Anisotropic exactly solvable models in the cold atomic systems Jiang, Guan, Wang & Lin Junpeng Cao

Content I.Spin-1/2 bose gas II.Spin-1 bose gas III.Spin-3/2 fermi gas

Li, EPL (2003) Zhou, JPA ; 2399 (1988) Anisotropic exactly solvable cold atomic model

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/r 2 : integrable. 2. Cold atoms : spin-½ bosons Spin-1/2 bosons : non-integrable.

Exact solutions

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 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 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

Dressed energy Strong repulsion Fermi-Dirac function

dimension d=1, critical exponent z= 2, correlation length exponent v= 1/2 Grand canonical ensemble Scaling behavior near the critical point Near critical point, density of particle number satisfies scaling law Focus on this model,

II. Anisotropic spin-1 Bose gas

repulsive interaction

Ground state In the case of strong repulsive interaction, the critical field hc is fully polarized state partially polarized states half of the atoms occupy the state with s z =0 and the rest stay on the state with s z =1 (or -1). Partially polarized states with s z =1 (or -1) and s z =0.

The compressibility at the ground state At the critical point, the compressibility is divergent. Compressibility can also be use to quantify the phase transition.

Entropy of the model with different temperature and external magnetic field

Scaling behavior near the critical point Grand canonical ensemble Near critical point, density of particle number satisfies scaling law when u<-|h|, the system is the vacuum state; when -|h| 0 (h<0), it is the fully upper (lower) polarized state; when u>0, it is the partially polarized state.

su(4) so(5) III. Anisotropic spin-3/2 Fermi gas Hamiltonian c c -c -c -c c so(4) so(3)=su(2) ?????

P. Schlottmann, PRL 68, 1916 (1992) Interacting fermions then form a spin singlet and orbital triplet (attractive interaction) or spin triplet and orbital singlet (repulsive interaction).

Symmetry =SO(4)

Spin-flipping operators

Conserved quantities

Wave function Scattering matrix Yang-Baxter equation

Bethe ansatz equations Energy and momentum Magnetization

spin changing -2 spin changing -2

|3/2> can not be directly flipped into |-3/2>

Ground state Solution of Bethe ansatz equations at T=0 k: 2 string & u: 2 string |2,2>, |2,-2> &|2,0> : singlet pair |2,2>|2,-2>+|2,-2>|2,2>-|2,0>|2,0>/3 1/2 |1,-1> + |-1,1> - |0,0>/3 1/2 Spin-1 singlet Not the four particle spin-singlet state

String hypothesis C C C C Real Imag Finite temperature

Integral Bethe ansatz equations for densities

Gibbs free energy Minimize the Gibbs free energy, we obtain the TBA

Define the dressed energy as Thermodynamic Bethe ansatz equations

Ground state h is not zero h = 0 k: 2 string & u: 2 string

Phase diagram: c>0 Vacuum Fully polarized state |3/2> Partially polarized state |3/2> & |2,2> Magnetic pair |2,2> k: 2-string, |2,2>, |2,-2> &|2,0> : singlet pair (B) (C) (D) k: real & 2-string, |2,2>, |2,-2>, |2,0>, |3/2>.

Phase diagram: c<0 Vacuum Fully polarized state |3/2> Partially polarized state |3/2> & |2,1> Magnetic pair |2,1> k: 2-string, |2,1>, |2,-1> &|0,0> : singlet pair (B) (C) (D) k: real & 2-string, |2,1>, |2,-1>, |0,0>, |3/2>.

Elementary excitation (h=0)

Effective interaction in partial channels

Discussions

Issue I: exactly solvable model of multicomponent anyons k=0, boson k=π, fermion

Boson: Lieb & Liniger, Phys. Rev. 130 (1963) 1605; Lieb, Phys. Rev. 130 (1963) Li, Gu, Ying & Eckern, Europhys. Lett. 61 (2003) 368. Zhou, J. Phys. A: Math. Gen. 21 (1988) 2391; Zhou, J. Phys. A: Math. Gen. 21 (1988) Fermion: Gaudin, Phys. Lett. A 24 (1967) 55. Yang, Phys. Rev. Lett. 19 (1967) Sutherland, Phys. Rev. Lett. 20 (1968) 98. Single-component anyon Kundu, Phys. Rev. Lett. 83, 1275 (1999). Batchelor, Guan & Oelkers, Phys. Rev. Lett. 96, (2006). Girardeau, Phys. Rev. Lett. 97, (2006). Hao, Zhang & Chen, Phys. Rev. A 78, (2008). Limit case I

c=0 & c=infinity Limit case II Unified exactly solvable models

Issue II: exactly solvable model for the cold atoms with gauge potential in a transverse field 1.Two-component bosons. For the fermion, the symmetry of wavefunction maybe wrong due to transverse field. 2. The particle numbers of pseudo-spin-up and down are not conserved. The total number of particle is conserved. 3. For the fermion, if the second order derivative is zero, the system is equivalent to the massive Thirring model and can be solved exactly. Where Ψ is the Dirac field operator. Done by Lijun Yang

First we set the interaction is zero, that is U = 0. Take the Fourier transformation In the momentum space, we have Bogoliubov transformation

To eliminate the nondiagonal term, we require

To diagonalize the total Hamiltonian H, for a 2-body system, assume the wavefunction has the following form: Compare with Bethe ansatz wave function Where

By acting the Hamiltonian on the wave function, we can get three parts: If,then

Hamiltonian acting on the assumed states, all the unwanted terms of H0 and interaction are canceled with each other. One set of the Bethe ansatz equations in the lower band.

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