One-dimensional spinless fermions in a confined system

According to the exact analytical Bethe-ansatz solution of the one-dimensional spin-1/2 XXZ model, we perform a numerical study of one-dimensional spinless fermions in an optical lattice in the presence of harmonic potential by using a local density approximation. We study the density profile, and obtain the (p) over tilde -u phase diagram (here (p) over tilde = N-f root V-2/t is the characteristic dimensionless density and u = V/t is the interaction strength scaled in units of the hopping parameter t). With the increases of particle density and nearest-neighbour interaction, the system undergoes five different quantum phases. Through calculating the thermodynamic stiffness S-p, we find that it can be used as an order parameter. Its singular points can measure the quantum phase transitions in such a confined system.