Beyond Hermiticity: Mapping Topology in Open Quantum Systems

Author: Denis Avetisyan

A new approach combines theoretical insights with Floquet-Monodromy Spectroscopy to characterize robust topological invariants in driven and dissipative quantum systems.

The framework divides the parameter space into Stokes Sectors-$\mathcal{V}_j$-and detects changes in basis via monodromy as a connection traverses Stokes Rays, effectively mapping jumps-$S_j$-in the solution space to reveal underlying geometric structure.

Researchers define and measure a quantum geometric tensor to resolve Stokes phenomena arising from exceptional points in non-Hermitian systems.

Conventional topological invariants fail to fully describe quantum systems operating far from equilibrium, particularly those that are open, driven, or non-Hermitian. In this work, ‘Quantum Geometric Tensor in the Wild: Resolving Stokes Phenomena via Floquet-Monodromy Spectroscopy’, we demonstrate that these limitations arise from essential singularities and resolve this crisis by introducing Floquet-Monodromy Spectroscopy (FMS) to extract previously hidden geometric data-the Stokes phenomenon-and map it to measurable time-domain observables. This allows for the rigorous reconstruction of non-perturbative physics and defines a new “Stokes invariant” for classifying quantum phases beyond conventional topology. Could this framework unlock a more complete understanding of complex quantum materials and dissipative systems?

Beyond the Standard Model: Embracing the Inevitable Imperfections

The foundation of traditional quantum mechanics rests upon the use of Hermitian Hamiltonians – mathematical operators ensuring that physical observables, like energy, remain real-valued. However, this principle fundamentally breaks down when considering systems that actively exchange energy and information with their surroundings – systems known as open quantum systems. Unlike their isolated counterparts, these systems exhibit non-Hermitian behavior, meaning their Hamiltonians lack the necessary symmetry. This isn’t a mathematical quirk, but a direct consequence of describing realities like optical cavities with loss, decaying particles, or any quantum system coupled to an external environment. Consequently, the standard tools of quantum mechanics become insufficient to accurately predict the behavior of these ubiquitous open systems, necessitating a shift towards a more generalized framework capable of handling non-Hermitian descriptions and unlocking the potential of these uniquely behaving quantum entities.

The foundations of quantum mechanics rest on the assumption of Hermitian operators, but many real-world systems constantly interact with their environment, violating this condition and giving rise to what are known as Exceptional Points. These points represent singularities in the parameter space of a quantum system, where standard notions of eigenvalues and eigenvectors-and thus, predictable behavior-break down. At an Exceptional Point, two or more eigenvalues coalesce, and the corresponding eigenvectors become identical, rendering the usual quantum description incomplete and potentially leading to inaccurate predictions about the system’s evolution. This isn’t merely a mathematical curiosity; the presence of Exceptional Points dramatically alters the system’s sensitivity to perturbations and can lead to enhanced or suppressed responses, opening up possibilities for novel device functionalities-but only if these singularities are understood and accounted for in theoretical models.

The ability to manipulate and harness the behavior of open quantum systems hinges on a thorough understanding of the singularities that arise when traditional quantum descriptions fail-specifically, at Exceptional Points. These points represent a breakdown in the conventional framework, but also offer a gateway to novel physics inaccessible in closed systems. This work introduces a theoretical framework designed to reconstruct the behavior of these systems even at and around these singularities, leveraging non-perturbative methods to move beyond approximations. By accurately describing the physics in these previously intractable regimes, researchers can potentially engineer systems with enhanced sensitivity, tailored energy transfer, and fundamentally new quantum functionalities-opening avenues for advancements in areas like laser design, sensing technologies, and topological photonics. The framework’s capacity to accurately model these complex interactions is therefore critical for translating theoretical possibilities into practical quantum devices.

This framework resolves the limitations of standard quantum geometry in characterizing singular phases by augmenting local metrics with topological invariants extracted via non-Abelian monodromy, enabling the reconstruction of non-perturbative wavefunctions from divergent series.

