Quantum phase transition and elementary excitations of a Bose-Fermi mixture in a one-dimensional optical lattice

Abstract

The mixture of the scalar bosonic and the spinless or polarized fermionic cold atoms in the one-dimensional optical lattice is studied. The system is modeled by the Bose-Fermi-Hubbard Hamiltonian, which shows different behavior from that of the Bose-Hubbard or the Fermi-Hubbard models. Because the $\text{SU}(1\ensuremath{\mid}1)$-supersymmetric Bethe ansatz solution gives an excellent approximation to this kind of mixed cold atomic systems, the ground-state properties of the system such as the densities of state in the momentum space are obtained based on the Bethe ansatz. If the number of bosons is equal to that of fermions and the filling factor is 1, it is found that there exists a critical on-site interaction ${U}_{c}$. If $U<{U}_{c}$, the ground state of the system is in the superfluid phase, while if $U>{U}_{c}$, the ground state is in the insulating phase. The superfluid-insulator transition occurs at ${U}_{c}$. From the analysis of the superfluid density, the value of the critical point is determined as ${U}_{c}=2.79256$, which is larger than the ${U}_{c}=0$ for the Fermi-Hubbard model and smaller than the ${U}_{c}=3.28$ for the Bose-Hubbard model. The elementary excitations and effective Hamiltonian in the strong-coupling limit are also discussed.