hidden symmetry and quantum phases in spin 3/2 cold atomic systems

1

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

Congjun Wu

Kavli Institute for Theoretical Physics, UCSB

Ref: 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.

OSU , 05/09/2006

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.

• L. Balents, UCSB.

Many thanks to 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.

• O. Motrunich, UCSB

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. New opportunity to study strongly correlated high spin systems.

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

• Optical traps and lattices: spin degrees of freedom are released; 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 pairing structures.

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

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

Spin-5/2 : 173Yb (quantum degeneracy has been achieved)

• Next focus: high spin fermions.

Fukuhara et al., cond-mat/06072

28

5

Difference from high-spin transition metal compounds

• New feature: counter-intuitively, large spin here means large quantum fluctuations.

eee

high-spin transition metal compounds

high-spin cold atoms

eee

6

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 ½ systems

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

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

Brief review: C. Wu, Mod. Phys. Lett. B 20, 1707 (2006)

7

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, nn43 , nn

5n

8

Quintet superfluidity and half-quantum vortex

• High symmetries do not occur frequently in nature, since every new symmetry brings with itself a possibility of new physics, 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 quantum entanglement.


9

Multi-particle clustering order

• Feshbach resonances: Cooper pairing superfluidity.

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

• Quartetting order in spin 3/2 systems.

k 1

k 1

k 2

k 2

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

4-fermion counter-part of Cooper pairing.

10

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 high SO(5) symmetry, quantum fluctuations here are even stronger than those in spin ½ systems.

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

11

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.

12

• Not accidental! 1/r Coulomb potential gives rise 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 in spin 3/2 systems?

L

• The energy level degeneracy n2 is mysterious.

H

A

L

A

L

A

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

13

Generic spin-3/2 Hamiltonian in the continuum

5~1

20

22

2/1,2/3

)()(2

)()(2

)()ˆ2

()(

aaa

d

rrg

rrg

rm

rrdH

• 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 )

00 | 3

2

3

2;

(r )

(r )

a(r )

2a | 3

2

3

2;

(r )

(r )quintet:

singlet:

14

Spin-3/2 Hubbard model in optical lattices

• The single band Hubbard model is valid away from resonances.

5~120

,,,,

,

)()()()(

.}.{

aaa

i

ii

ijij

i

iiUiiU

ccchcctH

U0.2

E 0.1,

l0 ~ 5000A, as,0,2 ~ 100aB

(V0

E r

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

E

t

U0,20V

0l

15

SO(5) symmetry: the single site problem

00 E

45 205 UUE

322

502

14 UUE

203 UE

222 UE

;

1E

,

,

,

;

,

,

;

;

,

,

;

,

,

,

,

,

,

,

,

.

SU(2) SO(5) degen

eracy

E0,3,5 singlet

scalar 1

E1,4 quartet spinor 4

E2 quintet vector 5

)1(2int nnU

H

• U0=U2=U, SU(4) symmetry.

24=16 states.

16

1,

Fx,Fy,Fz

ijaFiF j (a1 ~ 5) :

ijka FiF jFk (a1 ~ 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 abba

jiaij

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 ] (1a b5)

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

kjiaijkzyx FFFF and,,

;

;

;

2

1

2

1

ab

a

ab

a

L

n

n

1 density:

5 spin nematic:

3 spins + 7 spin cubic tensors:

Time Reversal

even

even

odd

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Outline


• Quintet superfluids and Half-quantum vortices.




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

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

d 4S

ai

a de ˆ

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

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

5d unit vector

)(: 1xy rd

)(: 522 rdyx

i

)(: 2 rd xz i

)(: 3 rd yz

irdrz

)( : 43 22

20

Superfluid with spin: half-quantum vortex (HQV)

,ˆˆ dd• Z2 gauge symmetry

• -disclination of as a HQV.

d

• Fundamental group of the manifold.

24

1 )( ZRP

244 /:ˆ ZSRPd

• is not a rigorous vector, but a directionless director.

ai

a de 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

}

{

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.

22 /:ˆ ZSd

x

z

y

n

• 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)

d

ii 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).

x

z

y

n

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 broken in a finite region, so it should be dynamically restored.

vortvortimdm )exp(

Sz mvort

m mvort

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

vortvortdm 0

• HQV pair or loop can carry spin quantum number.

z


25

Spin conservation by exciting Cheshire charge

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

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

vortpvortpdminit 0

vortp

vort

i

p

m

edfinal

1

• Both the initial and final states are product states. No entanglement!


