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

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.

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