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arXiv:2211.16565 · Physical review. B./Physical review. B

Scaling laws for non-Hermitian skin effect with long-range couplings

Yi-Cheng Wang, H. H. Jen, Jhih-Shih You

Published Aug 10, 2023• 16 citations
arXivHTMLPDF

Abstract

Recent years have witnessed a surge of research on the non-Hermitian skin effect (NHSE) in one-dimensional lattices with finite-range couplings. In this work, we show that the long-range couplings that decay as $1/{l}^{\ensuremath{\alpha}}$ at distance $l$ can fundamentally modify the behavior of NHSE and the scaling of quantum entanglement in the presence of nonreciprocity. At $\ensuremath{\alpha}=0$, the nonlocality of couplings gives rise to the scale-free skin modes, whose localization length is proportional to the system size. Increasing the exponent $\ensuremath{\alpha}$ drives a complex-to-real spectral transition and a crossover from a scalefree to constant localization length. Furthermore, the scaling of nonequilibrium steady-state entanglement entropy exhibits a subextensive law due to the nonlocality and the complex spectrum, in contrast to an area law arising from NHSE. Our results provide a theoretical understanding on the interplay between long-range couplings and non-Hermiticity.

DOI
10.1103/physrevb.108.085418

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Recent years have witnessed a surge of research on the non-Hermitian skin effect (NHSE) in one-dimensional lattices with finite-range couplings. In this work, we show that the long-range couplings that decay as 1/l\ensuremathα1/{l}^{\ensuremath{\alpha}}1/l\ensuremathα at distance lll can fundamentally modify the behavior of NHSE and the scaling of quantum entanglement in the presence of nonreciprocity. At \ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0, the nonlocality of couplings gives rise to the scale-free skin modes, whose localization length is proportional to the system size. Increasing the exponent \ensuremathα\ensuremath{\alpha}\ensuremathα drives a complex-to-real spectral transition and a crossover from a scalefree to constant localization length. Furthermore, the scaling of nonequilibrium steady-state entanglement entropy exhibits a subextensive law due to the nonlocality and the complex spectrum, in contrast to an area law arising from NHSE. Our results provide a theoretical understanding on the interplay between long-range couplings and non-Hermiticity.
Recent years have seen remarkable development in open quantum systems effectively described by non-Hermitian Hamiltonians. A unique feature of non-Hermitian topological systems is the skin effect, anomalous localization of an extensive number of eigenstates driven by nonreciprocal dissipation. Despite its significance for non-Hermitian topological phases, the relevance of the skin effect to quantum entanglement and critical phenomena has remained unclear. Here, we find that the skin effect induces a nonequilibrium quantum phase transition in the entanglement dynamics. We show that the skin effect gives rise to a macroscopic flow of particles and suppresses the entanglement propagation and thermalization, leading to the area law of the entanglement entropy in the nonequilibrium steady state. Moreover, we reveal an entanglement phase transition induced by the competition between the unitary dynamics and the skin effect even without disorder or interactions. This entanglement phase transition accompanies nonequilibrium quantum criticality characterized by a nonunitary conformal field theory whose effective central charge is extremely sensitive to the boundary conditions. We also demonstrate that it originates from an exceptional point of the non-Hermitian Hamiltonian and the concomitant scale invariance of the skin modes localized according to the power law. Furthermore, we show that the skin effect leads to the purification and the reduction of von Neumann entropy even in Markovian open quantum systems described by the Lindblad master equation. Our work opens a way to control the entanglement growth and establishes a fundamental understanding of phase transitions and critical phenomena in open quantum systems far from thermal equilibrium.
Spectral transitions in non-Hermitian lattices often arise from the competition between non-reciprocal skin accumulation and inter-component hybridization. In short-range systems formed by two coupled chains, this competition conventionally leads to the logarithmic critical-length law Nc∼ln⁡DN_c\sim\ln DNc​∼lnD, where DDD is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent ααα, the onset becomes algebraic, Nc∼Dα/3N_c\sim D^{α/3}Nc​∼Dα/3. When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for α<2α<2α<2, in which the criticality threshold equation depends only on the system aspect ratio Nc/DN_c/DNc​/D. At α=2α=2α=2 and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.
