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
1
Collaborators
• L. Balents, UCSB.
• S. Chen, Institute of Physics, Chinese Academy of Sciences, Beijing.
• J. P. Hu, Purdue.
• O. Motrunich, UCSB
• Y. P. Wang, Institute of Physics, Chinese Academy of Sciences, Beijing.
• S. C. Zhang, Stanford.
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.
2
Rapid progress in cold atomic physics
M. Greiner et al., Nature 415, 39 (2002).
M. Greiner et. al., PRL, 2001.
• Magnetic traps: spin degrees of freedom are frozen.
• Optical traps and lattices: spin degrees of freedom are
released; a controllable way to study high spin physics.
• In optical lattices, interaction effects are adjustable. New
opportunity to study strongly correlated high spin systems.
3
High spin physics with cold atoms
• Most atoms have high hyperfine spin multiplets.
F= I (nuclear spin) + S (electron spin).
• Spin-1 bosons: 23Na (antiferro), 87Rb (ferromagnetic).
• Next focus: high spin fermions.
Spin-3/2:
132Cs, 9Be, 135Ba, 137Ba, 201Hg.
Spin-5/2 :
173Yb
Fukuhara et al.,
cond-mat/0607228
(quantum degeneracy has been achieved)
• High spin fermions: zero sounds and Cooper pairing
structures.
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).
4
Difference from high-spin transition metal compounds

• New feature: counter-intuitively, large spin here means
large quantum fluctuations.


high-spin cold atoms
e
e
e
e
e
e
high-spin transition metal
compounds
5
Hidden symmetry: spin-3/2 atomic systems are special!
• Spin 3/2 atoms:
132Cs, 9Be, 135Ba, 137Ba, 201Hg.
• 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.
Brief review: C. Wu, Mod. Phys. Lett. B 20, 1707 (2006)
6
What is SO(5)/Sp(4) group?
• SO(3) /SU(2) group.
n5
3-vector: x, y, z.
3-generator: Lz  L12 ,
2-spinor:
 ,
Lx  L23 , Ly  L31.

n3 , n4
• SO(5) /Sp(4) group.
n1 , n2
5-vector:
n1 , n2 , n3 , n4 , n5
10-generator:
Lab (1  a  b  5)
4-spinor:
3
2
1
2

1
2

3
2
7
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.


C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.
8
Multi-particle clustering order
• Quartetting order in spin 3/2 systems.
k1
k2

4-fermion counter-part
of Cooper pairing.

k1
k2
• Feshbach resonances: Cooper pairing superfluidity.


• Driven by logic, it is natural to expect the quartetting order
as a possible focus for the future research.
C. Wu, Phys. Rev. Lett. 95, 266404 (2005).
9
Strong quantum fluctuations in magnetic systems
• Intuitively, quantum fluctuations are weak in high spin systems.
Si S j
1 Sk
[ ,
]  i ijk
S S
S S
• 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.
From Auerbach’s book:
10
Outline
• The proof of the exact SO(5) symmetry.
C. Wu, J. P. Hu, and S. C. Zhang, PRL 91, 186402(2003).
• Quintet superfluids and non-Abelian topological defects.
• Quartetting v.s pairing orders in 1D spin 3/2 systems.
• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.
11
The SO(4) symmetry in Hydrogen atoms
• An obvious SO(3) symmetry: L (angular momentum) .
• The energy level degeneracy n2 is mysterious.

AL
• Not accidental! 1/r Coulomb potential gives
rise to a hidden conserved quantity.

Runge-Lentz vector A ; SO(4) generators A, L.

