Computing a logarithm of a unitary matrix with general spectrum
For researchers needing robust matrix logarithms in numerical studies of topological insulators, this algorithm solves a known accuracy bottleneck.
The paper presents an algorithm for computing a skew-Hermitian logarithm of a unitary matrix that works well even with eigenvalues near -1, avoiding accuracy issues of other methods. It also introduces a modification for J-skew symmetric unitary matrices, with applications to topological insulators.
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.