Author: Denis Avetisyan
A new study reveals that discrepancies in non-Hermitian systems stem from focusing on eigenvalues rather than the full eigenstate spectrum, uncovering previously overlooked instabilities.
The research demonstrates that non-Hermitian topological phases can exhibit missing eigenstates due to the emergence of macroscopic Jordan blocks, challenging conventional interpretations of the bulk-boundary correspondence.
While conventional spectral analysis assumes a direct correspondence between eigenvalues and eigenstates, this connection breaks down in non-Hermitian systems, leading to spectral anomalies and misinterpretations of topological phases. In ‘When and why non-Hermitian eigenvalues miss eigenstates in topological physics’, we demonstrate that discrepancies arise not from fundamental failures of bulk-boundary correspondence, but from the inherent inability of the eigenvalue spectrum to detect a significant fraction of existing eigenstates-a consequence of macroscopic hidden exceptional points and the non-Hermitian skin effect. Our analysis, using the Hatano-Nelson model as a paradigm, reveals a systematic link between hidden modes and instabilities stemming from the proximity of macroscopic Jordan blocks. Does this necessitate a shift towards an eigenstate-centric approach for a more complete characterization of non-Hermitian topological phenomena?
Beyond Conventional Symmetry: A New Landscape in Quantum Mechanics
The foundations of quantum mechanics traditionally rest upon the use of Hermitian Hamiltonians – mathematical operators ensuring that measurable physical quantities, like energy, remain real numbers. However, this requirement inadvertently restricts the accurate modeling of a vast array of physical systems that are inherently ‘open’ – meaning they exchange energy and matter with their surroundings – or ‘dissipative’ – systems where energy is lost over time. These limitations become particularly evident when examining scenarios involving decay, gain, or interaction with an external environment, as the standard Hermitian framework struggles to accommodate the resulting complex dynamics. Consequently, a more generalized approach is needed to effectively describe these non-isolated systems, pushing the boundaries of quantum theory to encompass phenomena beyond the reach of conventional Hermitian mechanics and unlocking a deeper understanding of the quantum realm’s interplay with its surroundings.
A surprising number of physical systems deviate from the strict rules of traditional quantum mechanics, exhibiting behaviors described by non-Hermitian Hamiltonians. These aren’t merely theoretical curiosities; examples abound in optics – such as in parity-time symmetric structures – and condensed matter physics, where dissipation and gain play crucial roles. Conventional quantum theory, built upon the foundation of Hermitian operators guaranteeing real-numbered energy eigenvalues, struggles to accurately model these open systems that exchange energy and matter with their surroundings. Consequently, a broadened theoretical framework is essential, one capable of accommodating complex eigenvalues and describing phenomena like unidirectional invisibility, enhanced sensing, and non-Hermitian topological phases – properties fundamentally inaccessible within the confines of Hermitian quantum mechanics. This shift represents not a rejection of established principles, but rather an expansion, allowing for a more complete and nuanced understanding of the quantum world.
The exploration of non-Hermitian Hamiltonians fundamentally alters the landscape of quantum spectral analysis. Unlike traditional Hermitian systems where eigenvalues represent real-valued energies, these broadened frameworks allow for complex eigenvalues – numbers with both real and imaginary parts. The real component still corresponds to the energy, but the imaginary component signifies a gain or loss of probability, effectively describing decay or amplification processes within the quantum system. This leads to dramatic changes in spectral characteristics; what were once discrete energy levels in Hermitian systems can become lines with finite widths, or even coalesce into complex branching structures. Furthermore, the concept of conventional normalization is challenged, requiring modified approaches to ensure physical interpretations remain consistent. These spectral signatures are not merely mathematical curiosities, but directly reflect the dynamic, open nature of many physical systems, from lasers and optical resonators to decaying particles and driven-dissipative quantum systems, offering a richer and more accurate description of reality.
The Non-Hermitian Skin Effect: A Breakdown of Conventional Correspondence
The non-Hermitian skin effect is characterized by the exponential spatial localization of all eigenstates to the boundaries of a system when subjected to open boundary conditions. This means that, unlike systems described by Hermitian Hamiltonians where eigenstates are typically extended or localized within the bulk, non-Hermitian systems exhibit a concentration of probability density near the edges. The localization strength is determined by the system’s parameters and the degree of non-Hermiticity; a larger non-Hermitian component generally leads to stronger localization. This effect is not simply a consequence of finite size but rather an inherent property arising from the non-Hermitian nature of the Hamiltonian, H, and impacts the system’s overall behavior and spectral properties.
