Multi-component integrable models in cold atoms

The quantum gases are intensively studied for the inspiring advances in ultra-cold atomic physics. Various kinds of lattice and gas systems are created in different dimensions. The multi-component cold atomic systems have rich phase diagrams. Our works focus on the one-dimensional integrable quantum systems. We find a series of integrable models of $Sp(2s+1)$ fermions and $SO(2s+1)$ bosons and solve them via Bethe ansatz techniques, where $s$ the hyperfine spin of the atoms. We find the paired bosons exist in both repulsive and attractive $SO(3)$ integrable bosonic gases with hyperfine spin-1. For the $Sp(2s+1)$ repulsive fermions, there are no bound states in the ground state, while 2-string bound solutions appear in the spin sector. We also calculate the spin-wave velocities and low temperature specific heat of the repulsive fermions. These systems have spin-charge separation property of Luttinger liquid. Different from the $SU(2s+1)$ repulsive models, the spin-wave velocities of the $Sp(2s+1)$ models are no longer the same. The holes of 1-strings (real rapidities) of different branches in the spin sector have a same spin-wave velocity $v_1$ and the ones of 2-strings share a same spin-wave velocity $v_2$. The two velocities are different.