Monte-Carlo sampling of self-energy matrices within sigma-models derived from Hubbard-Stratonovich transformed coherent state path integrals

TL;DR

This paper presents a Monte-Carlo sampling method for calculating Green functions in Hubbard-Stratonovich transformed path integrals, avoiding direct matrix inversion by using random walks in self-energy matrices, applicable to sigma models and gauge theories.

Contribution

Introduces a Monte-Carlo sampling technique for self-energy matrices in sigma models, enabling efficient Green function computation without direct matrix inversion.

Findings

Effective around saddle point solutions of self-energy in sigma models.

Capable of sampling HS-transformed path integrals in fermionic gauge theories.

Applicable to matrices with eigenvalues less than one for Taylor expansion.

Abstract

The 'Neumann-Ulam' Monte-Carlo sampling is described for the calculation of a matrix inversion or a Green function in case of Hubbard-Stratonovich (HS-)transformed coherent state path integrals. We illustrate how to circumvent direct numerical inversion of a matrix to its Green function by taking random walks of suitably chosen matrices within a path integral of even- and complex-valued self-energy matrices. The application of a random walk sampling is given by the possible separation of the total matrix, e.g. that matrix which determines the Green function from its inversion, into a part of unity minus (or plus) a matrix which only contains eigenvalues with absolute value smaller than one. This allows to expand the prevailing Green function around the unit matrix in a Taylor expansion with a separated, special matrix of sufficiently small eigenvalues. The presented sampling method is particularly appropriate around the saddle point solution of the self-energy in a sigma model by using random number generators. It is also capable for random sampling of HS-transformed path integrals from fermionic fields which interact through gauge invariant bosons according to Yang-Mills theories.