## Time-Reversal Symmetry in Non-Hermitian Systems

Masatoshi Sato, Kazuki Hasebe, Kenta Esaki, Mahito Kohmoto

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 $$\Theta^2=-1$$, but no such a degeneracy exists when $$\Theta^2=+1$$. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when $$\Theta^2=+1$$. 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 NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.

http://arxiv.org/abs/1106.1806
Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Quantum Physics (quant-ph)