Computing Number Fluctuations

TL;DR

This paper analyzes the properties of a function related to momentum fluctuations in many-fermion systems, establishing formal connections and sum rules without relying on a specific Hamiltonian, aiding in spectral function computations.

Contribution

It introduces a formal analysis of the fluctuation function N(k,k') and relates it to momentum distributions in interacting and non-interacting fermion systems, independent of specific Hamiltonians.

Findings

N(k,k') satisfies elegant sum rules

The function is singular in some respects

Connections to spectral functions are established

Abstract

Here we try and delienate the properties of the function that corresponds to fluctuations in the momentum distribution. The quantity denoted by $ N(k,k^{'}) $ is quite an interesting object. It satisfies various elegant sum rules and is also quite singular in some respects. All these properties are brought out and a formal connection is found between this object and the momentum distribution of the interacting and non-interacting many-fermion systems. This exercise is quite general in that it does not refer to any particular hamiltonian. It is also quite useful since in an earlier preprint(cond-mat/9810043) we showed how to compute the spectral function and single-particle lifetime of homogeneous Fermi systems where the only undetermined quantity was this function $ N(k,k^{'}) $.