Thermodynamic and dynamic properties of many body boson and fermion system

Abstract

In part Ⅰ, we developed a new method that properly elucidates the effects of temperature on the free electron system and does not require the restriction of the Sommerfeld expansion method. Our method is based on the observation that the Fermi distribution function has an infinite number of poles in the complex momentum plane at finite temperature. By applying the Euler-Maclaurin formula we transform the residue contributions from the poles into the colsed form which is accurate up to the several times of the Fermi temperature $T_F$. Using our method we obtained the analytical results of the temperature dependent free electron susceptibilities $χ_d(q,T)$ and $χ_d(r,T)$ for all dimensiion $(d=1,2,3)$. Our analytical results show that irrespective of dimension, the singular behavior of $χ_d(q)$ at $q=±2k_F$ becomes suppressed at nonzero temperature as the singular points transit to complex wave vectors $2k_0^±$, and this transition makes $χ_d(r)$ to be exponentially damped with common damping factor $e^{-2η_0 sin φ_r k_F r~e^{-π T′ k_F r}$ for low enough temperature, where the exponent of damping factor corresponds to an imaginary part of wave vectors $2k_0^±$. Furthermore we showed that the poles of the Fermi distribution function are identical with those of the free electron temperature Green``s function by extending our theory to the Green``s function formalism. Finally we calculate the temperature dependent free electron dynamic response function for three dimension, which plays a very important role in the theory of the electron correlation. In part Ⅱ, we calculate the thermodynamic properties of 2D interacting Bose gases in semiclassical Hartree-Fock model and demonstrate the importance of the consistent treatment of mutual interaction between the condensate and the thermal component in determing the critical behavior of 2D Bose-Einstein condensation (BEC). In contrast to the case where the interaction from the thermal component is neglected, our results s...