Thermodynamics, density profiles, and correlation functions of the inhomogeneous one-dimensional spinor Bose gas

We investigate the finite-temperature properties of the one-dimensional two-component Bose gas (2CBG) with repulsive contact interaction in a harmonic trap. Making use of a lattice embedding for the 2CBG and the quantum transfer matrix, we derive a system of two nonlinear integral equations characterizing the thermodynamics of the uniform system for all values of the relevant parameters: temperature, strength of the interaction, chemical potential, and magnetic field. This system allows for an easy numerical implementation in stark contrast with the infinite number of equations obtained by employing the thermodynamic Bethe ansatz. We use this exact solution coupled with the local density approximation to compute the density profiles and local density correlation function of the inhomogeneous gas for a wide range of coupling strengths and temperatures. Our results show that the polarization in the center of the trap influences heavily the local correlator especially in the experimentally accessible Tonks-Girardeau regime.