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Eigenvalues and eigenfunctions of the perturbed non-Hermitian SSH Hamiltonian with PT symmetry

Tat'yana Sergeevna Tinyukova, Yu. P. Chuburin

Published Nov 1, 2023• 1 citation
Publisher

Abstract

In the paper we find near-zero eigenvalues (their physical meaning is the electron energy) and eigenfunctions describing the electronic states of the non-Hermitian SSH Hamiltonian for an infinite chain with PT symmetry, perturbed by a $\delta$-shaped potential. We prove that for a small non-Hermitian parameter there are two (generalized) eigenvalues of multiplicity one, and, in contrast to the Hermitian model, the corresponding (generalized) eigenfunctions depending on the parameters of the system can either increase exponentially (which corresponds to resonant, i.e., decaying states) or decrease exponentially (which corresponds to bound states) as $|n|\to\infty$.

DOI
10.35634/2226-3594-2023-62-07

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In the paper we find near-zero eigenvalues (their physical meaning is the electron energy) and eigenfunctions describing the electronic states of the non-Hermitian SSH Hamiltonian for an infinite chain with PT symmetry, perturbed by a δ\deltaδ-shaped potential. We prove that for a small non-Hermitian parameter there are two (generalized) eigenvalues of multiplicity one, and, in contrast to the Hermitian model, the corresponding (generalized) eigenfunctions depending on the parameters of the system can either increase exponentially (which corresponds to resonant, i.e., decaying states) or decrease exponentially (which corresponds to bound states) as ∣n∣→∞|n|\to\infty∣n∣→∞.
Abstract We analytically study zero-energy eigenstates for two non-Hermitian models based on the Su–Schrieffer–Heeger (SSH) chain. The first model is an infinite non-Hermitian SSH chain with nonreciprocal hoppings to a neighboring site and two barriers. We find analytical expressions for the eigenfunctions, including those that describe resonant states. For resonant states, an analogue of the non-Hermitian skin effect is observed. For the second model, a finite non-Hermitian SSH chain with nonreciprocal hoppings with an even or odd number of sites, we also find the eigenfunctions and observe the non-Hermitian skin effect. All results are obtained using the derived Green’s function of the infinite non-Hermitian SSH chain.
At present, non-Hermitian topological systems continue to be actively studed. In a rigorous approach, we study one of the key non-Hermitian systems — the Hatano–Nelson model HHH. We find the Green function for this Hamiltonian. Using the Green function, we analytically obtain the eigenvalues and eigenfunctions of HHH for finite and semi-infinite chains, as well as for an infinite chain with a local potential. We discuss the non-Hermitian skin effect for the models mentioned above. We also describe the boundary between localized and resonant eigenfunctions (for the zero spectral parameter, this is the boundary between non-Hermitian topological phases).
With a view to getting further insight into the solutions of one-dimensional analogous Schr?dinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT- symmetric solutions for an octic potential and its variant using extended complex phase space approach characterized by x=x1+ip2, p=p1+ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. Besides the complexity of the phase space, complexity of potential parameters is also considered. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalue and eigenfunction of a system. The imaginary part of energy eigenvalue of a non-hermitian Hamiltonian exist for complex potential parameters and reduces to zero for real parameters. However, in the present work, it is found that imaginary component of the energy eigenvalue vanishes even when potential parameters are complex, provided that PT-symmetric condition is satisfied. Thus PT- symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue.
We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number ℓ\ellℓ.