Polynomial method for canonical calculations

TL;DR

This paper introduces a polynomial formalism for efficiently calculating the canonical partition function in mesoscopic electron systems, simplifying complex many-body calculations to be comparable with grand canonical methods.

Contribution

A practical polynomial canonical formalism is developed that simplifies calculations for large N-electron systems, avoiding limitations on system size and temperature.

Findings

Exact canonical partition function expressed as a series of many-particle excitations.

Number of terms needed remains below 10 for T<10 inter-level spacings, even for N~100000.

Method makes canonical calculations as simple as grand canonical ones.

Abstract

A practical version of the polynomial canonical formalism is developed for normal mesoscopic systems consisting of N independent electrons. Drastic simplification of calculations is attained by means of proper ordering excited states of the system. In consequence the exact canonical partition function can be represented as a series in which the first term corresponds to the ground state whereas successive groups of terms belong to many particle-hole excitations (one particle-hole two particle-hole and so on). At small temperatures (T<10 inter-level spacings near the Fermi level) the number of terms which should be taken into account is weakly dependent on N and remains <10 even if N~100000. The elaborated method makes canonical calculations to be not more complicated than the grand canonical ones and is free from any limitations on N and T.