Decoding the Geometry of Dissipation: A New Quantum Metric

The Quantum Geometric Tensor (QGT) characterizes the state space of open quantum systems by generalizing the Fubini-Study metric, traditionally used for closed, pure quantum states. While the Fubini-Study metric relies on the projection onto a pure state, the QGT is defined for mixed states using the expectation value of the quantum Fisher information. This allows for geometric characterization even when the system is described by a density matrix $\rho$, rather than a state vector $|\psi\rangle$. The QGT is a rank-two tensor that determines an inner product on the space of states, defining distances and angles between them. Critically, the QGT is not necessarily positive definite for open systems, reflecting the non-Hermiticity and dissipation inherent in their dynamics, yet still provides a consistent geometric structure.

The Quantum Geometric Tensor (QGT) facilitates the definition of geometric quantities – specifically distances and curvature – for open quantum systems even when described by non-Hermitian Hamiltonians. Traditional geometric characterizations rely on Hermitian inner products to define distances between quantum states; however, the QGT employs a generalized inner product constructed from the system’s completely positive trace-preserving map, allowing for a consistent geometric structure in non-Hermitian contexts. This is achieved by defining the QGT, $Q_{ij} = \langle \partial_i \psi | \partial_j \psi \rangle – \langle \partial_i \psi | \psi \rangle \langle \psi | \partial_j \psi \rangle$, where $|\psi\rangle$ represents the system’s state and $\partial_i$ denotes differentiation with respect to a parameter. The resulting tensor then enables the computation of the Fubini-Study metric, $g_{ij} = Q_{ij}$, and associated geometric properties, even when the Hamiltonian is non-Hermitian, providing a means to analyze system geometry beyond the limitations of standard Hermitian treatments.

The Quantum Geometric Tensor (QGT) facilitates the analysis of open quantum system dynamics by exposing the underlying Berry curvature, a geometric phase arising from the adiabatic evolution of quantum states. This curvature, represented mathematically as the curl of the quantum geometric vector potential, directly influences the system’s response to external perturbations and can be experimentally accessed through measurable quantities. Furthermore, the QGT enables the calculation of integer-quantized topological invariants, such as the Chern number, which characterize the global geometric properties of the state space and are robust against small perturbations. These invariants provide a means to classify and understand topologically distinct phases of open quantum systems and predict their quantized response functions, offering potential for novel device design and control strategies.

The cQGT classification demonstrates robustness to Hamiltonian noise and confirms Sabbah scaling laws, revealing that non-perturbative geometry governs the dynamic universality class and distinguishes different topological regimes.

Reconstructing the Indeterminate: Beyond Perturbation Theory

The Complete Quantum Geometry Theory (QGT) represents an extension of the standard QGT framework designed to address singularities, specifically Exceptional Points (EPs), in the geometric description of quantum systems. Unlike the standard theory which breaks down at EPs, the Complete QGT provides a regularized description, maintaining continuity of geometric properties such as connections and curvatures even as parameters approach these singular points. This regularization is achieved through a refined mathematical treatment of the geometric phases and monodromy matrices, allowing for consistent calculations and predictions across the entire parameter space, including at and beyond EPs. Consequently, physical quantities derived from the geometric description, such as energy levels and transition probabilities, remain well-defined and continuous, avoiding the divergences typically encountered with standard perturbative approaches near EPs.

Resurgence Theory provides a mathematical framework for reconstructing solutions to differential equations even when standard perturbation methods fail, particularly around singularities. This is achieved by analyzing the non-perturbative contributions, often expressed as exponentially small terms, that are neglected in traditional asymptotic expansions. The theory posits that these seemingly insignificant terms are not merely corrections, but are essential components of a complete solution, and can be systematically recovered through the analysis of Borel transforms and associated monodromy properties. Specifically, Resurgence allows for the analytic continuation of solutions beyond the radius of convergence of perturbative series, effectively bypassing the limitations imposed by singularities and ensuring a globally defined solution space. This reconstruction is crucial for describing physical phenomena where non-perturbative effects, while small, can significantly alter system behavior and define the underlying structure.

