Anisotropic exactly solvable models in the cold atomic systems Junpeng Cao Jiang, Guan, Wang lin
Anisotropic exactly solvable models in the cold atomic systems Jiang, Guan, Wang & Lin Junpeng Cao
Content Spin-1/2 bose gas IL. Spin-1 bose gas Spin-3/2 fermi gas
Content I. Spin-1/2 bose gas II. Spin-1 bose gas III. Spin-3/2 fermi gas
Anisotropic exactly solvable cold atomic model N amiltonian H a2+ Wi G x -x (pseudo-)spin Interaction symmetry ★00os0)g=c U(l)Lieb,eta,P130.605(1963 ★( fermion)go=c SU(2) Yang, PRL19.1312(1967) ★12(og SU(2)Li,EPL61.368(2003 12(boson) g1 81=c2810=0.U(1) ★1( boson) go=C,82=c SU(3)Zhou,JPA21.2391:2399(99 ■1(0m)80=-c,82=2c.SU(2)actE179300020 1( boson)800=c,g21=0.821=0 U() g 2.0 g ★1( fermion) SU(3) Sutherland. PRL2098(1968) ★32( mion)g=c,g2=c SU(4)Sutherland, PRL2098(1968) ■32(emin)8o=3c,g2=c Sp(4) Jiang, eta/, EPL87 10006(2009) 3/2( fermion)go0=0,g22=c1,g2=c2,U(1) g 0,g2.0=0,g2-1=0 Integer s(boson) 8o=-(s-1/2)c, g24.=c SO(2s +1)Jiang, et al, JPA44.345001(2011) Half-odd s(fermion)go=(S+3/2)C,82.4.=C. Sp(2s +1)Jiang et al, JPA44. 345001(2011)
Li, EPL 61. 368 (2003) Zhou, JPA 21. 2391; 2399 (1988) Anisotropic exactly solvable cold atomic model
I. Anisotropic spin-1/ 2 bose gas Anisotropic spin-exchanging interaction Motivation 1. Kondo problems: spin-1 fermions =∑。+∑(07+0+△076(x-x) Contact interaction: non-integrable Heisenberg long range interactions i e 1/r&1/r2: integrable Spin-1/2 bosons: non-integrable 2. Cold atoms: spin -12 bosons =-∑+∑ (CIi )6(x-x)+∑ i≠ 202+21+2++()0+(x=x)-
I. Anisotropic spin-1/2 bose gas Anisotropic spin-exchanging interaction Motivation 1. Kondo problems : spin-½ fermions Contact interaction: non-integrable; Heisenberg & long range interactions, i.e. 1/r & 1/r2 : integrable. 2. Cold atoms : spin-½ bosons Spin-1/2 bosons : non-integrable
Exact so|utⅰons Sab() k+ic ab kV2 b +po,o k-icI ali k-ic2 pl 1,0 ab tpab E=∑∑k-MF,K=∑∑k i=1j=1 i=lj=l N +Ic e kg-ko-ici j=1,2,…,N-M,讠=1,2 M2=(N1-N2)/2
Exact solutions