On the Efetov-Wegner terms by diagonalizing a Hermitian supermatrix

TL;DR

This paper derives the supermatrix Bessel function including all Efetov-Wegner terms for Hermitian supermatrices, providing explicit formulas crucial for random matrix theory applications.

Contribution

It presents the first explicit and compact derivation of the supermatrix Bessel function with all Efetov-Wegner terms for arbitrary rotation invariant densities.

Findings

Derived the supermatrix Bessel function with Efetov-Wegner terms

Applied results to generating functions of Hermitian random matrices

Provided explicit formulas for integrals over eigenvalues

Abstract

The diagonalization of Hermitian supermatrices is studied. Such a change of coordinates is inevitable to find certain structures in random matrix theory. However it still poses serious problems since up to now the calculation of all Rothstein contributions known as Efetov-Wegner terms in physics was quite cumbersome. We derive the supermatrix Bessel function with all Efetov-Wegner terms for an arbitrary rotation invariant probability density function. As applications we consider representations of generating functions for Hermitian random matrices with and without an external field as integrals over eigenvalues of Hermitian supermatrices. All results are obtained with all Efetov-Wegner terms which were unknown before in such an explicit and compact representation.