arXiv subjects

Alexander Cerjan

Publications and source records attributed to Alexander Cerjan.

At least 19 recordsLinked to original sources

arXiv2026

Generating quantum error correcting codes from topological pre-thermal scars

Quantum many-body scars are rare nonthermal states embedded within otherwise thermalizing spectra, making them promising candidates for storing and manipulating quantum information. However, a general framework that can both identify such states without prior knowledge of their microscopic scarring mechanism and determine when they enable a quantum error correcting code has remained elusive. Here, we develop an end-to-end framework connecting nonthermal-state discovery to certifiable quantum error correction using a many-body spectral localizer. By searching for states jointly localized in energy and a diagnostic observable in systems with a many-body grading symmetry, the spectral localizer identifies candidate scars within dense many-body spectra in tandem with a topological index whose shifts determine their graded subspace. For suitable pairs of same-graded scars, the grading renders grading-odd errors exactly detectable, while their localization can suppress the logical action of sufficiently local grading-even error products, yielding an approximate Knill--Laflamme condition and an operator-algebra quantum error-correction structure. We demonstrate the scar-identification framework in one- and two-dimensional density-difference-dependent bosonic models and in the PXP model, and explicitly construct the resulting approximate quantum code in the one-dimensional bosonic system. Our results establish a systematic route from identifying nonthermal many-body states to exploiting them for protecting quantum information, and position pseudospectral methods as a broadly applicable tool for discovering useful behaviors in interacting quantum systems.

cond-mat.quant-gas↗

arXiv2026

Doubly topological harmonic generation

The proposition that band geometry alone can protect optical states against disorder has proven not merely theoretically elegant but experimentally incontrovertible. A key attribute of photonic topological systems is their capacity to simultaneously possess high-intensity excitations at multiple distinct frequencies that are confined to the same topological interface. However, exploiting this freedom to protect the interaction between at least two topological states has remained an open experimental challenge. Here, we report an interaction between two topological states, with one being precisely frequency-doubled to the other, supported in a hybrid plasmonic and photonic topological insulator via nonlinear phase matching. We find that the phase matching inherits a unique spin-momentum locking unseen in conventional nonlinear systems. This ability to bring two topological states into phase-matched nonlinear interaction at a single interface sets the stage for a new class of doubly protected nonlinear photonic devices, potentially finding implications in generating entangled photon pairs with enhanced resilience and robustness for secure quantum information technology.

physics.optics↗

arXiv2026

Band Unfolding via the Quadratic Pseudospectrum

Band theory provides the foundation for understanding electronic structure in crystalline materials, but its reliance on exact translational symmetry limits its applicability to systems with defects, disorder, incommensurate modulations, or large unit cells. Here, we introduce a band unfolding framework that directly generalizes traditional band theory to systems where exact periodicity is absent, and which remains well-defined for both aperiodic and finite systems. To do so, we employ a pseudospectral approach to identify approximate joint eigenvectors of a system's Hamiltonian and translation operators, thereby yielding an unfolded band structure whose features are directly connected to the manifestation of approximate extended states simultaneously localized in energy and crystalline momentum. To reveal bulk-only spectral phenomena in finite systems, we further show that this pseudospectral framework naturally accommodates additional operators that suppress contributions from boundary-localized states, enabling the systematic isolation of intrinsic bulk behavior. We benchmark the scheme on several representative systems in one and two dimensions, including a Fibonacci chain, where our approach is able to both reveal a dispersive envelope while preserving the underlying hierarchy of spectral gaps. Looking forward, this pseudospectral approach may yield a broad framework for predicting momentum-resolved material responses in aperiodic, disordered, and finite systems where conventional band-theoretic methods are not applicable.

