Class Of non-Hermitian Hamiltonians Research Articles

Non-Hermitian Spectral Flows and Berry-Chern Monopoles

We propose a non-Hermitian generalization of the correspondence between the spectral flow and the topological charges of band crossing points (Berry-Chern monopoles). A class of non-Hermitian Hamiltonians that display a complex-valued spectral flow is built by deforming an Hermitian model while preserving its analytical index. We relate those spectral flows to a generalized Chern number that we show to be equal to that of the Hermitian case, provided a line gap exists. We demonstrate the homotopic invariance of both the non-Hermitian Chern number and the spectral flow index, making explicit their topological nature. In the absence of a line gap, our system still displays a spectral flow whose topology can be captured by exploiting an emergent pseudo-Hermitian symmetry.

On a class of non-Hermitian Hamiltonians with tridiagonal matrix representation

We show that some non-Hermitian Hamiltonian operators with tridiagonal matrix representation may be quasi Hermitian or similar to Hermitian operators. In the class of Hamiltonian operators discussed here the transformation is given by a Hermitian, positive-definite, diagonal operator. We show that there is an important difference between open boundary conditions and periodic ones. We illustrate the theoretical results by means of two simple, widely used, models.

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 \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.

Supersymmetric Extension of Non-Hermitian su(2) Hamiltonian and Supercoherent States

A new class of non-Hermitian Hamiltonians with real spectrum, which are written as a real linear combination of su(2) generators in the form $ H=\omega J_{3}+\alpha J_{-}+\beta J_{+}$, $\alpha \neq \beta$, is analyzed. The metrics which allows the transition to the equivalent Hermitian Hamiltonian is established. A pseudo-Hermitian supersymmetic extension of such Hamiltonians is performed. They correspond to the pseudo-Hermitian supersymmetric systems of the boson-phermion oscillators. We extend the supercoherent states formalism to such supersymmetic systems via the pseudo-unitary supersymmetric displacement operator method. The constructed family of these supercoherent states consists of two dual subfamilies that form a bi-overcomplete and bi-normal system in the boson-phermion Fock space. The states of each subfamily are eigenvectors of the boson annihilation operator and of one of the two phermion lowering operators.

Pseudo supersymmetric partners for the generalized Swanson model

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., , where α, β are real constants, with α ≠ β, and and are generalized creation and annihilation operators. It is shown that the initial Hamiltonian H−, and its partner H+, are related by pseudo supersymmetry, and they share all the eigen energies except for the ground state. This pseudo supersymmetric extension enlarges the class of non Hermitian Hamiltonians H±, related to their respective Hermitian counterparts h±, through the same similarity transformation operator ρ:H± = ρ−1h±ρ. The formalism is applied to the entire class of shape-invariant models.

The diffusion–reaction (D–R) Hamiltonian and the solutions of certain types of linear and nonlinear D–R equations in one dimension

With a view to having further insight into the mathematical content of the non-Hermitian Hamiltonian associated with the diffusion–reaction (D–R) equation in one dimension, we investigate (a) the solitary wave solutions of certain types of its nonlinear versions, and (b) the problem of real eigenvalue spectrum associated with its linear version or with this class of non-Hermitian Hamiltonians. For case (a) we use the standard techniques to handle the quadratic and cubic nonlinearities in the D–R equation whereas for case (b) a newly proposed method, based on an extended complex phase space, is employed. For a particular class of solutions, an Ermakov system of equations is also found for the linear case. Further, corresponding to the 'classical' version of the above one-dimensional complex Hamiltonian, an equivalent integrable system of two, two-dimensional, real Hamiltonians is suggested.

A new class of non-Hermitian Hamiltonians with real spectra

We construct a new class of non-Hermitian Hamiltonians with real spectra. The Hamiltonians possess one explicitly known eigenfunction.