Exact Solution Of One-Dimensional Fermionic Gases With Attractive P-Wave Interaction

Quantum many-body physics is an important branch and an important research direction in the development of physics today.In quantum many-body systems,due to the complex interactions and huge internal degrees of freedom between particles,it is difficult for people to obtain the intrinsic energy spectrum of the system by directly solving the Hamiltonian.In high-dimensional systems,physicists have developed a series of theoretical methods such as ‘Fermi Liquid Theory’ and ‘Mean Field Theory’to calculate and describe multi-body systems;while in one-dimensional systems,due to its spatial dimensions the particularity of the high-dimensional system and the failure of the theory of the high-dimensional system.Based on the universal characteristics of the one-dimensional multi-body system,that is,the collective excitation mode and the linear excitation spectrum under low-energy scattering,people have developed the ‘Luttinger Liquid Theory’ to the excitation of one-dimensional multi-body system is accurately described.In a one-dimensional multi-body system,there is a special model that can be rigorously solved by Bethe Ansatz,so that the Luttinger liquid properties and other physical properties of the system can be described on the basis of accurate solutions.This type of model is called a one-dimensional strictly solvable model,also called a quantum integrable model.In recent years,with the rapid development of cold atom experimental technology,a‘quasi-one-dimensional’ multi-body physical system has been realized in the experiment,and the interaction strength between particles can be freely controlled through the ‘Feshbach resonance’.Many important theoretical predictions made in one-dimensional rigorously solvable models have been experimentally verified.The close combination of quantum integrable model and experimental discovery makes theoretical research in this field aroused the interest and attention of many scholars.The Fermi gas model of one-dimensional p-wave interaction is a special integrable model.In this system,there are two parameters that characterize the interaction between particles,‘scattering length’ and ‘effective range’.With the participation of the acting parameters,the model forms interaction areas of different strengths,which embodies rich and interesting physical connotations.In this paper,the Bethe Ansatz method is used to obtain the rigorous solution of the Fermi gas model under the one-dimensional p-wave attraction interaction.On the basis of the rigorous solution,the ground state,excited state and finite temperature properties of the system are discussed.The main research work is as follows:1.The pseudopotential of one-dimensional p-wave interaction Fermi gasStarting from the scattering theory of the one-dimensional system,the relationship between the contact boundary conditions of the one-dimensional p-wave interaction Fermi gas and the Hamiltonian pseudopotential is analyzed,and pseudopotential expression of the system both considering the interaction of the scattering length and the effective force range is obtained.2.Fermi gas under one-dimensional p-wave repulsive interactionThrough the interaction boundary conditions of the one-dimensional p-wave interaction Fermi gas model,the condition that the wave function of the system is in a bound state in the case of two-particle scattering is discussed,and present the research limitations of one-dimensional Fermionic gases with p-wave repulsive interaction.3.Rigorous solutions of Fermi gas under one-dimensional p-wave attraction interactionThrough the interaction boundary conditions of the one-dimensional p-wave interaction Fermi gas model and the introduction of periodic boundary conditions,the detailed derivation and demonstration of how to use the Bethe Ansatz method to obtain the Bethe Ansatz equations of the system is given,and present the Bethe ansatz equations for quasi-momentum of one-dimensional p-wave interacting Fermi gas under attractive potential.On the basis of the obtained Bethe Ansatz equations,the characteristics and properties of the rigorous solutions of the system are discussed.4.Ground state properties of Fermi gas under one-dimensional p-wave attraction interactionOn the basis of the rigorous solution,the system is under the thermodynamic limit.Using the continuous integral Bethe Ansatz equation,the ground state physical properties of the system under the interaction of the scattering length and the effective force range are given,such as a single particle Energy,Fermi point,and ground state energy density functions.At the same time,the quasi-momentum distribution density function when the scattering length is very large(that is,the strong interaction region discussed in this article based on the scattering length)is studied,and the corresponding analysis is given,it is found that it satisfies the law of semicircle,which reflects the distribution law of particles in the system.5.Finite temperature properties of Fermi gas under one-dimensional p-wave attraction interactionThrough the standardized Yang-Yang thermodynamics method,the derivation process of the thermodynamic Bethe Ansatz equation(TBA equation)of Fermi gas under one-dimensional p-wave attraction interaction is shown in detail,and the pressure expression of the system is given by the TBA equation.The pressure can then be used to obtain other thermodynamic quantities through the relationship in thermodynamic statistics.At the same time,this section also gives the chemical potential of the system based on the quasi-momentum distribution density function and the conjugate potential energy.6.Excited state properties of Fermi gas under one-dimensional p-wave attraction interactionBy analyzing the collective motion mode of the multi-body system under the one-dimensional system,the integral expressions describing the momentum and energy of the excited state of the system are obtained.Based on the integral Bethe Ansatz equation and the TBA equation of the system,the relationship between the momentum and energy in the state,the quasi-momentum distribution density and the conjugate potential energy,and the characteristics of the excitation spectrum in the region of different interaction strength under the participation of the two interaction parameters of the system are also given.