Source Galerkin Calculations in Scalar Field Theory

TL;DR

This paper applies the Source Galerkin method with polynomial expansions to scalar  theory, efficiently computing propagators and mass gaps on small lattices, while discussing limitations and alternative sparse matrix techniques.

Contribution

It extends the Source Galerkin method to scalar  theory using polynomial expansions and introduces a sparse matrix alternative for improved computational efficiency.

Findings

Calculations are rapid compared to Monte Carlo methods.

Polynomial expansions reveal limitations on system size due to computational complexity.

An alternative sparse matrix approach is proposed for the Galerkin procedure.

Abstract

In this paper, we extend previous work on scalar $\phi^4$ theory using the Source Galerkin method. This approach is based on finding solutions $Z[J]$ to the lattice functional equations for field theories in the presence of an external source $J$. Using polynomial expansions for the generating functional $Z$, we calculate propagators and mass-gaps for a number of systems. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. The use of polynomial expansions illustrates in a clear and simple way the ideas of the Source Galerkin method. But at the same time, this choice has serious limitations. Even after exploiting symmetries, the size of calculations become prohibitive except for small systems. The calculations in this paper were made on a workstation of modest power using a fourth order polynomial expansion for lattices of size $8^2$,$4^3$,$2^4$ in $2D$, $3D$, and $4D$. In addition, we present an alternative to the Galerkin procedure that results in sparse matrices to invert.