Modified discrete Laguerre polynomials for efficient computation of exponentially bounded Matsubara sums

TL;DR

This paper introduces modified discrete Laguerre polynomials to efficiently compute exponentially bounded Matsubara sums, significantly improving convergence rates for applications like finite-temperature Casimir force calculations.

Contribution

The authors develop a new orthogonal polynomial-based quadrature scheme that accelerates Matsubara sum computations with nearly temperature-independent convergence.

Findings

Achieves rapid convergence for Matsubara sums with exponentially decaying summands.

Demonstrates effectiveness in quantum field theory applications such as Casimir force calculations.

Proves convergence for polynomially decaying functions.

Abstract

We develop a new type of orthogonal polynomial, the modified discrete Laguerre (MDL) polynomials, designed to accelerate the computation of bosonic Matsubara sums in statistical physics. The MDL polynomials lead to a rapidly convergent Gaussian "quadrature" scheme for Matsubara sums, and more generally for any sum $F(0)/2 + F(h) + F(2h) + \cdots$ of exponentially decaying summands $F(nh) = f(nh)e^{-nhs}$ where $hs>0$. We demonstrate this technique for computation of finite-temperature Casimir forces arising from quantum field theory, where evaluation of the summand $F$ requires expensive electromagnetic simulations. A key advantage of our scheme, compared to previous methods, is that the convergence rate is nearly independent of the spacing $h$ (proportional to the thermodynamic temperature). We also prove convergence for any polynomially decaying $F$.