Metric operators for non-Hermitian quadratic su(2) Hamiltonians

TL;DR

This paper investigates non-Hermitian quadratic su(2) Hamiltonians with anti-linear symmetry, identifying conditions for real spectra and constructing similarity transformations to Hermitian forms using Lie algebraic methods.

Contribution

It extends the analysis of metric operators to quadratic su(2) Hamiltonians, exploring complex similarity transformations beyond linear cases and comparing with explicit diagonalization methods.

Findings

Identified classes of quadratic su(2) Hamiltonians with real spectra.

Constructed similarity transformations including quadratic exponents.

Compared transformations with exact diagonalization methods.

Abstract

A class of non-Hermitian quadratic su(2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realisations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here, Lie-algebraic methods which were used to investigate the generalised Swanson Hamiltonian are employed to identify the class of quadratic Hamiltonians that allow for such a mapping to the Hermitian counterpart. Whereas for the linear su(2) system every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su(2) generators, the situation is more complicated for quadratic Hamiltonians. Therefore, the possibility of more elaborate similarity transformations, including quadratic exponents, is also explored in detail. The existence of finite dimensional representations for the su(2) Hamiltonian, as opposed to the su(1,1) studied before, allows for comparison with explicit diagonalisation results for finite matrices. Finally, the similarity transformations constructed are compared with the analogue of Swanson's method for exact diagonalsation of the problem, establishing a simple relation between both approaches.