1

Hidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems

Congjun Wu

Kavli Institute for Theoretical Physics, UCSB

Ref: C. Wu, Mod. Phys. Lett. B 20, 1707, (2006);

C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402(2003);

C. Wu, Phys. Rev. Lett. 95, 266404 (2005);

S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005);

C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.

2

Collaborators

• S. Chen, Institute of Physics, Chinese Academy of Sciences, Beijing.

• J. P. Hu, Purdue.

• S. C. Zhang, Stanford.

• Y. P. Wang, Institute of Physics, Chinese Academy of Sciences, Beijing.

Many thanks to L. Balents, E. Demler, M. P. A. Fisher, E.

Fradkin, T. L. Ho, A. J. Leggett, D. Scalapino, J. Zaanen, and F.Zhou for very helpful discussions.

3

Rapid progress in cold atomic physics

M. Greiner et al., Nature 415, 39 (2002).

• Magnetic traps: spin degrees of freedom are frozen.

• In optical lattices, interaction effects are adjustable. Newopportunity to study strongly correlated high spin systems.

M. Greiner et. al., PRL, 2001.

• Optical traps and lattices: spin degrees of freedom arereleased; a controllable way to study high spin physics.

4

• Spin-1 bosons: 23Na (antiferro), 87Rb (ferromagnetic).

High spin physics with cold atoms

D. M. Stamper-Kurn et al., PRL 80, 2027 (1998); T. L. Ho, PRL 81, 742 (1998);

F. Zhou, PRL 87, 80401 (2001); E. Demler and F. Zhou, PRL 88, 163001 (2002);T. L. Ho and S. Yip, PRL 82, 247 (1999); S. Yip and T. L. Ho, PRA 59, 4653(1999).

• Most atoms have high hyperfine spin multiplets.

• High spin fermions: zero sounds and Cooper pairingstructures.

F= I (nuclear spin) + S (electron spin).

• Different from high spin transition metal compounds.

eee

5

Hidden symmetry: spin-3/2 atomic systems are special!

• Hidden SO(5) symmetry without fine tuning!

Continuum model (s-wave scattering); the lattice-Hubbard model.

Exact symmetry regardless of the dimensionality, lattice

geometry, impurity potentials.

SO(5) in spin 3/2 systems SU(2) in spin systems

• This SO(5) symmetry is qualitatively different from theSO(5) theory of high Tc superconductivity.

• Spin 3/2 atoms: 132Cs, 9Be, 135Ba, 137Ba, 201Hg.

C. Wu, J. P. Hu, and S. C. Zhang, PRL 91, 186402(2003).

6

What is SO(5)/Sp(4) group?

,

• SO(3) /SU(2) group.

.,,312312

LLLLLL yxz ===

3-vector: x, y, z.

3-generator:

2-spinor:

• SO(5) /Sp(4) group.

)51( < baLab

5-vector:

10-generator:

4-spinor:

54321,,,, nnnnn

2

3

2

1

2

1

2

3

21, nn

43, nn

5n

7

Quintet superfluidity and half-quantum vortex

• High symmetries do not occur frequently in nature, sinceevery new symmetry brings with itself a possibility of newphysics, they deserve attention.

---D. Controzzi and A. M. Tsvelik, cond-mat/0510505

• Cooper pairing with Spair=2.

• Half-quantum vortex (non-Abelian Alice string) and quantumentanglement.

C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.

+

8

Multi-particle clustering order

• Feshbach resonances: Cooper pairing superfluidity.

• Driven by logic, it is natural to expect the quartetting orderas a possible focus for the future research.

• Quartetting order in spin 3/2 systems.

r k 1

r k 1

r k 2

r k 2

C. Wu, Phys. Rev. Lett. 95, 266404 (2005).

4-fermion counter-part

of Cooper pairing.

9

Strong quantum fluctuations in magnetic systems

• Intuitively, quantum fluctuations are weak in high spin systems.

From Auerbach’s book:

S

S

Si

S

S

S

S kijk

ji 1],[ =

• However, due to the highSO(5) symmetry, quantumfluctuations here are evenstronger than those in spin systems.

large N (N=4) v.s. large S.

10

Outline

• The proof of the exact SO(5) symmetry.

• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.

• Quartetting v.s pairing orders in 1D spin 3/2 systems.

