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Theoretical and Mathematical Physics

Zero-energy states and non-Hermitian skin effect in the infinite double-barrier and finite Su–Schrieffer–Heeger models

Tat'yana Sergeevna Tinyukova, Yu. P. Chuburin

Published May 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s0040577926050053

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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.
We investigate domain-wall physics in a non-Hermitian Su-Schrieffer-Heeger model featuring the non-Hermitian skin effect and its higher-dimensional extensions, focusing on analytical eigenstate solutions under open boundary conditions. By gluing two Su-Schrieffer-Heeger chains with inverted coupling ratios together, we identify two distinct interface geometries, and derive closed-form quantization conditions for the complex wave number and wave functions using symmetry-based ansatzes. Besides conventional skin modes and topological zero modes, the domain walls generate additional localized states at finite energy that detach from the bulk open-boundary-condition continuum. We validate the analytical results by exact diagonalization, and further generalize the interface construction to a two-dimensional Lieb lattice, where competing nonreciprocities produce tunable funneling regimes towards codimension-one and codimension-two interfaces.
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).
We revisit the fate of the skin modes in many-body non-Hermitian fermionic systems. Contrary to the single-particle case, the many-body ground state cannot exhibit an exponential localization of all eigenstates due to the Pauli exclusion principle. However, asymmetry can still exist in the density profile, which can be quantified using the imbalance between the two halves of the system. Using the non-Hermitian Su-Schrieffer-Heeger (SSH) chain as an illustration, we show the existence of two distinct scaling regimes for the imbalance. In the first one, the imbalance grows linearly with the system size, as generically expected. In the second one, the imbalance saturates to a finite value. By combining high-precision exact diagonalization calculations and analytical arguments, we observe that the imbalance does not scale when the occupied bands can be deformed to their Hermitian limit. This suggests a direct connection between the corresponding bulk topological invariants and the skin effect in many-body systems. Importantly, this relation also holds for interacting systems.
The non-Hermitian skin effect, namely, that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamic limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We provide a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in noninteracting open quantum systems described by the master equation. We demonstrate that the evolution of the density matrix is independent of the boundary condition.
We investigate the interplay between the non-Hermitian skin effect and Aubry-André-Harper (AAH) quasiperiodic disorder in a one-dimensional Su-Schrieffer-Heeger (SSH) chain with nonreciprocal hopping. By exact diagonalization, transfer-matrix analysis, and an analytical similarity-transformation argument, we map the full ( , δδδ) phase diagram, where A is the AAH modulation strength and the nonreciprocity parameter. We identify five distinct regimes: ( ) topological with extended bulk, (II) AAH-localized, (III) skin-localized, (IV) fully localized, and a previously unreported (V) competition regime exhibiting reentrant partial delocalization, in which intermediate quasiperiodic disorder disrupts the directional skin accumulation before ultimately Anderson-localizing all states. Using phase-averaged diagnostics and finite-size scaling, we confirm that the reentrant regime is robust, characterized by a non-monotonic inverse participation ratio that sharpens with increasing system size. We derive an analytical expression for the modified localization boundary λc(δ)=2veffwλ_{c}(δ)=2\sqrt{v_{eff}w}λc​(δ)=2veff​w​ with vvff=v2−δ2v_{vff}=\sqrt{v^{2}-δ^{2}}vvff​=v2−δ2​, which agrees with numerical Lyapunov exponent calculations. We further show that quasiperiodic disorder progressively unwinds the complex spectral loops, destroying the point-gap topology at a critical strength distinct from the band-topological transition ; that the skin effect suppresses entanglement entropy to near-zero values while sufficiently strong AAH disorder partially restores it ; and that the SSH sublattice structure absent in the widely studied non-Hermitian AAH chain is essential for producing the five-phase landscape, as demonstrated by direct comparison with the non-dimerized limit.
We investigate the non-Hermitian skin effect (NHSE) and topological phases in a ladder model with staggered nonreciprocal inter-leg hopping, consisting of an Su-Schrieffer-Heeger (SSH) chain coupled to a normal tight-binding chain. By tuning the system parameters, we show that the direction of the NHSE under open boundary conditions can be reversed and can even become energy dependent. The NHSE is further characterized by the spectral winding number under periodic boundary conditions. We further demonstrate that the inter-leg coupling significantly enlarges the topologically nontrivial parameter regime. Remarkably, the zero-energy topological edge modes reside on different legs of the ladder and are characterized by real-space winding numbers W=±1W=\pm1W=±1. Furthermore, increasing the nonreciprocity of the inter-leg hopping modifies the topological phase boundaries and eventually drives the system into a topologically trivial phase. Our results establish staggered nonreciprocal inter-leg hopping as an effective mechanism for engineering both the non-Hermitian skin effect and topological phases in ladder systems.