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. When a grand-canonical ensemble is considered the free energy is replaced by the thermodynamic potential Ω. We can deduce from those quantities a number of static thermodynamic properties. The partition function is the trace of an operator

$$ U\left( \beta \right) = {{\text{e}}^{{\text{ - }}\beta {\text{H}}}},$$

(1)

where β = l/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.