The non-Hermitian skin effect originates from the presence of complex eigenvalues in the non-Hermitian Hamiltonian, specifically impacting the behavior of eigenstates. In traditional Hermitian systems, the bulk-boundary correspondence dictates that boundary states are localized and arise from topological properties of the bulk band structure. However, non-Hermitian Hamiltonians, due to their complex spectra, invalidate this correspondence; the complex eigenvalues lead to an asymmetric localization of all eigenstates at the system boundaries, even in the absence of topological edge modes. This means that states which would normally be extended in a Hermitian system are forced to localize, resulting in a breakdown of the relationship between bulk and boundary properties and fundamentally altering the system’s spectral characteristics.
The Hatano-Nelson and Su-Schrieffer-Heeger (SSH) models are frequently employed to investigate the non-Hermitian skin effect due to their relative simplicity and analytical tractability. The Hatano-Nelson model, characterized by non-reciprocal hopping terms, directly exhibits the exponential localization of eigenstates at the boundary when subjected to open boundary conditions. Similarly, the SSH model, a tight-binding chain with alternating hopping amplitudes, can be modified to incorporate non-Hermitian terms, leading to the same boundary-localized behavior. Analysis of these models provides concrete examples of how non-Hermitian Hamiltonians can lead to a breakdown of the conventional bulk-boundary correspondence, where boundary states are not necessarily dictated by bulk properties, and allows for quantitative investigation of the localization exponent and the resulting impact on system observables.
Beyond Eigenvalues: Characterizing the Full Spectrum of Non-Hermitian Systems
The conventional eigenvalue spectrum, defined as the set of solutions λ to the equation H|\psi\rangle = \lambda|\psi\rangle, proves inadequate for fully characterizing non-Hermitian systems due to the non-normality of their Hamiltonians. Unlike Hermitian operators where eigenvalues directly correspond to measurable energy levels, non-Hermitian Hamiltonians allow for complex eigenvalues, and the associated eigenvectors do not necessarily form a complete basis. This leads to discrepancies when attempting to interpret the eigenvalue spectrum in terms of physical observables or when using it to approximate system behavior. Specifically, the standard approach fails to accurately represent the decay rates and lifetimes of states in open quantum systems, necessitating the use of the eigenstate spectrum – the set of states |\psi\rangle that define the actual physical behavior – for a complete description.
While the eigenvalue spectrum is central to understanding Hermitian systems, it proves inadequate for non-Hermitian Hamiltonians due to the loss of spectral symmetry and the emergence of complex eigenvalues. The pseudospectrum, defined as the set of points in the complex plane with eigenvalues within a distance ε of a given energy, provides a more complete picture of spectral robustness by accounting for the sensitivity to perturbations. Similarly, the singular value spectrum, derived from the singular value decomposition of the Hamiltonian, characterizes the decay rates of eigenstates and offers insights into stability even when eigenvalues are ill-defined or complex. Both the pseudospectrum and singular value spectrum provide quantifiable metrics for assessing the qualitative changes in spectral properties arising from non-Hermiticity, offering alternatives to relying solely on eigenvalues for characterizing the system’s behavior.
The approximation of open boundary condition (OBC) eigenstates using periodic boundary condition (PBC) Bloch states exhibits a systematic error that scales inversely with the system size, denoted as O(1/L). This implies that as the characteristic length scale of the system, L, increases, the error in this approximation decreases. Specifically, doubling the system size reduces the error by a factor of two. This convergence behavior is crucial for numerical calculations where PBC are often employed to simplify computations; the O(1/L) scaling provides a quantifiable measure of the accuracy attainable with sufficiently large systems and allows for extrapolation to the OBC limit. The error primarily arises from the spurious states introduced by the PBC, which do not decay at the boundaries as they would in a true OBC system.
