Hidden Symmetry and Quantum Phases in Spin / Cold Atomic Systems Congjun Wu Kavli Institute for Theoretical Physics, UCSB Ref: C. Wu, Mod. Phys. Lett. B 0, 707, (006); C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 9, 8640(00); C. Wu, Phys. Rev. Lett. 95, 66404 (005); S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 7, 448 (005); C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0560.
Collaborators S. Chen, Institute of Physics, Chinese Academy of Sciences, Beijing. J. P. Hu, Purdue. Y. P. Wang, Institute of Physics, Chinese Academy of Sciences, Beijing. S. C. Zhang, Stanford. 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.
Rapid progress in cold atomic physics M. Greiner et al., Nature 45, 9 (00). M. Greiner et. al., PRL, 00. 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.
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- bosons: Na (antiferro), 87 Rb (ferromagnetic). High spin fermions: zero sounds and Cooper pairing structures. D. M. Stamper-Kurn et al., PRL 80, 07 (998); T. L. Ho, PRL 8, 74 (998); F. Zhou, PRL 87, 8040 (00); E. Demler and F. Zhou, PRL 88, 600 (00); T. L. Ho and S. Yip, PRL 8, 47 (999); S. Yip and T. L. Ho, PRA 59, 465(999). 4
Hidden symmetry: spin-/ atomic systems are special! Spin / atoms: Cs, 9 Be, 5 Ba, 7 Ba, 0 Hg. 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 / systems SU() in spin systems This SO(5) symmetry is qualitatively different from the SO(5) theory of high T c superconductivity. C. Wu, J. P. Hu, and S. C. Zhang, PRL 9, 8640(00). 5
What is SO(5)/Sp(4) group? SO() /SU() group. n 5 -vector: x, y, z. -generator: L z = L = L, Lx = L, Ly. -spinor:, SO(5) /Sp(4) group. n, n n, n 4 5-vector: n n, n, n4,, n 5 0-generator: L ab ( a < b 5) 4-spinor: 6
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/050505 Cooper pairing with S pair =. Half-quantum vortex (non-abelian Alice string) and quantum entanglement. C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0560. 7
Multi-particle clustering order Quartetting order in spin / systems. r k r k 4-fermion counter-part of Cooper pairing. r k r k 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, 66404 (005). 8
Strong quantum fluctuations in magnetic systems Intuitively, quantum fluctuations are weak in high spin systems. S S j [, ] = i S S i ijk S S S k However, due to the high SO(5) symmetry, quantum fluctuations here are even stronger than those in spin systems. large N (N=4) v.s. large S. From Auerbach s book: 9
Outline The proof of the exact SO(5) symmetry. C. Wu, J. P. Hu, and S. C. Zhang, PRL 9, 8640(00). Quintet superfluids and non-abelian topological defects. Quartetting v.s pairing orders in D spin / systems. SO(5) (Sp(4)) Magnetism in Mott-insulating phases. 0
The SO(4) symmetry in Hydrogen atoms r An obvious SO() symmetry: L (angular momentum). The energy level degeneracy n is mysterious. Not accidental! /r Coulomb potential gives rise to a hidden conserved quantity. Runge-Lentz vector ; SO(4) generators r A r A, L r. r A L r r L r A H Q: What are the hidden conserved quantities in spin / systems?