C. Wu, J. P. Hu, and S. C. Zhang, PRL 91, 186402(2003).

• Quintet superfluids and non-Abelian topological defects.

11

• Not accidental! 1/r Coulomb potential givesrise to a hidden conserved quantity.

• An obvious SO(3) symmetry: (angular momentum) .

The SO(4) symmetry in Hydrogen atoms

• Q: What are the hidden conserved quantities inspin 3/2 systems?

r L

• The energy level degeneracy n2 is mysterious.

H

r A

r L

r A

r L

r A

r A ,

r L .Runge-Lentz vector ; SO(4) generators

12

Generic spin-3/2 Hamiltonian in the continuum

=

++

±±=

+

++

=

5~1

20

22

2/1,2/3

)()(2

)()(2

)()ˆ2

()(

a

aa

d

rrg

rrg

rm

rrdH

rrrr

rhrrμ

• Pauli’s exclusion principle: only Ftot=0, 2

are allowed; Ftot=1, 3 are forbidden.

• The s-wave scattering interactions and spin SU(2) symmetry.

2

3

2

1

2

1

2

3

+(r r ) = 00 | 3

2

3

2; +(

r r ) + (

r r )

a+(

r r ) = 2a | 3

2

3

2; +(

r r ) + (

r r )quintet:

singlet:

13

Spin-3/2 Hubbard model in optical lattices

• The single band Hubbard model is valid away fromresonances.

=

++

++

++

+=

5~1

20

,,,

,

,

)()()()(

.}.{

a

aa

i

i

i

ij

ij

i

iiUiiU

ccchcctH μ

U0.2

E< 0.1,

l0 ~ 400nm, as,0,2 ~ 100aB

(V0Er

)1/ 4 1 ~ 2as: scattering length, Er: recoil energy

E

t

U0,20

V

0l

14

SO(5) symmetry: the single site problem

00=E

μ45205

+= UUE

μ322

5

02

1

4+= UUE

μ203

=UE

μ222

=UE

;

μ=1

E , , , ;

,± , ± + ;

;

, , ;, ,, ,, , ,,

.

5vectorquintetE2

4spinorquartetE1,4

1scalarsingletE0,3,5

degen

eracy

SO(5)SU(2)

)1(2

int = nnU

H

• U0=U2=U, SU(4)

symmetry.

24=16 states.

15

Quintet channel (S=2) operators as SO(5) vectors

=+ )(: 1xy rd

=+ )(: 522 rd

yx+i

=+ )(: 2 rd

xz i

=+ )(: 3 rd yz

+

irdrz

=+ )( : 43 22 +

d

4S

5-polarvector

• Kinetic energy has an obvious SU(4) symmetry; interactionsbreak it down to SO(5) (Sp(4)); SU(4) is restored at U0=U2.

trajectory under SU(2).

16

1,

Fx,Fy,Fz

ijaFiFj (a =1 ~ 5) :

ijka FiFjFk (a =1 ~ 7)

Particle-hole bi-linears (I)

• Total degrees of freedom: 42=16=1+3+5+7.

1 density operator and 3 spin operators are not complete.

rank: 0

1

2

3

• Spin-nematic matrices (rank-2 tensors) form five-

matrices (SO(5) vector) .

)5,1(,2},{, == baFF ab

ba

ji

a

ij

a

+M

Fx2 Fy

2, Fz2 5

4,

{Fx,Fy},{Fy,Fz},{Fz,Fx}M

17

Particle-hole channel algebra (II)

• Both commute with Hamiltonian.10 generators of SO(5): 10=3+7.

7 spin cubic tensors are the hidden conserved quantities.

ab=

i

2[ a , b ] (1 a < b 5)

• SO(5): 1 scalar + 5 vectors + 10 generators = 16

kji

a

ijkzyx FFFF and,,

;

;

;

2

1

2

1

ab

a

ab

a

L

n

n

=

=

=

+

+

+1 density:

5 spin nematic:

3 spins + 7 spincubic tensors:

Time Reversal

even

even

odd

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Outline

• The proof of the exact SO(5) symmetry.

• Quintet superfluids and Half-quantum vortices.

• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.

• Quartetting v.s pairing orders in 1D spin 3/2 systems.

C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.

