Keywords: 
• 热力学Bethe-ansatz方程 /
• 有限温度 /
• Gaudin-Yang模型 /
• 量子临界性质

Abstract: The one-dimensional system interacting via a delta-function interparticle interaction is a very important one in cold atomic systems and has fundamental importance in many-body physics. In one dimension, due to the geometric confinement induced quantum correlations and quantum fluctuations, there may exist a number of unusual phenomena, such as spin-charge separation, effective fermionization and quantum criticality. This paper studies the basic properties of a uniform one-dimensional Gaudin-Yang model for fermions by solving the thermodynamic Bethe-ansatz equations by a numerical method. Numerically, we use the many-variable Newton’s method to solve the coupled equations. We analyze the physical properties, including density, interaction, temperature and entropy at a given temperature and a given interaction, separately. We know that a lot of researches are limited to zero temperature. However, we cannot reach the absolute zero temperature in the real cold atomic experiment. So it is important to deal with the finite temperature problems. We study the density and entropy as a function of the chemical potential, temperature and interaction and, then give the phase diagrams, respectively. We found that there is a quantum critical zone in the phase diagram of entropy, including the high temperature zone with thermal fluctuations and the Luttinger liquid zone with quantum fluctuations. For a given temperature and low chemical potential, the thermal fluctuations are the main factor in the entropy. With the increase of chemical potential, the system enters the quantum critical zone where the competitive effect between the thermal fluctuations and the quantum fluctuations exists. When the chemical potential is large enough, the quantum fluctuations become the main factor in the system’s entropy, and we get the Luttinger liquid phase. Our results can be further used in the finite temperature density-functional theory and to analyze the collective phenomena at a finite temperature.