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

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.