Construction of Pseudo-hermitian matrices describing systems with balanced loss-gain

TL;DR

The paper addresses constructing finite-dimensional pseudo-Hermitian operators with entirely real spectra by introducing a positive-definite metric $\eta$ and a general algebraic framework based on $SU(N)$ generators. It provides an explicit construction of the operator ${\cal O}=a_0{\cal S}\eta + a_1{\cal M}$ decomposed into Hermitian and non-Hermitian parts, with the metric expanded in the $SU(N)$ basis and a sufficiency condition $\alpha_0>\alpha_0^{min}$ for positivity. The method is then used to build generic tight-binding lattice models of size $N$ with balanced loss-gain and NN/NNN couplings, including special cases with uniform coupling and an SSH-type chain. The work yields real spectra and unitary time evolution in the modified inner product, and sets the stage for exploring spectra, transport, and topological properties in these pseudo-Hermitian lattices.

Abstract

We present a general construction of pseudo-hermitian matrices in an arbitrary large, but finite dimensional vector space. The positive-definite metric which ensures reality of the entire spectra of a pseudo-hermitian operator, and is used for defining a modified inner-product in the associated vector space is also presented. The construction for an N dimensional vector space is based on the generators of SU (N ) in the fundamental representation and the identity operator. We apply the results to construct a generic pseudo-hermitian lattice model of size N with balanced loss-gain. The system is amenable to periodic as well as open boundary conditions and by construction, admits entirely real spectra along with unitary time-evolution. The tight binding and Su-Schrieffer-Heeger(SSH) models with nearest neighbour(NN) and next-nearest neighbour(NNN) interaction with balanced loss-gain appear as limiting cases.