Finite-Temperature Techniques

Abstract

Calculating the properties of a system of correlated electrons at finite temperatures is a demanding task. One must be able to describe the effect of correlations not only on the ground state but also on the low-energy excitations of the system. Of particular interest is the partition function Z or the free energy F of the electrons. We can deduce from both quantities a number of static thermodynamicproperties. The partition function is the trace of an operator

$$U\ (\beta)={\rm e}^{-\beta H}$$

((7.0.1))

where β = 1/k _~B T and k _B denotes Boltzmann’s constant. If one sets β = it, then U (it) is the time evolution operator of the system. When it is applied to a state |Φ⟩ it describes how this state evolves with time. This suggests treating β as imaginary time, which has the advantage of enabling us to develop a perturbation theory for finite temperatures, a straightforward generalization of the perturbation treatment of the time evolution operator at T = 0.