cond-mat.mtrl-sci↗

arXiv2026

A Spatial Localizer for Electrons in Insulators

The location of electrons governs phenomena ranging from chemical bonding and electric polarization to the topological classification of band insulators and the emergence of correlated states in quantum matter. While a prescription exists for finding local state representations of electrons in one-dimensional insulators, no comparably general theory exists in higher dimensions. Here, we introduce a general framework for finding the location of electrons in insulators in two and three dimensions based on the spectral properties of quantum-mechanical operators that we term Spatial Localizers. This framework naturally extends the notion of Wannier centers to insulators with boundaries, defects, and disorder, which we use to establish a position-space formulation of the bulk-defect correspondence for electronic charge. This framework also yields maximally localized electronic states. As two representative examples, we show that these states reduce to maximally localized Wannier functions in atomic insulators, whereas in Chern insulators they form coherent states that mirror the coherent-state structure of Landau levels in the quantum Hall effect.

cond-mat.mtrl-sci↗

arXiv2026

Clifford spectrum of three 2 by 2 matrices

We prove that the Clifford spectrum associated to three 2 by 2 matrices is nonempty. The structure of Clifford is described in terms "moving" level curves. We discuss some implication of a conjecture formulated for arbitrary size n by n of three matrices and give an example in the case of three self-adjoint operators in the infinite dimensional Hilbert space.

math.FA↗

arXiv2026

Efficient local classification of parity-based material topology

Although the classification of crystalline materials can be generally handled by momentum-space-based approaches, topological classification of aperiodic materials remains an outstanding challenge, as the absence of translational symmetry renders such conventional approaches inapplicable. Here, we present a numerically efficient real-space framework for classifying parity-based $\mathbb{Z}_2$ topology in aperiodic systems based on the spectral localizer framework and the direct computation of the sign of a Pfaffian associated with a large sparse skew-symmetric matrix. Unlike projector-based or momentum-space-based approaches, our method does not rely on translational symmetry, spectral gaps in the Hamiltonian's bulk, or gapped auxiliary operators such as spin projections, and instead provides a local, energy-resolved topological invariant accompanied by an intrinsic measure of topological protection. A central contribution of this work is the development of a scalable sparse factorization algorithm that enables the reliable determination of the Pfaffian's sign for large sparse matrices, making the approach practical to realistic physical materials. We apply this framework to identify the quantum spin Hall effect in quasicrystalline class AII systems, including gapless heterostructures, and to diagnose fragile topology in a large $C_2 \mathcal{T}$-symmetric photonic quasicrystal. Overall, our results demonstrate that the spectral localizer, combined with efficient sparse numerical methods, provides a unified and robust tool for diagnosing parity-based topological phases in aperiodic electronic, photonic, and acoustic materials where conventional band-theoretic indexes are inapplicable.

physics.comp-ph↗

arXiv2025

Probing Local Topology in a Disordered Higher-Order Topological Insulator

Higher-order topology is prized for its ability to realize lower-dimensional boundary states which are stable beyond fine-tuning. However, disorder presents a failure mechanism that can destroy topological in-gap states. Here, we investigate a disordered two-dimensional polariton lattice and employ the spectral localizer framework to define a real-space topological index rooted in crystalline spatial symmetries. This framework enables direct real-space mapping of topology beyond conventional momentum-space classifications, confirming the presence of corner and edge modes in this generalized Su-Schrieffer-Heeger model. Furthermore, it can directly quantify topological protection of a state. We leverage the versatility of our platform to experimentally realize normally distributed, random disorder and find that the corner states persist until the spectral gap closes. Experimentally, this corresponds to a disorder strength of approximately one quarter of the spectral gap. The spectral localizer accurately identifies the disorder strength at which the bandgap closes, establishing the framework as a predictive tool for every finite size system. Our results broaden the design principles for higher-order topological insulators and open the way towards imple menting disorder-resilient devices for robust lasing, light-routing, and quantum computation.

