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Non-Hermitian, tight-binding $\mathcal{PT}$-symmetric models are extensivelystudied in the literature. Here, we investigate two forms of non-HermitianHamiltonians to study the $\mathcal{PT}$-symmetry breaking thresholds andfeatures of corresponding surfaces of exceptional points (EPs). They includeone-dimensional chains with uniform or 2-periodic tunnelling amplitudes, onepair of balanced gain and loss potentials $\Delta\pm\i\gamma$ atparity-symmetric sites, and periodic or open boundary conditions. Byintroducing a Hermitian detuning potential, we obtain the dependence of the$\mathcal{PT}$-threshold, and therefore the exceptional-point curves, in theparameter space of detuning and gain-loss strength. By considering several suchexamples, we show that EP curves of a given order generically have cusp-pointswhere the order of the EP increases by one. In several cases, we obtainexplicit analytical expressions for positive-definite intertwining operatorsthat can be used to construct a complex extension of quantum theory byre-defining the inner product. Taken together, our results provide a detailedunderstanding of detuned tight-binding models with a pair of gain-losspotentials.

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