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g2<0: s-wave quintet (Spair=2) pairing

• BCS theory: polar condensation; order parameter formsan SO(5) vector.

d 4S

a

i

ade ˆ=

+

Ho and Yip, PRL 82, 247 (1999);

Wu, Hu and Zhang, cond-mat/0512602.

5d unit vector

=+ )(: 1xy rd

=+ )(: 522 rd

yx+i

=+ )(: 2 rd

xz i

=+ )(: 3 rd yz

+

irdrz

=+ )( : 43 22 +

20

Superfluid with spin: half-quantum vortex (HQV)

+,ˆˆ dd• Z2 gaugesymmetry

• -disclination of as a HQV.d

• Fundamental group of the manifold.

2

4

1 )( ZRP =

2

44/:ˆ ZSRPd =• is not a rigorous vector, but a

directionless director.

a

i

ade ˆ=

+ remains single-valued.

d

21

Stability of half-quantum vortices

• Single quantum vortex:

L

M

hE sf log

4 2=

})ˆ()({4

222

dM

drE spsf +=h

}

{

log2

log24 2

R

L

L

M

hE

spsf

spsf

+

+=

sfsp <• Stability condition:

• A pair of HQV:

R

22

Example: HQV as Alice string (3He-A phase)

• Example: 3He-A, triplet Cooper pairing.

2

2/:ˆ ZSd

xz

yn

• A particle flips the sign of its spin after it encircles HQV.

A. S. Schwarz et al., Nucl. Phys. B 208, 141(1982);F. A. Bais et al., Nucl. Phys. B 666, 243 (2003);

M. G. Alford, et al. Nucl. Phys. B 384, 251 (1992).P. McGraw, Phys. Rev. D 50, 952 (1994).Configuration space U(1)

dii

ee ,

=0,

,0)ˆ(

i

i

e

enU

23

The HQV pair in 2D or HQV loop in 3D

P. McGraw, Phys. Rev. D, 50, 952 (1994).

xz

yn

zSO(3) SO(2)

• Phase state.

vortzvortiS 0)exp( ==

ie

24

SO(2) Cheshire charge (3He-A)

• For each phase state, SO(2) symmetry is only brokenin a finite region, so it should be dynamically restored.

vortvortimdm )exp(=

Sz m vort= m m

vort

• Cheshire charge state (Sz) v.s phasestate.

vortvortdm == 0

• HQV pair or loop can carry spin quantum number.

z

P. McGraw, Phys. Rev. D, 50, 952 (1994).

25

Spin conservation by exciting Cheshire charge

• Initial state: particle spin up andHQV pair (loop) zero charge.

• Final state: particle spin down andHQV pair (loop) charge 1.

vortpvortpdminit === 0

vortp

vort

i

p

m

edfinal

1==

=

• Both the initial and final states areproduct states. No entanglement! P. McGraw, Phys. Rev. D, 50, 952 (1994).

26

Quintet pairing as a non-Abelian generalization

• HQV configuration space , SU(2) instead of U(1).

4e

n

S4

5,3,2,1e

4ˆ//)0(ˆ ed =

nend ˆsinˆcos)ˆ,(ˆ242

=

vortn

equator: )2(ˆ 3SUSn =

-disclination

27

SU(2) Berry phase

• After a particle moves around HQV, or passes a HQV pair:

=

2

1

,2

1

,2

3

,2

3

++=

+

2351

5123)ˆ(

0

0

inninn

inninnnW

W

W2

1

2

1

2

3

2

3 ,,

)2(3SUS

5533

2211

ˆˆ

ˆˆˆ

enen

enenn

++

+=nr

28

Non-Abelian Cheshire charge

• Construct SO(4) Cheshire chargestate for a HQV pair (loop) throughS3 harmonic functions.

vortTTTTSnvort

nnYndTTTT ˆ)ˆ(ˆ''; 33

3 '';

33 =

.,,)'( 1523,25,1335,12 LLLLLLTT ±±±=rr

4e

• Zero charge state.

vortSnvortnnd ˆˆ00;00

3=

SO(5) SO(4)

29

Entanglement through non-Abelian Cheshire charge!

• SO(4) spin conservation. For simplicity, onlyis shown.

init = 3

2 pzero charge

vort

final = 1

2 pSz =1

vort1

2 pSz = 2

vort

Sz = T3 +3

2T3'

• Generation of entanglement between the particle and HQVloop!

+

30

Outline

• The proof of the exact SO(5) symmetry.