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 3 SUSn

-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

W 2

1

2

1

2

3

2

3 ,,

)2(3 SUS 5533

2211

ˆˆ

ˆˆˆ

enen

enenn

n

28

Non-Abelian Cheshire charge

• Construct SO(4) Cheshire charge state for a HQV pair (loop) through S3 harmonic functions.

vortTTTTSnvortnnYndTTTT ˆ)ˆ(ˆ'';

333 '';

33

.,,)'( 1523,25,1335,12 LLLLLLTT

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, only

is shown.

init 3

2 p zero charge

vort

final 1

2 p Sz 1

vort 1

2 p Sz 2

vort

Sz T3 3

2T3

'

• Generation of entanglement between the particle and HQV loop!

30

Outline


• Alice string in quintet superfluids and non-Abelian Cheshire charge.


• 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 particles form bound states.

SU(4) singlet:

4-body maximally entangled 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 not possible 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

20 gg

SU(4)

B: Quartetting

C: Singlet pairing

0g

2g

20 gg

SU(4)

• Singlet pairing in purely repulsive regime.

• Ising transition between B and C.

• Quintet pairing is not allowed.

34

Competition between quartetting and pairing phases

• Phase locking problem in the two-band model.

.cos2

;cos24

21

21

ri

quar

ri

c

c

eO

e

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) channel in 1D.

2..2

1,

2

3

2

1,

2

3 cc

i eie

r

35

• 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

22rrrxrxeff a

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

36

Experiment setup and detection

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

• Array of 1D optical tubes.

• RF spectroscopy to measure the excitation gap.pair breaking:

quartet breaking:

37

Outline


• Alice string in quintet superfluids and non-Abelian Cheshire charge.

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


C. Wu, L. Balents, in preparation.

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

38

Spin exchange (one particle per site)

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

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

ijbaababex jnin

JJjLiL

JJH )()(

4

3)()(

42020

0,/4,/4

)()(

3122

202

0

2200

JJUtJUtJ

ijQJijQJHij

ex

3

23

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.

39

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

*

40

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

41

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

)4(20 SUJJ

spin gap dimer

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.

42

SU’(4) and SU(4) point at 2D

• J2>0, no conclusive results!

• At SU’(4) point (J2=0), QMC and 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), 2D Plaquette order at the SU(4) point?

Exact diagonalization on a 4*4 lattice

43

Exact result: SU(4) Majumdar-Ghosh ladder

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

H H i,i1,i2

i

, H i,i1,i2 J

2(S i

S i1

S i2)2

iJ

JJ2

1'

1i 2i

• 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 fractionalized domain walls.

sitessix

2

sitessix

2 )()( aabi nLH

44

Conclusion

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

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

• Quartetting order and its competition with the pairing order.

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

45

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

• Bond spin singlet:

• Level crossing:

2

1

3

4

4!

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?

46

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 the SU(4) point?

Exact diagonalization on a 4*4 lattice

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

J2

SU(4)

?

?

J0SU’(4)

C. Wu, L. Balents, in preparation.

47

Strong quantum fluctuations: high symmetry

• Different from the transition metal high spin compounds!

SU(4) singlet plaquette state: 4 sites order without any bond order.

S. Chen, C. Wu, Y. P. Wang, S. C. Zhang, cond-mat/0507234, accepted by PRB.

classical Neel order

SU(2): S=3/2

…

large N large S quantum: spin disordered states

Sp(4), SU(4): F=3/2

48

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

49

Phase C: the singlet pairing phase

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

.at wins

at wins

2

1,4

;2

12/12/12/32/3

cLLRRcdwk

c

KO

K

f

)4/(2 fkd

50

• 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 pairing structures.

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

• Different from high spin transition metal compounds.

eee

51

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

)(: 1xy rd

)(: 522 rdyx

i

)(: 2 rd xz i

)(: 3 rd yz

irdrz

)( : 43 22

d

4S

5-polar vector

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

trajectory under SU(2).

52

The SU’(4) model: dimensional crossover

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

Jy

Jx

• Competition between the dimer and Neel order.

• No frustration; transition accessible by QMC.

Jy /Jx

1

0

rc

T

dimerrenormalized

classical

quantum disorder

Neel orderC. Wu, O. Motrunich, L. Balents, in preparation .

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

hidden symmetry and quantum phases in spin 3/2 cold atomic systems

Documents

high spin physics

high spin systems

large spin

high spin fermions

nuclear spin s electron

cold atoms spin

spin degrees of freedom

high hyperfine spin