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
Densities distribution of quasi-momentum C1=1,c2=0.5,n=1andh=0. 0 Magnetization mz=(n1-12)/2 interaction 0.6 Spontaneous magnetization when h=o Phase transition from fully polarized 0.2 state to partially polarized state Critical points with strong repulsion hc+=m2n2-8m2n3/3c1, he=n2n2-8m2n3/3 -0.500.511.5 The critical points are different because the couplings cl and c2 are different
Densities distribution of quasi-momentum Magnetization Critical points with strong repulsion Spontaneous magnetization when h=0 Phase transition from fully polarized state to partially polarized state. The critical points are different because the couplings c1 and c2 are different. interaction
The pressures magnetization in the strong coupling limit h22 h n+ 12C1 (3n-57n2) 3n+5|+01/c2)+01/c2 12C2 h h mz= n+ 3n2 +O(1/c1)+O(1/c2) (b n=08 n=0.04 0 02 =n=0.8 n=0.6 n=0.04 10 0 510 10 0 h When the external field h is zero, the pressure takes its minimum. With the increasing h, the pressure increases. At the fully polarized state, the pressure arrives at its maximum
The pressures & magnetization in the strong coupling limit When the external field h is zero, the pressure takes its minimum. With the increasing h, the pressure increases. At the fully polarized state, the pressure arrives at its maximum
The ground state energy density 0.02 (上) 0.02 0.04 0.06 -0.08 n=1 -0.1 n=1 n=08 n=0.8 n=06 -0.12 =!=n=0.6 11n=0 -0.14 ……n=04 3 10 510 C1=100andc2=50 At the critical point, the second E=∑∑k-hM2 order derivative of the ground state i=1j=1 energy density is not continue, thus it is a second order phase transition
The ground state energy density At the critical point, the second order derivative of the ground state energy density is not continue, thus it is a second order phase transition
Entropy at finite temperature 0.1 0.6 008 05 006 04 03 004 0.2 002 0.1 -0.5 k0.5 15 C1=1,C2=0.5andn=1
Entropy at finite temperature
Dressed energy Ei(k)=TIn h()/ni(k) 0.4 (a) △E1 0.2 E2 0.8 △E2 C2=0.5 04 =1 0 2 0.5 0.5 k K/2 Strong repulsion E1(k)=k2-1-c17T32m12F1/2(u/T)+Oc12 Fermi-Dirac function (x)=T1(+1)0y/eyx+1dy
Dressed energy Strong repulsion Fermi-Dirac function
