Computing Floquet Hamiltonians with Symmetries

Unitary matrices arise in many ways in physics, in particular as a time evolution operator. For a periodically driven system one frequently wishes to compute a Floquet Hamilonian that should be a Hermitian operator $H$ such that $e^{-iTH}=U(T)$ where $U(T)$ is the time evolution operator at time corresponding the period of the system. That is, we want $H$ to be equal to $-i$ times a matrix logarithm of $U(T)$. If the system has a symmetry, such as time reversal symmetry, one can expect $H$ to have a symmetry beyond being Hermitian. We discuss here practical numerical algorithms on computing matrix logarithms that have certain symmetries which can be used to compute Floquet Hamiltonians that have appropriate symmetries. Along the way, we prove some results on how a symmetry in the Floquet operator $U(T)$ can lead to a symmetry in a basis of Floquet eigenstates.