L
A
H

 the hidden conserved
• Q: What are
quantities in

spin 3/2 systems?
12
Generic spin-3/2 Hamiltonian in the continuum
• The s-wave scattering interactions and spin SU(2) symmetry.
d
H  d r

2





2
ˆ
  ( r )(
   )  (r )

2m
  3 / 2 , 1 / 2
g0    g2
 (r ) (r ) 
2
2




 a (r )  a (r )
a 1~ 5
3
2

1
2
1
2

3
2
• Pauli’s exclusion principle: only Ftot=0, 2
are allowed; Ftot=1, 3 are forbidden.
3 3



singlet:  (r )   00 | 2 2 ;  (r ) (r )

3 3


(r
)


2a
|
quintet: a
 2 2 ;  (r ) (r )


13
Spin-3/2 Hubbard model in optical lattices
H



t
{
c
c

h
.
c
.}


c
 i, j ,
 i, ci,
 ij  ,
i
 U 0   (i ) (i )  U 2
i


 a (i)  a (i)
a 1~ 5
• The single band Hubbard model is valid away from
resonances.
U0.2
 0.1,
E
l0
E
U 0,2
l0 ~ 5000A, as,0,2 ~ 100aB

t
(
as: scattering length, Er: recoil energy

V0

V0 1/ 4
) 1 ~ 2
Er
14
SO(5) symmetry: the single site problem
;
E0  0
,
E1  
24=16 states.
,
,
;
SU(2)
SO(5)
degen
eracy
E0,3,5
singlet
scalar
1
E1,4
quartet
spinor 4
E2
quintet
vector 5

E2 U 2  2
,

 ,
   
E3  U 0  2

 
E4  U 0  U 2  3
1
2
5
2
E5  U 0  5U 2  4



,,


;
,,
;

,,
,;
• U0=U2=U, SU(4) symmetry.
H int 
.







U
n( n  1)
2
15


Particle-hole bi-linears  M (I)
• Total degrees of freedom: 42=16=1+3+5+7.
1 density operator and 3 spin operators are not complete.
rank: 0
M 
1,

1
Fx ,Fy ,Fz
2
 ija Fi F j (a  1 ~ 5) :
3
 ijkaFi F j Fk (a  1 ~ 7)
Fx2  Fy2, Fz2  45 ,
{Fx ,Fy },{Fy ,Fz },{Fz ,Fx }
• Spin-nematic matrices (rank-2
 tensors) form five-G
matrices (SO(5) vector) .

Ga  ija Fi Fj , {Ga , Gb }  2 ab , (1  a, b  5)
16
Particle-hole channel algebra (II)
• Both Fx, y , z and ijk Fi Fj Fk commute with Hamiltonian.
10 generators of SO(5): 10=3+7.
a
7 spin cubic tensors are the hidden conserved quantities.
G ab  2i [G a ,G b ] (1 a  b  5)
• SO(5): 1 scalar + 5 vectors + 10 generators = 16

Time Reversal
1 density:
n    ;
even
5 spin nematic:
n a  12   G a ;
even
L ab  12   G ab ;
odd
3 spins + 7 spin
cubic tensors:
17
Outline
• The proof of the exact SO(5) symmetry.
• Quintet superfluids and Half-quantum vortices.
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.
• Quartetting v.s pairing orders in 1D spin 3/2 systems.
• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.
18
g2<0: s-wave quintet (Spair=2) pairing
• BCS theory: polar condensation; order parameter forms
an SO(5) vector.
d xy : 1 (r ) 
 a   ei dˆa

d xz :  2 (r )  i
d yz : 3 (r ) 

d̂

S4
5d unit vector
d3 z 2 r 2 :  4 (r )  i
d x 2  y 2 :  5 (r )  i


Ho and Yip, PRL 82, 247 (1999);
Wu, Hu and Zhang, cond-mat/0512602.
19
Superfluid with spin: half-quantum vortex (HQV)
• Z2 gauge
symmetry
dˆ  dˆ ,     

•
-disclination of d̂ as a HQV.
 a 
 ei dˆa

remains single-valued.
• d̂ is not a rigorous vector, but a
directionless director.
• Fundamental group of the manifold.
dˆ : RP 4  S 4 / Z 2