Floquet Monodromy Spectroscopy (FMS) provides experimental verification of the geometric features predicted by the Quasi-Exactly Solvable (QES) system description, specifically through the observation of monodromy matrices. These matrices, obtained via analysis of the Floquet spectrum, directly encode information about the system’s geometric properties, allowing for the measurement of Stokes multipliers. Crucially, experimental results consistently demonstrate that these measured Stokes multipliers are integer-valued, confirming a key prediction of the QES framework and validating the regularization achieved even in the presence of Exceptional Points. The integer nature of these multipliers is a robust signature of the underlying geometric structure and distinguishes the QES approach from standard perturbation theories.

Leveraging complete quantum group theory, the Functional Mock-up System (FMS) accurately resolves the resurgence problem by recovering the non-perturbative solution (black) and overcoming the divergence issues of standard perturbation theory (red) and the limitations of optimal truncation (blue).

Mapping the Singular Landscape: Topology as a Guiding Principle

A profound connection arises when the Complete Quantum Geometry and Topology (QGT) is considered alongside the Wild Riemann-Hilbert Correspondence, revealing a deep relationship between a system’s singular Hamiltonian – the operator describing its total energy – and its underlying geometric characteristics. This interplay isn’t merely observational; it establishes a robust link, meaning alterations in the Hamiltonian’s singularities directly correspond to predictable changes in the system’s geometry, and vice versa. Specifically, the Correspondence allows for the extraction of geometric information from the algebraic properties of the Hamiltonian, even when traditional geometric methods fail due to the system’s singularities. This framework permits the precise mapping of complex, non-smooth landscapes defined by the Hamiltonian, offering a pathway to understand how energy levels and wave functions are shaped by these geometric features, and providing a tool to predict system behavior based on its intrinsic structural properties. The implications extend to fields where singular geometries are prevalent, such as the study of disordered materials and complex quantum systems.

The robust link between the Complete Quantum Geometry and the Wild Riemann-Hilbert Correspondence facilitates the identification of Topological Invariants, characteristics of a system that persist regardless of continuous alterations to its shape or form. These invariants are not merely abstract mathematical curiosities; they directly reveal fundamental properties of the quantum system under investigation. Importantly, the ability to pinpoint these invariants allows for empirical verification of predicted scaling laws, such as the relationship between energy difference, $ΔE$, and frequency, $ω$, expressed as $ΔE ∝ ω^(1+1/k)$. This confirms theoretical predictions and provides a powerful tool for characterizing the behavior of complex quantum phenomena, establishing a bridge between abstract mathematical descriptions and observable physical realities.

The geometry of open quantum systems, those interacting with their environment, is elegantly captured by the Dissipative Mixed Hodge Module, a sophisticated mathematical construction extending traditional Hodge theory. This framework allows researchers to move beyond the limitations of closed, Hermitian systems and effectively analyze non-Hermitian Hamiltonians – those where energy is not necessarily conserved. By characterizing the singular points and topological features within these systems, the module reveals how energy dissipation influences quantum behavior, providing tools to understand phenomena like resonance decay and tunneling. Crucially, it establishes a rigorous link between the algebraic properties of the Hamiltonian and the geometric properties of the corresponding quantum state space, enabling the prediction and interpretation of observable phenomena in dissipative environments, and offering insights into the broader landscape of non-equilibrium quantum mechanics.

Open quantum systems can be classified by their singularity rank, differentiating 'tame' systems with regular behavior from 'wild' systems exhibiting more complex, irregular characteristics and associated invariants. — Open quantum systems can be classified by their singularity rank, differentiating ‘tame’ systems with regular behavior from ‘wild’ systems exhibiting more complex, irregular characteristics and associated invariants.

Towards Inherently Robust Quantum Technologies

Quantum technologies, despite their potential, are notoriously susceptible to environmental noise, which introduces errors and degrades performance. Recent investigations reveal that a key to overcoming this fragility lies in leveraging the geometric and topological characteristics of open quantum systems – those interacting with their surroundings. Rather than shielding a quantum system entirely, researchers are discovering methods to encode information in its geometric properties, making it resilient to local disturbances. This approach treats noise not simply as a detractor, but as a force that can be navigated and even exploited. By carefully manipulating the system’s topology – its fundamental shape and connectivity – and understanding how quantum states evolve on these geometric landscapes, it becomes possible to create robust quantum bits, or qubits, that maintain coherence – the delicate quantum state essential for computation – for extended periods. This control is achieved by designing systems where certain quantum states are intrinsically protected by their geometric arrangement, effectively creating ‘safe zones’ against environmental perturbations, promising a new generation of reliable quantum devices.

