Winding Paths to Chirality: Beyond Eigenmodes in Non-Hermitian Systems

Author: Denis Avetisyan

A new study reveals a robust form of chirality arising not from the properties of individual states, but from the way paths loop around singularities in non-Hermitian physics.

The study demonstrates that exceptional points (EPs) exhibit a topological chirality defined by the relationships between loops on a Riemann surface, where concatenated loops and their mirror images reduce to trivial braids and satisfy the condition $w_0w_\alpha = e$ within a capped fundamental group, indicating a fundamental connection between algebraic properties and topological characteristics of these singularities.

This research introduces ‘topological chirality’ defined by non-Abelian loops encircling exceptional points, offering a topologically protected property independent of specific eigenmodes.

Chirality, typically understood through geometric handedness, finds surprising limitations when extended to the growing field of non-Hermitian physics. In the work ‘Singularity Selector: Topological Chirality via Non-Abelian Loops around Exceptional Points’, we introduce ‘topological chirality’-a robust invariant arising from the non-commutative nature of paths encircling exceptional points in these systems. This chirality is determined by the winding of these loops, acting as a singularity selector independent of specific eigenmodes, and revealing a fundamental two-sheeted topology governing exceptional point pairs. Could this framework unlock new loop-sensitive observables and pave the way for topologically protected devices in diverse platforms like optics, spintronics, and metamaterials?

Unveiling the Asymmetry: Beyond Conventional Physics

For generations, physics has been fundamentally built upon the principle of Hermitian symmetry, a mathematical condition ensuring real-valued energies and probabilities – the bedrock of predictable physical behavior. However, a growing body of research reveals that this symmetry is often an idealization, rarely perfectly realized in actual physical systems. Many real-world scenarios, from open quantum systems constantly interacting with their environment to dissipative optical setups and even certain electronic circuits, inherently lack this perfect symmetry. These deviations aren’t merely imperfections; they unlock a realm of previously unseen phenomena. The lack of Hermitian symmetry allows for energy gain and loss, leading to effects like unidirectional propagation and enhanced sensitivity, challenging conventional understandings of stability and response in physical systems. Consequently, a paradigm shift is underway, recognizing that embracing non-Hermitian physics is crucial for accurately describing and potentially harnessing the behavior of a vast range of physical phenomena.

Non-Hermitian systems, deviating from the conventional requirement of Hermitian symmetry in quantum mechanics, reveal a landscape of unusual physical behaviors centered around what are known as exceptional points. These points represent singularities in the parameter space of a system, where two or more eigenstates coalesce, leading to drastic changes in the system’s sensitivity and response. Unlike traditional systems, non-Hermitian systems at exceptional points exhibit enhanced controllability; a small perturbation can induce a large change in the system’s properties, offering potential for novel sensing and switching applications. This sensitivity isn’t simply a limitation, however-it’s a feature actively being investigated for manipulating wave phenomena, designing advanced lasers, and even creating topologically protected devices with unprecedented robustness. The exploration of exceptional points, therefore, provides a pathway towards harnessing unconventional physics for technological innovation, moving beyond the limitations imposed by traditional Hermitian constraints.

The pursuit of non-Hermitian physics represents a significant shift in how physicists approach understanding and potentially utilizing physical systems. While conventional physics often assumes a symmetry dictated by Hermitian operators, many real-world scenarios-from open quantum systems to driven-dissipative processes-inherently lack this property. This divergence unlocks a realm of unconventional behaviors, including phenomena like unidirectional invisibility, enhanced sensing capabilities, and topologically protected edge states. Investigating these systems isn’t merely an academic exercise; it offers the potential to engineer devices with functionalities impossible within the constraints of Hermitian physics, promising advancements in areas such as laser design, metamaterial engineering, and even quantum information processing. The ability to harness these non-Hermitian effects-to control and manipulate systems operating outside the bounds of traditional symmetry-is therefore becoming a central focus of contemporary research.

Exceptional point (EP) pairs, characterized by a square root dependence of the spectrum, universally appear in diverse physical systems-including optical microcavities and non-Hermitian Dirac models-and are evidenced by specific conditions defining Fermi-arc behavior in parameter space.