The non-Hermitian skin effect (NHSE), characterized by the collapse of bulk-band eigenstates into the localized boundary modes of the systems, is one of most striking properties in the field of non-Hermitian physics. Unique physical phenomena related to the NHSE have attracted a great deal of interest; however, their experimental realization usually requires nonreciprocal hopping, which faces a great challenge in ultracold-atom systems. In this work we propose the realization of the NHSE in a one-dimensional optical lattice by periodically driven ultracold atoms in the presence of staggered atomic loss. By studying the effective Floquet Hamiltonian in the high-frequency approximation, we reveal the underlying mechanism for the periodic-driving-induced NHSE. We find that the robust NHSE can be tuned by the driving phase, which is manifested by the dynamical localization. Most remarkably, we uncover the periodic-driving-induced critical skin effect for two coupled chains with different driving phases, accompanied by the appearance of size-dependent topological in-gap modes. Our study provides a feasible way to observe the NHSE and explore the corresponding unique physical phenomena due to the interplay of non-Hermiticity and many-body statistics in ultracold-atom systems.
We study the one-dimensional non-Hermitian lattices with linearly varying nonreciprocal hopping, where the non-Hermitian skin effect (NHSE) is found to be dissolved gradually as the strength of nonreciprocity increases. The energy spectrum under the open boundary condition is composed of real and imaginary eigenenergies when the nonreciprocal hopping is weak. Interestingly, the real eigenenergies form an equally spaced ladder, and the corresponding eigenstates are localized at the boundary with a Gaussian distribution due to NHSE. By increasing the nonreciprocity, the number of real eigenenergies will decrease while more and more eigenenergies become imaginary. Accompanied by the real-imaginary transition in the spectrum, the eigenstates are shifted from the boundary into the bulk of the lattice. When the nonreciprocity gets strong enough, the whole spectrum will be imaginary and the NHSE disappears completely in the system; i.e., all the eigenstates become Gaussian bound states localized inside the bulk. Our work unveils the exotic properties of non-Hermitian systems with spatially varying nonreciprocal hopping.
Non-Hermitian skin effect is one of the most captivating phenomena in non-Hermitian systems, characterized by the extensive localization of eigenstates near open boundaries. In its conventional form, the localization length of a skin mode is independent of the system size. Remarkably, however, when the open boundaries are coupled to realize a generalized boundary condition, the localization length can undergo a drastic transformation, becoming proportional to the system length. This intriguing regime is known as the scale-free skin effect (SFSE). Although SFSE has been observed in numerous one-dimensional non-Hermitian models through case-by-case studies, a unified theoret ical framework capable of predicting its emergence and characteristic properties remains absent. In this work, we take a firm step forward by establishing a model-independent framework for the ana lytic determination of scale-free localization lengths in certain regimes. Our key insight is to treat generalized boundary conditions as a perturbation of the periodic boundary conditions, thereby circumventing the singular response-to-perturbation that would otherwise arise if they were pertur batively treated relative to open boundary conditions. Our work sheds new light on understanding SFSE in non-Hermitian systems.
The non-Hermitian skin effect (NHSE), featured by the collapse of bulk-band eigenstates into the localized boundary modes of the systems, is one of most striking properties in the fields of non-Hermitian physics. Unique physical phenomena related to the NHSE have attracted a lot of interest, however, their experimental realizations usually require nonreciprocal hopping, which faces a great challenge in ultracold-atom systems. In this work, we propose to realize the NHSE in a 1D optical lattice by periodically-driven ultracold atoms in the presence of staggered atomic loss. By studying the effective Floquet Hamiltonian in the high-frequency approximation, we reveal the underlying mechanism for the periodic-driving-induced the NHSE. We found that the robust NHSE can be tuned by driving phase, which is manifested by the dynamical localization. Most remarkably, we uncover the periodic-driving-induced critical skin effect for two coupled chains with different driving phases, accompanied by the appearance of size-dependent topological in-gap modes. Our studies provide a feasible way for observing the NHSE and exploring corresponding unique physical phenomena due to the interplay of non-Hermiticity and many-body statistics in ultracold-atom systems.