Computing a logarithm of a unitary matrix with general spectrum

TL;DR

This paper presents a simple, robust algorithm for computing the logarithm of a unitary matrix that works well even with challenging spectra, including cases with eigenvalues near -1, and extends to special matrix classes with applications in topological insulators.

Contribution

It introduces a straightforward algorithm for skew-Hermitian logarithms of unitary matrices that avoids accuracy issues common in existing methods, including modifications for J-skew symmetric matrices.

Findings

Algorithm performs well without spectral restrictions.

Handles matrices with eigenvalues near -1 effectively.

Applicable to topological insulator studies.

Abstract

We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix. This algorithm is very easy to implement using standard software and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near -1, lead to very non-Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force an Hermitian output creates accuracy issues which are avoided in the considered algorithm. A modification is introduced to deal properly with the $J$-skew symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed.