Time-Reversal Symmetry in Non-Hermitian Systems

For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys � 2 = −1, but no such a degeneracy exists when � 2 = +1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when � 2 = +1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split-quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail N × N Hamiltonians with N = 2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.