Non-Hermitian Hamiltonian beyond PT-symmetry for time-dependant SU(1,1) and SU(2) systems -- exact solution and geometric phase in pseudo-invariant theory

TL;DR

This paper provides exact solutions and analyzes the geometric phase for time-dependent non-Hermitian SU(1,1) and SU(2) systems, extending beyond PT-symmetry by employing pseudo-invariant theory.

Contribution

It introduces a unified method to solve non-Hermitian SU(1,1) and SU(2) Hamiltonians and derives their geometric phases, even when PT-symmetry is broken.

Findings

Exact solutions for non-Hermitian SU(1,1) and SU(2) systems.

Pseudo-Hermitian invariants with real eigenvalues.

Analytical geometric phase matches Hermitian case results.

Abstract

We investigate in this paper time-dependent non-Hermitian Hamiltonians, which consist respectively of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1)and SU(2)systems in a unified manner. Then, we derive the LR phase, which can be separated to the dynamic phase and the geometrical phase. The analytical results are exactly in agreement with those of corresponding Hermitian Hamiltonians in the literature.