physics.optics↗

arXiv2025

Semiconductor Meta-Graphene and Valleytronics

Nano-patterned semiconductor interfaces offer a versatile platform for creating quantum metamaterials and exploring novel electronic phenomena. In this study, we illustrate this concept using artificial graphene--a metamaterial featuring distinctive properties including Dirac and saddle points. We demonstrate that introducing additional nano-patterning can open a Dirac band gap, giving rise to what we term artificial hexagonal boron nitride (AhBN). The calculated valley Chern number of AhBN indicates the presence of topological valley Hall states confined to Dirac-gap domain walls. A key question is whether these one-dimensional edge states are topologically protected against disorder, given their vulnerability to Anderson localization. To this end, we perform band structure and electronic transport simulations under experimentally relevant disorder, including charge puddles and geometric imperfections. Our results reveal the resilience of the domain wall states against typical experimental disorder, particularly while the AhBN band gap remains open. The localization length along the domain wall can reach several microns--several times longer than the bulk electron mean free path--even though the number of bulk transport channels is greater. To enhance the effectiveness of the low-dissipation domain wall channel, we propose ribbon geometries with a large length-to-width ratio. These findings underscore both the potential and challenges of AhBN for low-energy, power-efficient microelectronic applications.

cond-mat.mes-hall↗

arXiv2025

Dynamically reconfigurable topological routing in nonlinear photonic systems

The propagation path of topologically protected states is bound to the interface between regions with different topology, and as such, the functionality of linear photonic devices leveraging these states is fixed during fabrication. Here, we propose a mechanism for dynamic control over a driven dissipative system's local topology, yielding reconfigurable topological interfaces and thus tunable paths for protected routing. We illustrate our approach in non-resonantly pumped polariton lattices, where the nonlinear interaction between the polaritons and the exciton reservoir due to non-resonant pumping can yield a dynamical change of the topology. Moreover, using a continuous model of the polariton system based on a driven-dissipative Gross-Pitaevskii equation alongside the spectral localizer framework, we show that the local changes in the nonlinear non-Hermitian system's topology are captured by a local Chern marker. Looking forward, we anticipate such reconfigurable topological routing will enable the realization of novel classes of topological photonic devices.

physics.optics↗

arXiv2025

Efficient first-principles inverse design of nanolasers

We develop and demonstrate a first-principles approach, based on the nonlinear Maxwell-Bloch equations and steady-state ab-initio laser theory (SALT), for inverse design of nanostructured lasers, incorporating spatial hole-burning corrections, threshold effects, out-coupling efficiency, and gain diffusion. The resulting figure of merit exploits the high-$Q$ regime of optimized laser cavities to perturbatively simplify the nonlinear model to a single linear ''reciprocal'' Maxwell solve. The consequences for laser-cavity design, and in particular the strong dependence on the nature of the gain region, are demonstrated using topology optimization of both 2d and full 3d geometries.

physics.optics↗

arXiv2025

Detecting local topology with the spectral localizer

The spectral localizer is a predictive framework for the computation of topological invariants of natural and artificial materials. Here, three crucial improvements on the criterion for the validity of the framework are reported: first, merely a properly defined local spectral gap of the Hamiltonian is required, second, only relative bounds on the Hamiltonian and its noncommutative derivative are relevant, and, third, the numerical constant in a tapering estimate is improved. These developments further stress the local nature of the spectral localizer framework, enabling more precise predictions in heterostructures, aperiodic, and disordered systems. Moreover, these results strengthen the bounds on the spectral localizer's spectral flow when crossing topological phase boundaries.

math-ph↗

arXiv2025

Dimensional crossover of class D real-space topological invariants

The topological properties of a material depend on its symmetries, parameters, and spatial dimension. Changes in these properties due to parameter and symmetry variations can be understood by computing the corresponding topological invariant. Since topological invariants are typically defined for a fixed spatial dimension, there is no existing framework to understand the effects of changing spatial dimensions via invariants. Here, we introduce a framework to study topological phase transitions as a system's dimensionality is altered using real-space topological markers. Specifically, we consider Shiba lattices, which are class D materials formed by magnetic atoms on the surface of a conventional superconductor, and characterize the evolution of their topology when an initial circular island is deformed into a chain. We also provide a measure of the corresponding protection against disorder. Our framework is generalizable to any symmetry class and spatial dimension, potentially guiding the design of materials by identifying, for example, the minimum thickness of a slab required to maintain three-dimensional topological properties.

