Time-reversal and its effect in the eigenfunctions of the Floquet operator

Question

When time-reversal is present in a kicked Hamiltonian system (e.g. kicked rotator) it can be shown that there exists a basis in which the associated Floquet operator $U$ is symmetric. One can then try to find the eigenvalues and eigenvectors of this operator

$$ U |v\rangle = e^{i\phi}|v\rangle. $$

If time-reversal holds, is it possible to classify the eigenfunctions in muttually exclusive sets? By this I mean the following: for example when parity is present in Hamiltonian system, then for an eigenfunction $|w\rangle$ of the Hamiltonian $H$ one can classify the eigenfunctions as even or odd according to the parity operator $P$, i.e.

$$ P|w\rangle = \pm |w\rangle. $$

Can something similar exists for time-reversal?

Answer 1

There are two types of time reversal invariance. One leads to the Floquet operator $U$ being a symmetric unitary matrix, and it will have real eigenvectors. The other leads to a different symmetry in $U$ that has eigenvectors that appear in time-reversal pairs. This means one can find a Floquet Hamiltonian (a static Hamiltonian) that has the correct time reversal symmetry.

Full details are in a recent paper by myself and Fredy Vides [1].

[1] Loring, Terry A., and Fredy Vides. "Computing Floquet Hamiltonians with symmetries." Journal of Mathematical Physics 61.11 (2020): 113501.