Robustness Through Topology: A New Paradigm for Chirality

Conventional chirality, arising from the handedness of molecular structures or the spin of particles, is susceptible to disruption from external influences such as electromagnetic fields or mechanical stress, leading to a loss of chiral signal or functionality. In contrast, topological chirality-a property arising from the topology of a system’s energy landscape-exhibits inherent robustness against these perturbations. This resilience originates from the system’s global topological features, which are protected from local distortions. Specifically, chiral information is encoded not in the instantaneous shape of an object or state, but in the connectivity of its pathways in parameter space, making it insensitive to small changes that would affect conventional chirality. This difference in stability is critical for applications requiring consistent chiral responses in noisy or dynamic environments.

The inherent robustness of topological chirality arises from the non-commutative properties of loops tracing paths around exceptional points in the system’s parameter space. Unlike conventional systems where a continuous deformation can alter chirality, the order in which loops encircle these points matters; reversing the order yields a different, and potentially opposite, chiral response. This non-commutativity effectively prevents continuous deformations from eliminating the chiral signature. Mathematically, this is reflected in the representation of these loops by elements of a braid group, where the group’s non-abelian nature guarantees that different looping sequences are distinguishable and lead to stable, topologically protected chirality, resistant to small perturbations.

Systems exhibiting an orbifold order of 2 modify the expected angular response due to topological constraints. Specifically, this order effectively bisects the local angle experienced by a chiral entity within the system. This arises from the topological properties imposed by the orbifold, which alters the way loops are defined and measured around exceptional points. Consequently, the chiral response is not simply a scaled version of a conventional chiral system but represents a fundamentally different interaction, with a halved angular dependence compared to systems without this topological constraint. This alteration is not due to a change in the physical properties of the interacting entity but a direct consequence of the system’s altered topology.

Realizing the potential of topological chirality necessitates a strong understanding of the underlying mathematical framework, specifically braid group representation. The braid group, denoted $B_n$, describes the configurations of $n$ strands that can be interwoven without cutting or passing through each other. In the context of topological chirality, each loop encircling an exceptional point can be represented as a braid. The non-commutative nature of the braid group means that the order in which these loops are traversed is significant, directly impacting the resulting chiral response. Analyzing the braid group allows for the prediction and control of these responses, as the braid word associated with a specific loop configuration determines the resulting topological charge and, consequently, the robustness of the chiral state against perturbations. This mathematical formalism provides a rigorous means to design and analyze systems exhibiting enhanced chiral stability.

Loops on the Riemann surface, projecting onto exceptional points, demonstrate a bijection with their lifts as time increases, revealing the homotopy-lifting process for both clockwise and counterclockwise encirclements.

From Theory to Experiment: Realizing Non-Hermitian Systems

Non-Hermitian Dirac Hamiltonians extend the standard Dirac equation by incorporating complex potentials, allowing for the description of systems with gain and loss. This modification introduces exceptional points – singularities in the parameter space where both eigenvalues and eigenvectors coalesce – and alters the topological properties of the system. Specifically, these Hamiltonians can exhibit topological chirality, characterized by a non-zero Chern number even without a magnetic field. The presence of complex potentials breaks the conventional Hermiticity, leading to asymmetric energy spectra and the emergence of unidirectional propagation of states, which are hallmarks of topologically protected transport. The mathematical form of a generalized non-Hermitian Dirac Hamiltonian in 2D is typically expressed as $H = \alpha_x p_x + \alpha_y p_y + \beta m + i\gamma$, where $p_x$ and $p_y$ are momentum operators, $m$ is the mass, and $\gamma$ represents the non-Hermitian term inducing gain or loss.

Optical microcavities facilitate the physical realization of non-Hermitian Dirac Hamiltonians by engineering the flow of light within a confined space. These cavities, typically fabricated using techniques like wet etching of III-V semiconductors, support high-Q resonances that act as pseudospin states. By introducing controlled losses and gain within the cavity, often through coupling to waveguides or quantum dots, one can effectively simulate the non-Hermitian terms in the Hamiltonian. The spatial distribution of light at resonance then directly maps onto the solutions of the Dirac equation, enabling experimental observation of phenomena predicted by the theory, such as exceptional points and topological edge states. The small size of microcavities – often on the order of tens of micrometers – allows for precise control over the system parameters and high spatial resolution of the observed effects.

