On the evaluation of Matsubara sums

TL;DR

This paper introduces a method to evaluate complex multidimensional Matsubara sums, associated with graphs, in closed form by applying a linear operator to related integrals, simplifying their computation.

Contribution

It provides a novel approach to evaluate Matsubara sums in closed form using linear operators on associated integrals, applicable to arbitrary graphs.

Findings

Matsubara sums can be expressed as linear operators on integrals.

The method applies to sums constructed from arbitrary connected graphs.

Closed-form evaluations of Matsubara sums are achievable with this approach.

Abstract

Given a connected (multi)graph G, consisting of V vertices and I lines, we consider a class of multidimensional sums constructed in the following way: - orient the lines of the graph in some (arbitrary) fashion - assign to each line i a positive variable q_i and an integer summation variable n_i - assign to each vertex v an integer variable N_v - construct the following rational function: -- the denominator is a product of factors (n^2+q^2), one for each line of the graph; -- the numerator is a product of Kronecker deltas, one for each vertex of the graph. For each vertex, the Kronecker delta imposes a linear constraint among the summation variables n_i of the lines incident upon the vertex, requiring that the sum of the variables n_i of the lines coming out of vertex minus the sum of the variables n_i of the lines coming into the vertex be equal to the integer variable N assigned to that vertex. - sum over all the n_i variables from minus infinity to infinity The sums thus constructed, called Matsubara sums, are functions of the I real positive variables q_i and the V integer variables N_v. It is shown any Matsubara sum can be evaluated in closed form by applying a linear operator to an integral closely associated with the sum.