑ 6= N↓

• Add a constraint to the Heisenberg model that enforces

Heisenberg Model with imbalance

Effective magnetic field (Lagrange multiplier):

H =J

2

Xhjki

Sj · Sk −Xi

B · (Si −m)

m = (0, 0,mz)

mz = SN↑ −N↓N↑ +N↓

(fermions: )S = 12

N↑ = N↓ = N/2

hSi =m

B

8

• ground state is antiferromagnetic (Néel state)Two sublattices:

• Linearize Hamiltonian:

• Magnetization:

• Néel order parameter:

• Obtain the on-site free energy subject to the constraint (eliminates )

J > 0⇒

Mean field analysis

A, B

m =hSAi+ hSBi

2

n =hSAi− hSBi

2

SA(B)i = hSA(B)i i+ δS

A(B)i

f(n,m;B)

∇Bf = 0 B

B

A B

A

A

B

AB

B

B

A

A

9

Phase Diagram in three dimensions

0 Tc0

0.5

T

⟨n⟩

Add imbalance0.0

0.2

0.4mz0 0.3 0.6

0.91.2kBT�J

0.0

0.2

0.4n

mz

k B T

/J

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

m

nIsing:

m

n

Canted:

n 6= 0

n = 0

10

Spin waves (magnons)

• Spin dynamics can be found from:

• Imbalance splits the degeneracy:

−π2

π

2kd

h̄ω/Jz

00

0.1

0.2

0.3

0.4

0.5No imbalance: Doubly degenerate antiferromagnetic dispersion

Antiferromagneticmagnons: ω ∝ |k|

ω ∝ k2Ferromagneticmagnons:

dS

dt=i

~[H,S]

Gap:(Larmorprecessionof n)

11

Long-wavelength dynamics: NLσM

• Dynamics are summarised a non-linear sigma model with an action

• The equilibrium value of is found from the Landau free energy:

• NLσM admits spin waves but also topologically stable excitations in the local staggered magnetisation .

S[n(x, t)] =

Zdt

Zdx

dD

½1

4Jzn2

µ~∂n(x, t)

∂t− 2Jzm× n(x, t)

¶2− Jd

2

2[∇n(x, t)]2

¾

F [n(x),m] =

Zdx

dD

½Jd2

2[∇n(x)]2 + f [n(x),m]

¾

• lattice spacing:• number of nearest neighbours:• local staggered magnetization:

d = λ/2z = 2Dn(x, t)

n(x, t)

n(x, t)

12

• The topological excitaitons are vortices; Néel vector has an out-of-easy-plane component in the core

• In two dimensions, these are merons:• Spin texture of a meron:

• Ansatz:

• Merons characterised by:Pontryagin index ±½VorticityCore size λ

Topological excitations

n =

⎛⎝ pn2 − [nz(r)]2 cosφ

nvpn2 − [nz(r)]2 sinφ

nz(r)

⎞⎠nv = 1

nv = −1nv = ±1

nz(r) =n

[(r/λ)2 + 1]2

13

Meron size

• Core size λ of meron found by plugging the spin texture into and minimizing (below Tc):

• The energy of a single meron diverges logarithmically with the system area A at low temperature as

merons must be created in pairs.

0 0.1 0.20.3

0.4mz0.3

0.60.9

1.21.5

kBT�J0

2

4

6Λ

d

F [n(x),m]

Jn2π

2ln

A

πλ2

mz

k B T

/J

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

Meronspresent

Meron core size

14

Meron pairs

Low temperatures:A pair of merons with opposite vorticity, has a finite energy since the deformation of the spin texture cancels at infinity:

Higher temperatures:Entropy contributions overcome the divergent energy of a single meronThe system can lower its free energy through the proliferation of single merons

15

Kosterlitz-Thouless transition

• The unbinding of meron pairs in 2D signals a KT transition. Thisdrives down Tc compared with MFT:

• New Tc obtained by analogy to an anisotropic O(3) model (Monte Carlo results: [Klomfass et al, Nucl. Phys. B360, 264 (1991)] )

mzk B

T/J

0 0.05 0.10

0.02

0.04

0.06

n 6= 0

mz

k B T

/J

0 0.2 0.40

0.2

0.4

0.6

0.8

1

KT transition

MFT in 2D

16

Experimental feasibility

• Experimental realisation:Imbalance: drive spin transitions with RF fieldNéel state in optical lattice: adiabatic cooling [AK et al. PRA77, 023623 (2008)]

• Observation of Néel stateCorrelations in atom shot noiseBragg reflection (also probes spin waves)

• Observation of KT transitionInterference experiment [Hadzibabic et al. Nature 441, 1118 (2006)]In situ imaging [Gericke et al. Nat. Phys. 4, 949 (2008); Würtz et al.arXiv:0903.4837]

17

Conclusion

• Tc calculated for entering an antiferromagnetically ordered state in mean field theory

• Topological excitations give rise to a KT transition in 2D which significantly lowers Tc compared to MFT.

• The imbalanced antiferrromagnet is a rich systemferro- and antiferromagnetic propertiescontains topological excitationsmodels quantum magnetism, bilayers, etc.merons possess an internal Ising degree of freedom associated toPontryagin index — possible application to topological quantum computation

• Future work:include fluctuations beyond MFT for better accuracy in three dimensionsinvestigate topological excitations in 3D (vortex rings)incorporate equilibrium in the NLσMgradient of n gives rise to a magnetization

18

• On-site free energy:

where• Constraint equation:

• Critical temperature:

• Effective magnetic field below the critical temperature:

Results

f(n,m;B) =Jz

2(n2 −m2) +m ·B

− 12kBT ln

∙4 cosh

µ |BA|2kBT

¶cosh

µ |BB |2kBT

¶¸BA (B) = B− Jzm± Jzn

B = 2Jzm

m =1

4

∙BA|BA|

tanh

µ |BA|2kBT

¶+BB|BB |

tanh

µ |BB |2kBT

¶¸

Tc =Jzmz

2kBarctanh(2mz)

19

Anisotropic O(3) model

• Dimensionless free energy of the anisotropic O(3) model [Klomfass et al, Nucl. Phys. B360, 264 (1991)] :

• KT transition:

• Numerical fit:

βf3 = −β3Xhi,ji

Si · Sj + γ3Xi

(Szi )2

γ3(β3) =β3

β3 − 1.06exp[−5.6(β3 − 1.085)].

1.0 1.2 1.4 1.6 1.8 2.00.0

0.2

0.4

0.6

0.8

1.0

b3

g 3êH1+g

3L

20

Analogy with the anisotropic O(3) model

• Landau free energy:

0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

m

gHm,bLê

J

βF =− βJ

2

XhI,ji

ni · nj + βXi

f(m,ni,β)

'− βJn2

2

XhI,ji

Si · Sj + βn2γ(m,β)Xi

(Szi )2

β3 =Jβn2

2

γ3 =2β3Jγ(m,β)

Mapping of our model toAnisotropic O(3) model:

Numerical fit parameter

0.02

1/Jβ =

0.2

0.4

0.60.8

Tc