• Alice string in quintet superfluids and non-AbelianCheshire charge.

• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.

• Quartetting and pairing orders in 1D systems.

C. Wu, Phys. Rev. Lett. 95, 266404(2005).

31

Multiple-particle clustering (MPC) instability

• Pauli’s exclusion principle allows MPC. More than two particlesform bound states.

SU(4) singlet:

4-body maximallyentangled states

• Spin 3/2 systems: quartetting order.

)()()()( 2/32/12/12/3 rrrrOqt

++++=

• Difficulty: lack of a BCS type well-controlled mean field theory.

trial wavefunction in 3D: A. S. Stepanenko and J. M. F Gunn, cond-mat/9901317.

baryon (3-quark); alpha particle (2p+2n); bi-exciton (2e+2h)

32

1D systems: strongly correlated but understandable

P. Schlottmann, J. Phys. Cond. Matt 6, 1359(1994).

• Competing orders in 1D spin 3/2 systems with SO(5) symmetry.

• Bethe ansatz results for 1D SU(2N) model:

2N particles form an SU(2N) singlet; Cooper pairing is notpossible because 2 particles can not form an SU(2N) singlet.

Both quartetting and singlet Cooper pairing are allowed.

Transition between quartetting and Cooper pairing.

C. Wu, Phys. Rev. Lett. 95, 266404(2005).

33

Phase diagram at incommensurate fillings

• Gapless charge sector.

• Spin gap phases B and

C: pairing v.s. quartetting.

A: Luttinger liquid

20gg =

SU(4)

B: Quartetting

C: Singlet pairing

0g

2g

20gg =

SU(4)

• Singlet pairing in purely

repulsive regime.

• Ising transition between

B and C.

• Quintet pairing is not

allowed.

34

Phase B: the quartetting phase

• Quartetting superfluidity v.s. CDW of quartets (2kf-CDW).

)2/(2 fkd =

Kc: the Luttinger parameter in the charge channel.

.2at wins

;2at wins

2

2/32/12/12/3

<=

>=

+

++++

cLRk

cqt

KN

KO

f

35

Phase C: the singlet pairing phase

• Singlet pairing superfluidity v.s CDW of pairs (4kf-CDW).

.at wins

at wins

2

1

,4

;2

1

2/12/12/32/3

<=

>=

++

+++++

cLLRRcdwk

c

KO

K

f

)4/(2 fkd =

36

Competition between quartetting and pairing phases

• Phase locking problem in the two-band model.

+= 1

+

2+ ei ccos r;

Oquar = 1+

2+ ei 4 ccos2 r .

1+

= 3 / 2+

3 / 2+

2+

= 1/ 2+

1/ 2+

overall phase; relative phase.

dual field of the relative phase

c r

A. J. Leggett, Prog, Theo. Phys. 36, 901(1966); H. J. Schulz, PRB 53, R2959 (1996).

• No symmetry breaking in the overall phase (charge) channelin 1D.

2..2

1,

2

3

2

1,

2

3 +++

±±±± cc

ieie

r

37

• Ising symmetry: .

Ising transition in the relative phase channel

++++

±±±±2

1

2

1

2

3

2

3 , ii

quartquart OO

,

relative phase: ±rr rr

• Ising ordered phase:

Ising disordered phase:

)2cos2cos(2

1})(){(

2

121

22

rrrxrxeffa

H +++=

• the relative phase is pinned: pairing order;

the dual field is pinned: quartetting order.

21>

21<

Ising transition: two Majorana fermions with masses: 21±

38

Experiment setup and detection

M. Greiner et. al., PRL, 2001.

• Array of 1D optical tubes.

• RF spectroscopy to measurethe excitation gap.

pairbreaking:

quartet breaking:

39

Outline

• The proof of the exact SO(5) symmetry.

• Alice string in quintet superfluids and non-Abelian Cheshirecharge.

• SO(5) (Sp(4)) magnetism in Mott-insulating phases.

• Quartetting v.s pairing orders in 1D spin 3/2 systems.

S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005).

40

Spin exchange (one particle per site)

• Spin exchange: bond singlet (J0), quintet (J2). No exchangein the triplet and septet channels.

