Effects of detuning on $\mathcal{PT}$-symmetric, tridiagonal, tight-binding models

• URL: http://arxiv.org/abs/2302.13204v1
• Date: Sun, 26 Feb 2023 01:36:59 GMT
• Title: Effects of detuning on $\mathcal{PT}$-symmetric, tridiagonal, tight-binding models
• Authors: Jacob L. Barnett and Yogesh N. Joglekar
• Abstract summary: Non-Hermitian, tight-binding $mathcalPT$-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the $mathcalPT$-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs) Taken together, our results provide a detailed understanding of detuned tight-binding models with a pair of gain-loss potentials.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Non-Hermitian, tight-binding $\mathcal{PT}$-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the $\mathcal{PT}$-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs). They include one-dimensional chains with uniform or 2-periodic tunnelling amplitudes, one pair of balanced gain and loss potentials $\Delta\pm\i\gamma$ at parity-symmetric sites, and periodic or open boundary conditions. By introducing a Hermitian detuning potential, we obtain the dependence of the $\mathcal{PT}$-threshold, and therefore the exceptional-point curves, in the parameter space of detuning and gain-loss strength. By considering several such examples, we show that EP curves of a given order generically have cusp-points where the order of the EP increases by one. In several cases, we obtain explicit analytical expressions for positive-definite intertwining operators that can be used to construct a complex extension of quantum theory by re-defining the inner product. Taken together, our results provide a detailed understanding of detuned tight-binding models with a pair of gain-loss potentials.

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