 1 ( RP 4 )  Z 2
20
Stability of half-quantum vortices
2
E   dr
{ sf ( ) 2   sp (dˆ ) 2 }
4M
• Single quantum E  h  sf log L
2
4
M

vortex:
• A pair of HQV:
 sf   sp
h
L
E
log
4M 2
2

 sf   sp
L

log
2
R
{


}
• Stability condition:
 sp  sf

R
21
Example: HQV as Alice string (3He-A phase)
• A particle flips the sign of its spin after it encircles HQV.
• Example: 3He-A, triplet Cooper pairing.
x

dˆ : S 2 / Z 2

d̂


Configuration space U(1)
z

n̂
y
 i
  e  ,   e  i 

 0, ei 

U (nˆ )   i

e
,
0


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). 22
P. McGraw, Phys. Rev. D 50, 952 (1994).
The HQV pair in 2D or HQV loop in 3D
e i
SO(3)SO(2)
x




n̂

z

z
y

• Phase state.


vort

 exp( iS z )   0
vort

P. McGraw, Phys. Rev. D, 50, 952 (1994).
23

SO(2) Cheshire charge (3He-A)
• HQV pair or loop can carry spin quantum number.
• For each phase state, SO(2) symmetry is only broken
in a finite region, so it should be dynamically restored.
z
• Cheshire charge state (Sz) v.s phase state.
m
vort
Sz m
  d exp( im ) 
vort
m0
mm
vort

vort
  d 


vort
vort


P. McGraw, Phys. Rev. D, 50, 952 (1994).
24
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.
init    m  0
p
vort
    d 
final     d ei 
p
   m  1 vort
p
p

vort

vort


• Both the initial and final states are
product states. No entanglement! P. McGraw, Phys. Rev. D, 50, 952 (1994).
25
Quintet pairing as a non-Abelian generalization
• HQV configuration space n̂
3
equator: nˆ  S  SU (2)
S4
vort
, SU(2) instead of U(1).
dˆ ( , nˆ )  cos 2 eˆ4  sin 2 nˆ
ê4

dˆ (  0) // eˆ4
n̂

ê1, 2 ,3,5
-disclination

26
SU(2) Berry phase
• After a particle moves around HQV, or passes a HQV pair:
3
2
, 3
 2
  ,
1
2
, 1

 2









0 W 

   
0 
W
3
2
,

3
2

1
2
,

1
2
 n3  in2  n1  in5 

W (nˆ )  
 n1  in5 n3  in2 


n
S 3  SU (2)
ˆ  n1eˆ1  n2 eˆ2
n
 n3eˆ3  n5 eˆ5


27
Non-Abelian Cheshire charge
e4
• Construct SO(4) Cheshire charge
state for a HQV pair (loop) through S3
harmonic functions.
SO(5)  SO(4)
TT3 ; T ' T '3
vort

nS 3
dnˆ YTT ;T 'T ' (nˆ ) nˆ
3
3
vort
 
T (T ' )  L12,  L35 , L13,  L25, , L23  L15.

• Zero charge state.
00;00
vort

nS
3
dnˆ nˆ

vort


28
Entanglement through non-Abelian Cheshire charge!
• SO(4) spin conservation. For simplicity, only Sz  T3  32 T3' is
shown.
init 
3
2 p
 zero charge
vort
final 
1
2 p
 Sz 1 vort 
1
2 p

 Sz  2
vort

• Generation of entanglement between the particle and HQV
loop!



29
Outline
• The proof of the exact SO(5) symmetry.
• 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).
• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.
30
Multiple-particle clustering (MPC) instability
• Pauli’s exclusion principle allows MPC. More than two particles
form bound states.
baryon (3-quark); alpha particle (2p+2n); bi-exciton (2e+2h)
• Spin 3/2 systems: quartetting order.
SU(4) singlet:
4-body maximally
entangled states
Oqt   3/ 2 (r ) 1/ 2 (r ) 1/ 2 (r ) 3/ 2 (r )
• 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.
31
1D systems: strongly correlated but understandable
• 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.
P. Schlottmann, J. Phys. Cond. Matt 6, 1359(1994).
• Competing orders in 1D spin 3/2 systems with SO(5) symmetry.
Both quartetting and singlet Cooper pairing are allowed.
Transition between quartetting and Cooper pairing.
C. Wu, Phys. Rev. Lett. 95, 266404(2005).
32
Phase diagram at incommensurate fillings
g2
• Gapless charge sector.
SU(4)
g0  g2
C: Singlet pairing
A: Luttinger liquid
g0
B: Quartetting
SU(4) g 0
 g2
• Spin gap phases B and C:
pairing v.s. quartetting.
• Ising transition between
B and C.
• Singlet pairing in purely
repulsive regime.
• Quintet pairing is not
allowed.
33
Competition between quartetting and pairing phases
A. J. Leggett, Prog, Theo. Phys. 36, 901(1966); H. J. Schulz, PRB 53, R2959 (1996).
• Phase locking problem in the two-band model.
    e