Reconciling Boundaries: A Generalized Brillouin Zone for Non-Hermitian Systems
The conventional bulk-boundary correspondence, a cornerstone of condensed matter physics, dictates that the properties of a material’s interior-its ‘bulk’-determine the behavior of states existing at its edges or surfaces-the ‘boundary’. However, systems exhibiting the non-Hermitian skin effect defy this principle. This phenomenon, arising from non-reciprocal hopping or gain-loss imbalances, causes a macroscopic number of states to accumulate at the boundary, fundamentally altering the relationship between bulk and boundary characteristics. Unlike traditional systems where boundary states are merely a consequence of the bulk’s properties, in non-Hermitian systems, the boundary actively shapes the entire system’s behavior, leading to exponentially localized states and a breakdown of the established correspondence. This challenges the very foundation of how physicists understand the connection between a material’s interior and its exterior, necessitating new theoretical frameworks to accurately describe these unusual systems.
The emergence of the non-Hermitian skin effect challenged established principles linking a material’s interior – its ‘bulk’ – to its edges, a correspondence vital for understanding many physical phenomena. Recognizing this breakdown, researchers proposed a significant extension to the conventional Brillouin zone – the fundamental unit defining a crystal’s momentum space. This ‘generalized Brillouin zone’ isn’t merely a larger version of the original; it’s a fundamentally altered space constructed to account for the non-Hermitian nature of the system and the accumulation of states at the boundaries. By redefining the momentum space in this way, the framework allows for a consistent description of both bulk and boundary states, restoring a modified, yet powerful, bulk-boundary correspondence. This innovative approach doesn’t discard prior understandings but rather expands upon them, providing a crucial tool for investigating and predicting the behavior of non-Hermitian systems and paving the way for novel applications in areas like topological photonics and metamaterials.
The generalized Brillouin zone offers a revised understanding of how electronic states are confined within a material, particularly when subjected to open boundary conditions – situations where the material isn’t infinitely extended. Traditional definitions of localization, where a state is considered confined if it decays exponentially away from a specific point, falter in non-Hermitian systems exhibiting the skin effect. This new framework, however, redefines localization not by the decay of the wavefunction itself, but by its behavior within this extended Brillouin zone. States are now considered localized if their spectral projection – essentially, the portion of the wavefunction that contributes to a specific energy – is confined to a finite region of this generalized zone. This allows for a consistent description of both bulk and boundary states, even when the conventional bulk-boundary correspondence breaks down, and provides a powerful tool for predicting and controlling the behavior of electrons in complex, open systems.
The study meticulously carves away at assumptions surrounding non-Hermitian systems, revealing a landscape often obscured by incomplete observation. It demonstrates that focusing solely on the eigenvalue spectrum can mask the presence of crucial eigenstates, leading to misinterpretations of system behavior. This pursuit of essential elements echoes Blaise Pascal’s sentiment: “The dignity of man lies in thought.” The research doesn’t add complexity; instead, it refines understanding by stripping away misleading indicators-specifically, the overreliance on eigenvalues-to reveal the underlying eigenstates and instabilities linked to macroscopic Jordan blocks. This distillation of information highlights the core mechanics at play, a process akin to revealing the structure beneath superfluous ornamentation.
The Road Ahead
The persistence of macroscopic Jordan blocks, revealed by a shift in focus from eigenvalues to eigenstates, suggests a certain architectural flaw in how these non-Hermitian systems have been traditionally approached. They called it a framework to hide the panic – the panic of dealing with instabilities that weren’t absent, merely obscured by a convenient mathematical trick. The field now faces a choice: continue refining the eigenvalue-centric view, endlessly patching over the cracks, or embrace the inherent complexity of the eigenstate spectrum.
A natural progression lies in understanding how these hidden eigenstates manifest in observable phenomena. The bulk-boundary correspondence, a cornerstone of topological physics, may require re-evaluation. Is it truly a correspondence, or simply a fortunate alignment of perspectives? Further work must explore the role of these states in transport, localization, and the non-Hermitian skin effect, seeking to connect the abstract mathematical picture to concrete physical realities.
Ultimately, the simplification sought by physicists is not about reducing complexity, but about identifying the essential elements. This work hints that the essential elements were staring back at them, masked by a preference for tidy spectra. The future belongs to those who are willing to accept a little messiness, a little ambiguity – for it is in those shadows that the most interesting physics often resides.
Original article: https://arxiv.org/pdf/2601.05234.pdf
Contact the author: https://www.linkedin.com/in/avetisyan/
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2026-01-10 02:20