Generic spin-/ Hamiltonian in the continuum The s-wave scattering interactions and spin SU() symmetry. H = d g0 d r =± /, ± / r r ( ) ( ) g r h ( )( m a= ~5 a ˆ μ) r r ( ) ( ) a r ( ) Pauli s exclusion principle: only F tot =0, are allowed; F tot =, are forbidden. singlet: quintet: ( r ) = a ( r ) = 00 a ; ( r ) ( r ) ; ( r ) ( r )
Spin-/ Hubbard model in optical lattices H = U 0 ij, i t{ c i, c j, ( i) ( i) U h. c.} μ a= ~5 a i c ( i) i, a c ( i) i, The single band Hubbard model is valid away from resonances. E t l 0 V0 U 0, a s: scattering length, E r: recoil energy U 0. E < 0., l 0 ~ 400nm, a s,0, ~ 00a B ( V 0 E r ) /4 ~
SO(5) symmetry: the single site problem E 0 = 0 ; 4 =6 states. E E = μ,,, ; = U E = U 0 μ μ, ±, ± ; ; SU() SO(5) degen eracy E 0,,5 singlet scalar E,4 quartet spinor 4 E quintet vector 5 E 5 4 = U 0 U E 5 = U 0 5U μ 4μ,,,,,,,,, ;. U 0 =U =U, SU(4) symmetry. U Hint = n( n ) 4
Quintet channel (S=) 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 U 0 =U. d ( r) = xy : d xz : ( r) = i 5-polar vector dˆ d d yz : ( r) = : ( r) = i z r 4 S 4 d : ( ) y 5 r = x i trajectory under SU(). 5
Particle-hole bi-linears M (I) Total degrees of freedom: 4 =6=57. density operator and spin operators are not complete. rank: 0 M, F x,f y,f z ij a F i F j (a = ~ 5) : a ijk F i F j F k (a =~ 7) F x F y, F z 5 4, {F x,f y },{F y,f z },{F z,f x } Spin-nematic matrices (rank- tensors) form five- matrices (SO(5) vector). a = a ij F i F j a b, {, } =, ( a, b ab 5) 6
F Particle-hole channel algebra (II) and a F F F Both x, y, z ijk i j k commute with Hamiltonian. 0 generators of SO(5): 0=7. 7 spin cubic tensors are the hidden conserved quantities. ab = i [a, b ]( a < b 5) SO(5): scalar 5 vectors 0 generators = 6 Time Reversal density: n = ; even 5 spin nematic: spins 7 spin cubic tensors: n L a ab = = a ; ab ; even odd 7
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/0560. Quartetting v.s pairing orders in D spin / systems. SO(5) (Sp(4)) Magnetism in Mott-insulating phases. 8
g <0: s-wave quintet (S pair =) pairing BCS theory: polar condensation; order parameter forms an SO(5) vector. d ( r) = xy : = a e i dˆ a d d xz : ( r) = i d yz : ( r) = : ( r) = i z r 4 4 dˆ S 5d unit vector d : ( ) y 5 r = x i Ho and Yip, PRL 8, 47 (999); Wu, Hu and Zhang, cond-mat/0560. 9
Superfluid with spin: half-quantum vortex (HQV) Z gauge symmetry dˆ d ˆ, -disclination of dˆ as a HQV. = a e i d ˆ a remains single-valued. dˆ is not a rigorous vector, but a directionless director. ˆ 4 d : RP = S 4 / Z Fundamental group of the manifold. ( RP = Z 4 ) 0
E = 4 h M Stability of half-quantum vortices h E = dr { sf ( ) 4M Single quantum vortex: L sf log sp ( dˆ) } A pair of HQV: E = 4 h M sf { sp sf log L R sp } L log Stability condition: sp < sf R
Example: HQV as Alice string ( He-A phase) A particle flips the sign of its spin after it encircles HQV. Example: He-A, triplet Cooper pairing. dˆ : S / dˆ Z e, x z nˆ y e i i U ( nˆ) = 0, e i e,0 i Configuration space U() A. S. Schwarz et al., Nucl. Phys. B 08, 4(98); F. A. Bais et al., Nucl. Phys. B 666, 4 (00); M. G. Alford, et al. Nucl. Phys. B 84, 5 (99). P. McGraw, Phys. Rev. D 50, 95 (994).
The HQV pair in D or HQV loop in D i e SO()SO() z x z nˆ y Phase state. = exp( is vort z) = 0 vort P. McGraw, Phys. Rev. D, 50, 95 (994).