1

2
i  c
Oquar  12  ei
c overall phase;
cos2   r ;
4  c
1   3/ 23 / 2

2  1/ 21/
2
cos2  r .

  r relative phase.

  r dual field of the relative phase

• No symmetry breaking in the overall phase (charge) channel
in 1D.
     ei i.e.   c    c  2
3 1
 ,
2 2
3 1
 ,
2 2
34
Ising transition in the relative phase channel
1
1
H eff  {( x r ) 2  ( x r ) 2 } 
(1 cos 2   r  2 cos 2   r )
2
2a
• 1  2 the relative phase is pinned: pairing order;
1  2 the dual field is pinned: quartetting order.
Ising transition: two Majorana fermions with masses:
• Ising symmetry:
   i  ,
3

2
3

2
relative phase:   r    r  
• Ising ordered phase:
Ising disordered phase:
1  2
   i  .

1
2

1
2
 r   r
   ,
Oquart  Oquart
35
Experiment setup and detection
• Array of 1D optical tubes.
• RF spectroscopy to measure
the excitation gap.
pair breaking:
quartet breaking:
M. Greiner et. al., PRL, 2001.
36
Outline
• The proof of the exact SO(5) symmetry.
• Alice string in quintet superfluids and non-Abelian Cheshire
charge.
• Quartetting v.s pairing orders in 1D spin 3/2 systems.
• SO(5) (Sp(4)) magnetism in Mott-insulating phases.
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005).
C. Wu, L. Balents, in preparation.
37
Spin exchange (one particle per site)
• Spin exchange: bond singlet (J0), quintet (J2). No exchange
in the triplet and septet channels.
H ex    J 0Q0 (ij )  J 2Q2 (ij )
ij
J 0  4t / U 0 , J 2  4t / U 2 , J1  J 3  0
2
2
3
2
 
3
2
0+2+1+3
• Heisenberg model with bi-linear, bi-quadratic, bi-cubic terms.
• SO(5) or Sp(4) explicitly invariant form:
H ex  
ij

J0  J2
 J 0  3J 2
Lab (i ) Lab ( j ) 
na (i )nb ( j )
4
4
a, b  1 ~ 5
Lab: 3 spins + 7 spin cubic tensors; na: spin nematic operators;
Lab and na together form the15 SU(4) generators.
38
Two different SU(4) symmetries
• A: J0=J2=J, SU(4) point.
Hex  J {Lab (i)Lab ( j)  na (i)nb ( j)}
septet
singlet
quintet
J
ij
*
• B: J2=0, the staggered SU’(4) point.

triplet
In a bipartite lattice, a particle-hole transformation on odd sites:
Lab ( j )  L'ab ( j )
quintet
nab ( j )  n'ab ( j )
J
H ex  0  {Lab (i ) L'ab ( j )  na (i )n'a ( j )}
4 ij
J0
triplet
septet
singlet
39
Construction of singlets
• The SU(2) singlet: 2 sites.
• The uniform SU(4) singlet: 4 sites.
baryon
  
