Dispersion Relation Of Strongly Interacting Fermion Matter And Low Temperature Expansion Of Thermodynamics

With Feshbach resonance technology development, the many-body community raises the unprecedented research upsurge to ultra-cold atomic fermions, especially for the condensed thermodynamic properties of strong interaction unitary Fermi system. For the interaction of quantum many-body problems, the basic content is the dispersion relation of the excitation mode. As the Fermi gas is polarized, the exotic phase separation of a two-component Fermi gas has been paid much attention.Firstly, we analyze the dispersion relation of the excitation mode in non-relativistic interacting fermion matter. The polarization tensor is calculated with the random phase approximation in the framework of finite temperature field theory. The influences of the temperature, particle number density and interaction strength on the dispersion relation determined by the polarization tensor are discussed in detail. It is found that the collective effects are qualitatively more important in the unitary fermions than those in the finite contact interaction matter.In addition to the fundamental ground state energy or the universal constantξ, the Landau effective mass m~* is an another universal parameter, which is usually considered to be determined experimentally. In this thesis, we want to make an analytical attempt to fix the ratio of the effective mass m~* to the bare fermion mass m. Based on our previous works for unitary Fermi gas, the low temperature expansions are performed for the thermodynamic quantities such as the energy density, chemical potential and pressure. For the ultra-cold unitary Fermi system, compared to the low temperature expansion of the strong interaction of pressure, we can obtain the universal constant B of the equation of state for symmetric with equal up and down fermions. According to the thermodynamic properties and Landau theory of Fermi-Liquid, as one of the Landau parameters, the effective fermion mass is m~*/m = 10/9. This exact result is in agreement with some quantum Monte-Carlo attempts.