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Critical non-Hermitian Skin Effect in a Single Closed non-Hermitian Chain

Zhuo Bin Siu, S. M. Rafi‐Ul‐Islam, M. B. A. Jalil

Published Feb 11, 2022• 1 citation
Publisher

Abstract

Abstract A critical non-Hermitian skin effect (CNHSE) was recently discovered in which the eigenenergy spectrum of a coupled system comprising two 1D non-Hermitian (NH) chains exhibits an abrupt change when the length of the system increases beyond a critical length. Here, we overturn the conventional wisdom that multiple chains are required for the CNHSE by showing that a similar abrupt transition in the eigenenergy spectrum occurs with increasing length in a single finite 1D chain when the two ends of the chain are connected by a weak terminal coupling to form a closed loop. We illustrate this phenomenon using the classic Hatano-Nelson and Su-Schrieffer-Heeger NH systems, and show that it can co-exist with the conventional CNHSE in coupled NH loops.

DOI
10.21203/rs.3.rs-1312995/v1

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Abstract A critical non-Hermitian skin effect (CNHSE) was recently discovered in which the eigenenergy spectrum of a coupled system comprising two 1D non-Hermitian (NH) chains exhibits an abrupt change when the length of the system increases beyond a critical length. Here, we overturn the conventional wisdom that multiple chains are required for the CNHSE by showing that a similar abrupt transition in the eigenenergy spectrum occurs with increasing length in a single finite 1D chain when the two ends of the chain are connected by a weak terminal coupling to form a closed loop. We illustrate this phenomenon using the classic Hatano-Nelson and Su-Schrieffer-Heeger NH systems, and show that it can co-exist with the conventional CNHSE in coupled NH loops.
We investigate the critical non-Hermitian skin effect (CNHSE) and the coupling-driven morphing of scale-free localization (SFL) in two Hatano-Nelson chains coupled locally at two bulk sites separated by an inclusive distance d. At weak interchain coupling, competition between the opposite skin accumulations of the two chains generates a geometrically selected critical branch with a complex loop-like spectrum and scale-free skin states localized near the physical boundaries. Remarkably, this SFL branch persists as the coupling becomes strong, although its underlying mechanism and spatial profile change qualitatively. Strong local hybridization generates high-energy impurity states and effectively divides the low-energy Hilbert space into geometry-selected inner and outer segments, with the former supporting SFL states near the interchain-coupled sites and the latter retaining conventional skin states. The number and localization scale of the SFL states are controlled by the separation between the coupling sites. Consequently, increasing the interchain coupling continuously relocates the SFL states from the physical boundaries to the coupling-defined internal interfaces. Our results demonstrate that bilocal coupling can induce a CNHSE and further drive its scale-free states into a segmentation-induced localization regime.
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).
The non-Hermitian skin effect is nonreciprocity-induced localization phenomena in which a macroscopic number of eigenstates accumulate anomalously at the boundary, accompanied by the extreme sensitivity to boundary conditions. Here, we develop a geometric characterization of the non-Hermitian skin effect. We demonstrate that the localization length scale associated with the skin effect is encoded in the quantum metric defined solely from right eigenstates, but not in the biorthogonal quantum metric. Moreover, we show that the quantum metrics can exhibit the power-law divergences at gapless points that depend on the different boundary conditions. We also reveal that cusps of the generalized Brillouin zone in non-Bloch band theory are signaled by discontinuities in the quantum metrics. We illustrate these behavior using prototypical non-Hermitian models, such as the Hatano-Nelson model and the non-Hermitian, nonreciprocal Su-Schrieffer-Heeger model.
Abstract Criticality in non-Hermitian systems unveils unique phase transitions and scaling behaviors beyond Hermitian paradigms, offering new insights into the interplay between gain/loss, non-reciprocity, and complex energy spectra. In this paper, we uncover a new class of many-body critical non-Hermitian skin effect (CSE) originating from the interplay between multiple non-Hermitian pumping channels and Hubbard interactions. In particular, criticality in the real-to-complex transitions can selectively emerge within the subspace of bound states or scattering states, as well as their interacting admixtures. These mechanisms possess no single-particle analog and can be diagnosed through a specially defined correlation function. As more particles are involved, higher-order CSEs naturally arise, with greatly enhanced effective coupling strengths and hence greater experimental accessibility. Our results reveal an enriched landscape of non-Hermitian critical phenomena in interacting many-body systems, and pave the way for investigating unconventional non-Hermitian criticality in the context of various interaction-induced particle clustering configurations.
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.