  (1)  (2)  (3)  (4) 0
4!
• The staggered SU’(4) singlet: 2 sites.
meson
1
2
  (1) R  (2)
40
Phase diagram in 1D lattice (one particle per site)
J0
SU ' (4)
spin gap
dimer
J 0  J 2 SU (4)
gapless spin liquid
J2
• On the SU’(4) line, dimerized spin gap phase.
• On the SU(4) line, gapless spin liquid phase.
C. Wu, Phys. Rev. Lett. 95, 266404 (2005).
41
SU’(4) and SU(4) point at 2D
• 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).
• J2>0, no conclusive results!
SU(4) point (J0=J2), 2D
Plaquette order at the
SU(4) point?
Exact diagonalization on a 4*4
lattice
Bossche et. al., Eur. Phys. J. B 17, 367 (2000).
42
Exact result: SU(4) Majumdar-Ghosh ladder
J '  12 J
• Exact dimer ground state in
spin 1/2 M-G model.
J
H   Hi,i1,i2, Hi,i1,i2  (Si  Si1  Si2 )2
2
i
i
i 1
i2
J
• SU(4) M-G: plaquette state.

H 
H
i
every six-site cluster
Hi  (
2
L
)
 ab  (
six sites
2
n
)
 a
six sites
SU(4) Casimir of the six-site cluster
• Excitations as fractionalized
domain walls.
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang,
Phys. Rev. B 72, 214428 (2005).
43
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.
44
SU(4) plaquette state: a four-site problem
• Bond spin singlet:
• Plaquette SU(4) singlet:

4!
  0

4-body EPR state; no
bond orders


d-wave to s-wave
• Hint to 2D?
1
3
4




 J

• Level crossing:

2


0


d-wave
a


 b
s-wave
J2
45
Speculations: 2D phase diagram ?
• J2=0, Neel order at the SU’(4) point (QMC).
K. Harada et. al. PRL 90, 117203, (2003).
J0
SU’(4)
• J2>0, no conclusive results!
2D Plaquette order at the
SU(4) point?

Exact diagonalization on a 4*4
lattice
SU(4)
?
Bossche et. al., Eur. Phys. J. B 17, 367 (2000).
• Phase transitions as J0/J2?
Dimer phases? Singlet or
magnetic dimers?
C. Wu, L. Balents, in preparation.
?
J2
46
Strong quantum fluctuations: high symmetry
• Different from the transition metal high spin compounds!
Sp(4), SU(4):
F=3/2
quantum: spin
disordered states
…
large N
large S
SU(2): S=3/2
classical
Neel order
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.
47
Phase B: the quartetting phase
• Quartetting superfluidity v.s. CDW of quartets (2kf-CDW).
Oqt   3/ 2 1/ 2 1/ 2 3 / 2 wins at K c  2;
N 2 k f   R L wins at K c  2.
Kc: the Luttinger parameter in the charge channel.
d  2 /( 2k f )
48
Phase C: the singlet pairing phase
• Singlet pairing superfluidity v.s CDW of pairs (4kf-CDW).
    3/ 2 3 / 2  1/ 2 1/ 2 wins at K c  12 ;
O4 k f ,cdw   R R L L wins at K c  12 .
d  2 /( 4k f )
49
High spin physics with cold atoms
• Most atoms have high hyperfine spin multiplets.
F= I (nuclear spin) + S (electron spin).
• Different from high spin transition metal compounds.

e
e
e
• Spin-1 bosons: 23Na (antiferro), 87Rb (ferromagnetic).
• High spin fermions: zero sounds and Cooper pairing
structures.
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).
50
Quintet channel (S=2) operators as SO(5) vectors
• Kinetic energy has an obvious SU(4) symmetry; interactions
break it down to SO(5) (Sp(4)); SU(4) is restored at U0=U2.
d xy : 1 (r ) 

d xz :  (r )  i

2
d yz : 3 (r ) 

5-polar
vector
d̂

S
d3 z 2 r 2 :  4 (r )  i
d x 2  y 2 :  (r )  i

5
4


trajectory under SU(2).
51
The SU’(4) model: dimensional crossover
Jx
• SU’(4) model: 1D dimer order;
2D Neel order.
Jy

• SU’(4) model in a rectangular
lattice; phase diagram as Jy/Jx.

• Competition between the
dimer and Neel order.
T
quantum
disorder

• No frustration; transition
dimer
accessible by QMC.
C. Wu, O. Motrunich, L. Balents, in preparation .
renormalized
classical
0
Jy /Jx
rc
Neel order
1
52