SO() Cheshire charge ( He-A) HQV pair or loop can carry spin quantum number. For each phase state, SO() symmetry is only broken in a finite region, so it should be dynamically restored. Cheshire charge state (S z ) v.s phase state. z m vort = d exp( im) vort S z m vort = mm vort m = 0 vort = d vort P. McGraw, Phys. Rev. D, 50, 95 (994). 4
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. init = m = 0 p vort = p d vort final = p i = d e p m = vort vort Both the initial and final states are product states. No entanglement! P. McGraw, Phys. Rev. D, 50, 95 (994). 5
Quintet pairing as a non-abelian generalization HQV configuration space nˆ, SU() instead of U(). vort equator: n ˆ S = SU () dˆ(, nˆ) = cos eˆ 4 sin nˆ S 4 nˆ ê 4 d ˆ( = 0) // e ˆ4 ê,,,5 -disclination 6
7 SU() Berry phase After a particle moves around HQV, or passes a HQV pair: =,,, = 5 5 ( ˆ) 0 0 in n in n in n in n n W W W,, () S SU 5 5 ˆ ˆ ˆ ˆ ˆ e n e n e n n e n = n r
Non-Abelian Cheshire charge Construct SO(4) Cheshire charge state for a HQV pair (loop) through S harmonic functions. TT ; T ' T ' = dn Y vort n S TT ; T ' T ' ˆ ( nˆ) nˆ vort SO(5) SO(4) e 4 r r T ( T ') = L, ± L5, L, ± L5,, L ± L5. Zero charge state. 00;00 vort = n S dnˆ nˆ vort 8
Entanglement through non-abelian Cheshire charge! SO(4) spin conservation. For simplicity, only is shown. S z = T T ' init = p zero charge vort final = p S z = vort p S z = vort Generation of entanglement between the particle and HQV loop! 9
Outline The proof of the exact SO(5) symmetry. Alice string in quintet superfluids and non-abelian Cheshire charge. Quartetting and pairing orders in D systems. C. Wu, Phys. Rev. Lett. 95, 66404(005). SO(5) (Sp(4)) Magnetism in Mott-insulating phases. 0
Multiple-particle clustering (MPC) instability Pauli s exclusion principle allows MPC. More than two particles form bound states. baryon (-quark); alpha particle (pn); bi-exciton (eh) Spin / systems: quartetting order. SU(4) singlet: 4-body maximally entangled states O qt = / ( r) / ( r) / ( r) / ( r) Difficulty: lack of a BCS type well-controlled mean field theory. trial wavefunction in D: A. S. Stepanenko and J. M. F Gunn, cond-mat/9907.
D systems: strongly correlated but understandable Bethe ansatz results for D SU(N) model: N particles form an SU(N) singlet; Cooper pairing is not possible because particles can not form an SU(N) singlet. P. Schlottmann, J. Phys. Cond. Matt 6, 59(994). Competing orders in D spin / 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, 66404(005).
Phase diagram at incommensurate fillings g C: Singlet pairing SU(4) g 0 = g A: Luttinger liquid g 0 Gapless charge sector. Spin gap phases B and C: pairing v.s. quartetting. Ising transition between B and C. B: Quartetting Singlet pairing in purely repulsive regime. SU(4) g 0 = g Quintet pairing is not allowed.
Phase B: the quartetting phase Quartetting superfluidity v.s. CDW of quartets (k f -CDW). O qt = / / / / wins at K c > ; N k f = R L wins at K c <. K c : the Luttinger parameter in the charge channel. d = /(k f ) 4
5 Phase C: the singlet pairing phase Singlet pairing superfluidity v.s CDW of pairs (4k f -CDW).. wins at wins at, 4 ; / / / / < = > = c L L R R cdw k c K O K f ) /(4 f k d =
Competition between quartetting and pairing phases A. J. Leggett, Prog, Theo. Phys. 6, 90(966); H. J. Schulz, PRB 5, R959 (996). Phase locking problem in the two-band model. = e i c cos r ; O quar = e i 4 c cos r. = / / = / / c overall phase; relative phase. r r dual field of the relative phase No symmetry breaking in the overall phase (charge) channel in D. i e i. e. ±, ± ±, ± c c 6
Ising transition in the relative phase channel H eff = {( x r ) ( xr ) } ( cos r a cos ) r > the relative phase is pinned: pairing order; < the dual field is pinned: quartetting order. Ising transition: two Majorana fermions with masses: ± Ising symmetry: i, i. ± ± ± ± relative phase: ± r r r r Ising ordered phase: Ising disordered phase:, Oquart O quart 7
Experiment setup and detection Array of D optical tubes. RF spectroscopy to measure the excitation gap. pair breaking: quartet breaking: M. Greiner et. al., PRL, 00. 8
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 D spin / systems. SO(5) (Sp(4)) magnetism in Mott-insulating phases. S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 7, 448 (005). 9
40 Spin exchange (one particle per site) Spin exchange: bond singlet (J 0 ), quintet (J ). No exchange in the triplet and septet channels. SO(5) or Sp(4) explicitly invariant form: = ij b a ab ab ex j n i n J J j L i L J J H ) ( ) ( 4 ) ( ) ( 4 0 0 0, / 4, / 4 ) ( ) ( 0 0 0 0 = = = = = J J U t J U t J ij Q J ij Q J H ij ex = 0 L ab : spins 7 spin cubic tensors; n a : spin nematic operators; L ab and n a together form the5 SU(4) generators., = ~ 5 b a Heisenberg model with bi-linear, bi-quadratic, bi-cubic terms.
