Time-reversal symmetry in non-Hermitian systems

Abstract

For ordinary Hermitian Hamiltonians, the states support the mathematical structure of quaternion and show the Kramers degeneracy when the system has a half-odd-integer spin and the time-reversal operator obeys Θ 2 = - 1, while no such structure exists when Θ 2 = +1. Here, we point out that for non-Hermitian systems, there exists a similar mathematical structure called split-quaternion even when Θ 2 = +1. It is also found that a degeneracy similar to the Kramers degeneracy follows from the split-quaternion structure if the Hamiltonian is pseudo-Hermitian with the metric operator η satisfying {η,Ζ} = 0. Furthermore, we 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.