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, Θ} = 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.
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.
In a recent paper [M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012)] Sato and his collaborators established a generalization of the Kramers degeneracy structure to pseudo-Hermitian Hamiltonian systems, admitting even time-reversal symmetry, |
T2=1|. This extension is achieved using the mathematical structure of split-quaternions instead of quaternions, usually adopted in the case of Hermitian Hamiltonians with odd time-reversal symmetry, |
T2=−1|. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian |
H| that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such |
H| with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart |
h|, so that the generalized Kramers degeneracy of |
H| is in fact crypto-Hermitian Kramers degeneracy. We study in greater detail a new example of the pseudo-Hermitian split-quaternionic four-level Hamiltonian system, which admits an indefinite metric operator and time-reversal symmetry and, as a consequence, a generalized Kramers degeneracy structure. We provide a complete solution of the eigenvalue problem, construct pseudo-Hermitian ladder operators closing the normal and abnormal pseudo-fermionic algebras, and show that this system fulfills a crypto-Hermitian degeneracy.
We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under
PT, where
P is the parity and
T denotes time reversal, for the case that time-reversal symmetry is odd (${\mathcal{T}}^{2}=\ensuremath{-}1$), generalizing prior work for the even case (
T2=1). We discover an analog of Kramer's theorem for
PT quantum mechanics, present a prototypical example of a
PT quantum system with odd time reversal, and discuss potential applications of the formalism.
Kramers' theorem ensures double degeneracy in the energy spectrum of a time-reversal symmetric fermionic system with half-integer total spin. Here, we are trying to go beyond the closed system and discuss Kramers' degeneracy in open systems out of equilibrium. A natural way to extend the Kramers' degeneracy in open quantum systems is by the degeneracy of different spins' spectra together with the vanishing interspin spectrum. We find the violation of Kramers' degeneracy in time-reversal symmetric open quantum systems is locked with whether the system reaches thermal equilibrium. After a general coupling to an environment in a time-reversal symmetry-preserving way, the Kramers doublet experiences an energy splitting at a short time and then a recovery process. We verified the violation and revival of Kramers' degeneracy in a concrete model of interacting fermions and we find Kramers' degeneracy is restored after the local thermalization time. By contrast, for time-reversal symmetry $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}$ with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}}^{2}=1$, we find that although there is a violation and revival of spectral degeneracy for different spins, the inter-spin spectral function is always nonzero. We also prove that the degeneracy in spectral function protected by unitary symmetry can always be maintained.
This thesis examines the quantum-mechanical geometric phase with a view toward time reversal symmetry considerations. The idea of time reversal in quantum mechanics is investigated, disagreements and inconsistencies in the literature are examined, and the action of the time reversal operator is extended to time-dependent Hamiltonians. With this background, and using a definition of time reversal symmetry based on the evolution operator, I demonstrate that the existence of a non-zero geometric phase can in all cases be attributed to a breakdown of time reversal symmetry in some form. This result holds for both adiabatic and nonadiabatic evolutions, and for arbitrary dimensional parameter spaces. I explore the role of the geometric phase in a two-level Kramers system described by a parameter-dependent Hamiltonian such that the two levels can become degenerate for some value of the parameters, and discuss, from a mathematical point of view, the monopole geometric potential that results. I then extend this analysis by considering a pair of Kramers doublets, each doublet degenerate due to time reversal symmetry, where the parameters can be chosen so that each of the pair of doublets becomes degenerate with the other. I find the explicit functional forms for the two resulting nonabelian geometric gauge potentials and show that they can be identified exactly with the only two gauge-inequivalent SU(2) monopole potentials of Yang. Furthermore, following a conformal transformation these potentials can be mapped to those of the SU(2) instanton/anti-instanton pair. Finally I examine the relevance of the geometric phase to the molecular physics of time-odd systems. Time-odd coupling in molecular physics is a much under-studied area, with many potentially interesting results. Specifically I study time-odd coupling in Jahn-Teller systems under the Born-Oppenheimer approximation, where the electronic position states are coupled to the lattice momentum rather than the usual time-even lattice position. As an example I solve the E ⊗ (b₁ ⊕ b₂ ⊕ a₂) Jahn-Teller system exactly, showing that once again monopole-like geometric potentials arise, and comment on how this affects the angular momentum of the lattice subsystem.