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arXiv:2401.01126

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

Pijush K. Ghosh

Published Jan 2, 2024• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2401.01126

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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.
A class of pseudo-Hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators that are Hermitian with respect to a pre-determined positive-definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-Hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark (Zeeman) effect with non-Hermitian interaction, non-Hermitian general quadratic form of N boson (fermion) operators, symmetric and asymmetric XXZ spin chain in the complex magnetic field, non-Hermitian Haldane–Shastry spin chain and Lipkin–Meshkov–Glick model.
Abstract The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.
A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered.
We consider a Su-Schrieffer-Heeger(SSH)-type trimer model with next-nearest-neighbor(NNN) interaction and balanced loss-gain(BLG) to study the combined effect of lattice symmetries, topology, non-hermiticity and general boundary conditions(GBC)on the existence of flat band and the nature of Bulk-Boundary correspondence(BBC). We derive the necessary and sufficient conditions for the existence of an entirely real spectrum under the periodic boundary condition(PBC). The exact expressions for the compact localized states(CLS) and energy eigenvalues corresponding to flat bands are derived analytically under the PBC. We establish topological phase transitions(TPT) for PT-symmetry and pseudo-chiral symmetry through the computation of the Zak phase and sub-lattice Zak phase, respectively. The Hamiltonian under the open boundary condition(OBC) is studied numerically, and edge states are observed in the topologically non-trivial phase, thereby establishing the non-hermitian BBC. The CLS exists in both bulk and the boundary for systems having only pseudo-chiral symmetry, and an additional PT-symmetry destroys the CLS at the boundary. We generalize a known formalism to study the same Hamiltonian under GBC, and derive analytic expressions for the energy and eigenstates for a class of boundary conditions in parametric ranges which admit flat band under the PBC. The edge states for these boundary conditions, including the OBC, are obtained analytically in the topologically non-trivial phase, thereby establishing BBC. The non-hermitian skin effect(NHSE) is seen in the model with reciprocal bulk interaction and strongly non-reciprocal boundary terms. The winding number based on spectral topology is computed analytically.
Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space ( pip -space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extent they preserve spectral properties. Then we apply some of the previous results to operators on a particular pip -space, namely a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.