The Stokes phenomenon, long considered a disruptive element in quantum systems, is increasingly understood not as an obstruction, but as a powerful tool for state manipulation. Originally identified in classical optics as a discontinuity in polarization, this phenomenon manifests in open quantum systems as a sudden change in the evolution of quantum states due to the flow of information between the system and its environment. Recent research demonstrates that by carefully engineering the conditions that give rise to the Stokes phenomenon – specifically, by controlling the geometry of the system’s interaction with its surroundings – it becomes possible to steer quantum states along desired trajectories. This offers a novel pathway for implementing robust quantum gates and controlling entanglement, effectively transforming a traditionally problematic effect into a resource for precision quantum control and potentially enhancing the resilience of quantum technologies against decoherence.

Recent progress in quantum information processing hinges on a geometric understanding of quantum states and their evolution, a concept now substantiated by both rigorous theoretical models and increasingly precise experimental results. This approach moves beyond traditional methods by focusing on the shape and topology of quantum states, allowing for greater resilience against environmental disturbances – a major hurdle in building stable quantum computers. Crucially, the successful measurement of integer-valued Milnor numbers – topological invariants characterizing the ‘twisting’ of quantum states – provides concrete evidence that these geometric principles aren’t merely abstract concepts, but directly reflect measurable properties of quantum systems. This ability to precisely characterize and control these topological features opens avenues for designing quantum devices with unprecedented levels of stability and accuracy, potentially ushering in a new era of fault-tolerant quantum computation and advanced quantum technologies.

Observation of the Stokes phase transition reveals a jump in the topological invariant (red) concurrent with a sustained, finite spectral gap (dashed), confirming its non-traditional, or 'phantom', nature. — Observation of the Stokes phase transition reveals a jump in the topological invariant (red) concurrent with a sustained, finite spectral gap (dashed), confirming its non-traditional, or ‘phantom’, nature.

The pursuit of topological invariants in non-Hermitian systems, as detailed in this work, reveals a fascinating truth about modeling reality. It isn’t merely about achieving mathematical elegance, but about acknowledging the inherent imperfections and dissipation present in any physical process. As Niels Bohr observed, “It is the theory that decides what can be observed.” This sentiment resonates deeply with the development of Floquet-Monodromy Spectroscopy; the theoretical framework dictates how robust topological features can be extracted even amidst exceptional points and driven dissipation. The article demonstrates that seemingly intractable systems yield to analysis when the proper lens-a carefully constructed theory-is applied, transforming a chaotic landscape into one where meaningful invariants emerge.

Where Do We Go From Here?

The pursuit of topological invariants in non-Hermitian systems isn’t about finding order, but about charting the precise ways in which systems fail to be orderly. This work, by offering a method-Floquet-Monodromy Spectroscopy-to navigate the Stokes phenomenon in dissipative quantum systems, doesn’t solve the problem of exceptional points; it merely provides a more refined instrument for cataloging their influence. Humans crave categorization, a soothing illusion of control over inherent instability. The true challenge lies not in defining robust invariants, but in acknowledging the limits of that very definition.

Future iterations will undoubtedly focus on extending this framework to higher-dimensional systems and more complex driving protocols. But a more pressing concern is the translation of these theoretical tools into genuinely predictive models. It’s easy to demonstrate the existence of a topological invariant; it is far harder to anticipate its effects on measurable quantities, especially given the inherent sensitivity of these systems to even minor perturbations. The temptation will be to chase ever-increasing precision, to refine the map while ignoring the shifting terrain.

Ultimately, the value of this approach may not lie in its ability to create perfectly stable quantum devices, but in its capacity to reveal the underlying fragility of all systems-physical, economic, or otherwise. The Stokes phenomenon isn’t a bug; it’s a feature. It is a reminder that change is the only constant, and that any attempt to impose absolute order is, at best, a temporary reprieve from the inevitable chaos.

Original article: https://arxiv.org/pdf/2512.20253.pdf

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