Direct experimental verification of non-Hermitian topological phenomena is achieved through observation of predicted signatures in optical microcavity systems. Specifically, spatial mode imbalance-where the energy distribution between clockwise and counterclockwise propagating modes differs-serves as a key indicator. Furthermore, the detection of local spatial chirality, manifested as a spatially varying handedness of the field, provides direct evidence of the topologically non-trivial band structure. These observations, quantitatively matching theoretical predictions derived from the non-Hermitian Dirac Hamiltonian, validate the approach of utilizing optical microcavities for the physical realization and study of these effects. The magnitude and spatial distribution of these features are consistent with parameter regimes predicted by the theoretical model.

The observation of the relationship $a^2 = e$ serves as a critical verification of the non-Hermitian topology within the system. Here, ‘a’ represents the geometric phase accumulated during a closed loop encircling the exceptional point in parameter space, and ‘e’ is the complex exponential. This equation signifies that a single traversal around the exceptional point results in a phase shift equivalent to the complex exponential, effectively demonstrating a nontrivial topological charge. Experimentally, this is confirmed by monitoring the spatial modes of the optical microcavity; a spatial mode imbalance observed after a full loop confirms the predicted phase accumulation and validates the topological nature of the system. Deviations from this relationship would indicate a failure to properly realize the non-Hermitian Hamiltonian or the presence of uncontrolled perturbations.

The Geometry of Chirality: Singularities and Topological Invariants

Non-Hermitian systems, unlike their conventional Hermitian counterparts, often exhibit topological features defined by the presence of singularities within their parameter space. These singularities, manifesting as branch points and, crucially, cone points, are not merely mathematical curiosities but fundamentally alter the system’s behavior. A cone point, characterized by a specific “orbifold order”, effectively introduces a geometric twist, impacting how the system responds to changes in its parameters. This isn’t simply a local distortion; the presence of such singularities dictates the overall topology of the system, influencing the flow of energy and information. The mathematical description of these points requires tools from complex analysis, treating the parameter space as a Riemann surface where these singularities define its non-trivial structure and ultimately determine the system’s unique properties. These singularities aren’t defects to be avoided, but rather the very features that give rise to interesting and potentially useful physical phenomena.

The peculiar behavior of non-Hermitian systems around their singularities-points where standard physical descriptions break down-necessitates the tools of complex analysis. These systems aren’t adequately described by real-valued functions; instead, their properties are best understood by mapping them onto Riemann surfaces, which are complex manifolds that locally resemble the complex plane. These surfaces allow for a consistent description of the system’s behavior even where traditional notions of a single-valued function fail. By extending the system’s description into the complex domain, researchers can analyze the winding of wavefunctions around these singular points, revealing crucial information about their topological properties. The geometry of these Riemann surfaces directly dictates the system’s spectral features, allowing for the prediction and control of chiral phenomena through careful manipulation of the complex energy landscape, and providing a powerful framework for understanding systems that defy conventional Hermitian descriptions.

The chiral characteristics of non-Hermitian systems aren’t merely qualitative features; they are quantifiable through topological invariants, most notably the spectral winding number. This number, derived from analyzing the eigenvalues of the system’s non-Hermitian Hamiltonian in the complex energy plane, effectively counts how many times the eigenvalues ‘wind’ around a specific point. A non-zero winding number signals the presence of chiral edge states-unique states localized at the boundaries of the system-and directly correlates with the system’s topological charge. Essentially, the spectral winding number provides a robust, mathematical signature of chirality, remaining unchanged under smooth deformations of the system that don’t close the gap in the energy spectrum, thereby offering a powerful tool for classifying and understanding these complex physical phenomena and guaranteeing the stability of associated chiral effects. The magnitude of this number determines the number of protected chiral modes present, offering a direct link between abstract topology and observable physical properties like conductance.

The emergence of a cone point, specifically one exhibiting an orbifold order of 2, represents a dramatic reshaping of the system’s geometric landscape. This isn’t merely a local distortion; the cone point fundamentally alters how paths around it wind and connect, introducing a topological defect. Imagine smoothing the sharp tip of the cone – this process would necessitate a change in the underlying connectivity, signaling a shift in the system’s chiral character. The orbifold order of 2 signifies that a full circuit around the cone point effectively ‘flips’ the system’s state, and this winding is quantifiable as a topological invariant. Consequently, the presence of this cone point doesn’t just indicate a singularity, but actively defines the system’s topological chirality, establishing a robust and measurable signature of its non-Hermitian nature and influencing the behavior of wavefunctions within its vicinity.