• SO(5) or Sp(4) explicitly invariant form:

++

+=

ij

baababex jninJJ

jLiLJJ

H )()(4

3)()(

4

2020

0,/4,/4

)()(

312

2

20

2

0

2200

====

=

JJUtJUtJ

ijQJijQJHij

ex

3

2

3

2= 0+2+1+3

Lab: 3 spins + 7 spin cubic tensors; na: spin nematic operators;Lab and na together form the15 SU(4) generators.

5~1, =ba

• Heisenberg model with bi-linear, bi-quadratic, bi-cubic terms.

41

In a bipartite lattice, a particle-hole transformation on odd sites:

Two different SU(4) symmetries

• A: J0=J2=J, SU(4) point.

Hex = J {Lab (i)Lab ( j) + na (i)nb ( j)}ij

• B: J2=0, the staggered SU’(4) point.

)(')()(')( jnjnjLjL abababab ==

+=ij

aaababex jninjLiLJ

H )}(')()(')({4

0

singlet quintet

triplet septet

J

singlet

quintet triplet septet

J0

*

42

Construction of singlets

• The SU(2) singlet: 2 sites.

• The uniform SU(4) singlet: 4 sites.

• The staggered SU’(4) singlet: 2 sites.

baryon

meson

0)4()3()2()1(!4

++++

)2()1(2

1 ++R

43

Phase diagram in 1D lattice (one particle per site)

)4(20 SUJJ =

spin gapdimer

gapless spin liquid

0J

2J

)4( 'SU

C. Wu, Phys. Rev. Lett. 95, 266404 (2005).

• On the SU’(4) line, dimerized spin gap phase.• On the SU(4) line, gapless spin liquid phase.

44

SU’(4) and SU(4) point at 2D

• J2>0, no conclusive results!

• At SU’(4) point (J2=0), QMCand large N give the Neel order,but the moment is tiny.

Read, and Sachdev, Nucl. Phys. B 316(1989).

K. Harada et. al. PRL 90, 117203, (2003).

Bossche et. al., Eur. Phys. J. B 17, 367 (2000).

SU(4) point (J0=J2), 2DPlaquette order at theSU(4) point?

Exact diagonalization on a 4*4lattice

45

Exact result: SU(4) Majumdar-Ghosh ladder

• Exact dimer ground state inspin 1/2 M-G model.

H = Hi,i+1,i+2i

, Hi,i+1,i+2 =J

2(v S i +

r S i+1 +

r S i+2)

2

iJ

JJ2

1'=

1+i 2+i

• SU(4) M-G: plaquette state.

=cluster site-sixevery

iHH

SU(4) Casimir of the six-site cluster

S. Chen, C. Wu, S. C. Zhang and Y. P. Wang,

Phys. Rev. B 72, 214428 (2005).

• Excitations as fractionalizeddomain walls.

+=sitessix

2

sitessix

2 )()(aabi

nLH

46

SU(4) plaquette state: a four-site problem

• Bond spin singlet:

• Level crossing:

= ++2 1

3 44!

+ + + + 0

• Plaquette SU(4) singlet:

J2

+a ( ) + b

J0

d-wave

s-wave

4-body EPR state;

no bond orders

d-wave to s-wave

• Hint to 2D?

47

Speculations: 2D phase diagram ?

• J2>0, no conclusive results!

• J2=0, Neel order at the SU’(4) point (QMC).

K. Harada et. al. PRL 90, 117203, (2003).

Bossche et. al., Eur. Phys. J. B 17, 367 (2000).

2D Plaquette order at theSU(4) point?

Exact diagonalization on a 4*4lattice

• Phase transitions as J0/J2?Dimer phases? Singlet ormagnetic dimers? J2

SU(4)

?

?

J0 SU’(4)

48

The SU’(4) model: dimensional crossover

• SU’(4) model: 1D dimer order;2D Neel order. Jy

Jx

• Competition between thedimer and Neel order.

• No frustration; transitionaccessible by QMC. Jy /Jx

10 rc

T

dimer

renormalized classical

quantum disorder

Neel order

• SU’(4) model in a rectangularlattice; phase diagram as Jy/Jx.

49

Conclusion

• Spin 3/2 cold atomic systems open up a new opportunityto study high symmetry and novel phases.

• Quintet Cooper pairing: the Alice string and topologicalgeneration of quantum entanglement.

• Quartetting order and its competition with the pairing order.

• Strong quantum fluctuations in spin 3/2 magnetic systems.