Two different SU(4) symmetries A: J 0 =J =J, SU(4) point. triplet septet H ex = J ij {L ab (i)l ab ( j) n a (i)n b ( j)} J singlet quintet B: J =0, the staggered SU (4) point. * In a bipartite lattice, a particle-hole transformation on odd sites: L ab ( j) = L' ( j) n ( j) = n' ( j) ab J 0 H ex = { Lab( i) L' ab ( j) na ( i) n' a ( j)} 4 ij ab ab J 0 quintet triplet septet singlet 4
Construction of singlets The SU() singlet: sites. The uniform SU(4) singlet: 4 sites. baryon 4! () () () (4) 0 The staggered SU (4) singlet: sites. meson () R () 4
Phase diagram in D lattice (one particle per site) J 0 SU ' (4) J 0 = J SU (4) spin gap dimer gapless spin liquid J On the SU (4) line, dimerized spin gap phase. On the SU(4) line, gapless spin liquid phase. C. Wu, Phys. Rev. Lett. 95, 66404 (005). 4
SU (4) and SU(4) point at D At SU (4) point (J =0), QMC and large N give the Neel order, but the moment is tiny. Read, and Sachdev, Nucl. Phys. B 6(989). K. Harada et. al. PRL 90, 70, (00). J >0, no conclusive results! SU(4) point (J 0 =J ), D Plaquette order at the SU(4) point? Exact diagonalization on a 4*4 lattice Bossche et. al., Eur. Phys. J. B 7, 67 (000). 44
Exact result: SU(4) Majumdar-Ghosh ladder Exact dimer ground state in spin / M-G model. H = H i,i,i, H i,i,i = J ( S v i S r i S r i ) i i i i ' J = J J SU(4) M-G: plaquette state. H = H i = H i every six-site cluster Lab) ( ( na six sites 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 7, 448 (005). 45
SU(4) plaquette state: a four-site problem Bond spin singlet: Plaquette SU(4) singlet: 4! 0 = 4 4-body EPR state; no bond orders J 0 Level crossing: d-wave to s-wave d-wave a ( ) b Hint to D? s-wave J 46
Speculations: D phase diagram? J =0, Neel order at the SU (4) point (QMC). K. Harada et. al. PRL 90, 70, (00). J >0, no conclusive results! D Plaquette order at the SU(4) point? Exact diagonalization on a 4*4 lattice Bossche et. al., Eur. Phys. J. B 7, 67 (000). J 0 SU (4)? SU(4) Phase transitions as J 0 /J? Dimer phases? Singlet or magnetic dimers? J? 47
The SU (4) model: dimensional crossover J x SU (4) model: D dimer order; D Neel order. J y SU (4) model in a rectangular lattice; phase diagram as J y /J x. Competition between the dimer and Neel order. No frustration; transition accessible by QMC. T dimer quantum disorder 0 r c Neel order renormalized classical J y /J x 48
Conclusion Spin / 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 / magnetic systems. 49