Future Directions: Harnessing Topological Chirality

The manipulation of topological chirality – a property dictating how waves twist and propagate – receives a novel approach through the dynamical encircling of exceptional points. These exceptional points, singularities in a system’s parameter space, act as sensitive control knobs; by carefully navigating around them, researchers can effectively ‘steer’ the topology of wave propagation. This isn’t merely a theoretical exercise, as encircling an exceptional point fundamentally alters the system’s chiral characteristics, allowing for precise control over how waves interact with matter. The process relies on subtly changing system parameters – like refractive index or gain – to trace a closed loop around the exceptional point, inducing a robust and predictable change in the wave’s topological charge. This dynamic control offers exciting possibilities beyond static manipulation, promising new avenues for designing devices with tailored wave behavior, and opening pathways towards advanced functionalities in fields like photonics and metamaterials.

The capacity to dynamically control topological chirality presents compelling opportunities for advancements in photonics and integrated optics. Specifically, researchers are intensely focused on leveraging this control to achieve robust waveguiding – the ability to confine and direct light around sharp bends and through complex structures with minimal loss – and asymmetric mode switching, where light is selectively directed into different pathways. These capabilities rely on manipulating the flow of light at the nanoscale, and topological effects promise to protect these pathways from imperfections and disturbances. Successful implementation could lead to more stable and efficient optical devices, potentially revolutionizing fields like optical computing, sensing, and communications by creating circuits that are less susceptible to noise and fabrication errors. The precise control offered by dynamically encircling exceptional points is therefore paramount to realizing these next-generation technologies.

The ability to dynamically control topological chirality represents a significant leap forward in the manipulation of both light and matter. This newfound control isn’t merely about steering photons or particles; it unlocks possibilities for creating devices with unprecedented functionality. Imagine robust waveguiding systems immune to defects, or optical switches that favor one direction of light transmission over another with exceptional precision. Beyond optics, this approach could influence the design of novel materials with tailored electromagnetic properties, and even pave the way for advanced sensing technologies. The implications extend to fields as diverse as telecommunications, where efficient and reliable signal transmission is paramount, and materials science, where the creation of designer materials with unique characteristics is a constant pursuit. Ultimately, harnessing topological chirality promises a transformative impact, offering a versatile toolkit for innovation across numerous scientific and technological disciplines.

The exploration of non-Hermitian systems, as detailed in the study, reveals a fascinating departure from traditional understandings of chirality. It demonstrates how encircling exceptional points-singularities where standard physical properties break down-can give rise to a novel topological chirality. This concept resonates with Albert Einstein’s observation: “The important thing is not to stop questioning.” The research embodies this spirit by questioning the conventional link between chirality and specific eigenmodes, instead revealing a more fundamental, topologically protected form arising from the non-commutative nature of these encircling paths. By meticulously mapping these ‘non-Abelian loops’, the study showcases how seemingly abstract mathematical concepts directly manifest in physical observables, echoing the power of rigorous logic to unlock hidden patterns within complex systems.

Where Do We Go From Here?

The identification of topological chirality through non-Abelian loops around exceptional points offers a compelling, if unsettling, shift in perspective. The conventional reliance on eigenstates to define chirality now appears… provincial. The robustness promised by this topology hinges, of course, on the consistent definition and experimental verification of these non-Abelian structures. The mathematics, while elegant, demands a corresponding physicality; simply demonstrating the existence of a braid group structure is insufficient without observable consequences beyond the abstract.

A critical limitation resides in extending this concept beyond simplified, often artificially constructed, non-Hermitian systems. Real-world materials rarely present the pristine conditions necessary to isolate and manipulate exceptional points with such precision. The challenge, then, lies in identifying or engineering disorder – or indeed, utilizing inherent material properties – that naturally give rise to these topological features. Furthermore, a connection to established frameworks like Berry phase theory remains elusive; bridging that gap would solidify this chirality’s place within the broader landscape of topological physics.

The potential for device applications – robust chiral transport, topologically protected sensors – is frequently invoked. However, these remain speculative until practical material systems are realized. Ultimately, if a pattern cannot be reproduced or explained, it doesn’t exist. The elegance of the mathematics must yield to the scrutiny of experiment, or it remains merely a beautiful curiosity.

Original article: https://arxiv.org/pdf/2512.18789.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/