2011/06/30 by Masatoshi Sato, Kazuki Hasebe, Kenta Esaki +1 · 4 citations
Mathematics · Physics and Astronomy · #cond-mat.stat-mech #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1143/ptp.127.937
published as Prog. Theor. Phys. 127 (2012), 937-974 · 40 pages, 2 figures; typos fixed, references added
arxiv created 2011/07/15 · arxiv updated 2012/07/03
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 Θ2=-1, but no such a degeneracy exists when Θ2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when Θ2=+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.