cond-mat.mes-hall↗

arXiv2025

Classification of Fragile Topology Enabled by Matrix Homotopy

The moire flat bands in twisted bilayer graphene have attracted considerable attention not only because of the emergence of correlated phases but also due to their nontrivial topology. Specifically, they exhibit a new class of topology that can be nullified by the addition of trivial bands, termed fragile topology, which suggests the need for an expansion of existing classification schemes. Here, we develop a Z2 energy-resolved topological marker for classifying fragile phases using a system's position-space description, enabling the direct classification of finite, disordered, and aperiodic materials. By translating the physical symmetries protecting the system's fragile topological phase into matrix symmetries of the system's Hamiltonian and position operators, we use matrix homotopy to construct our topological marker while simultaneously yielding a quantitative measure of topological robustness. We show our framework's effectiveness using a C2T-symmetric twisted bilayer graphene model and photonic crystal as a continuum example. We have found that fragile topology can persist both under strong disorder and in heterostructures lacking a bulk spectral gap, and even an example of disorder-induced re-entrant topology. Overall, the proposed scheme serves as an effective tool for elucidating aspects of fragile topology, offering guidance for potential applications across a variety of experimental platforms from topological photonics to correlated phases in materials.

cond-mat.mes-hall↗

arXiv2025

Topological phenomena in artificial quantum materials revealed by local Chern markers

A striking example of frustration in physics is Hofstadter's butterfly, a fractal structure that emerges from the competition between a crystal's lattice periodicity and the magnetic length of an applied field. Current methods for predicting the topological invariants associated with Hofstadter's butterfly are challenging or impossible to apply to a range of materials, including those that are disordered or lack a bulk spectral gap. Here, we demonstrate a framework for predicting a material's local Chern markers using its position-space description and validate it against experimental observations of quantum transport in artificial graphene in a semiconductor heterostructure, inherently accounting for fabrication disorder strong enough to close the bulk spectral gap. By resolving local changes in the system's topology, we reveal the topological origins of antidot-localized states that appear in artificial graphene in the presence of a magnetic field. Moreover, we show the breadth of this framework by simulating how Hofstadter's butterfly emerges from an initially unpatterned 2D electron gas as the system's potential strength is increased, and predict that artificial graphene becomes a topological insulator at the critical magnetic field. Overall, we anticipate that a position-space approach to determine a material's Chern invariant without requiring prior knowledge of its occupied states or bulk spectral gaps will enable a broad array of fundamental inquiries and provide a novel route to material discovery, especially in metallic, aperiodic, and disordered systems.

cond-mat.mes-hall↗

arXiv2025

Topological photonic crystal fibre

Photonic crystal fibres (PCFs) are optical fibres that guide light using a modulated dielectric medium. They provide an exceptionally versatile platform for various applications, thanks to the flexibility with which light-guiding can be customised by modifying the fibre geometry. Here, we realise a PCF with guided modes produced by photonic bandstructure topology rather than conventional mode-trapping mechanisms. The design, which is compatible with the stack-and-draw fabrication process, consists of a cross-sectional photonic topological crystalline insulator with a disclination. A bulk-defect correspondence produces degenerate topological modes, lying below the cladding light line. We use various theoretical methods to confirm their topological origins, including a spectral localiser that makes minimal assumptions about the bandstructure. Our experiments on the fabricated topological fibre show it transmitting visible to near-infrared light with low losses of 10--20 dB/km, which do not increase much when the fibre is bent. A comparable solid-core PCF of conventional design exhibits substantially higher bending losses. Optical fibres based on topological modes thus hold promise for improved performance and novel functionalities.

physics.optics↗

arXiv2025

Bi-stability and period-doubling cascade of frequency combs in exceptional-point lasers

Recent studies have demonstrated that a laser can self-generate frequency combs when tuned near an exceptional point (EP), where two cavity modes coalesce. These EP combs induce periodic modulation of the population inversion in the gain medium, and their repetition rate is independent of the laser cavity's free spectral range. In this work, we perform a stability analysis that reveals two notable properties of EP combs, bi-stability and a period-doubling cascade. The period-doubling cascade enables halving of the repetition rate while maintaining the comb's total bandwidth, presenting opportunities for the design of highly compact frequency comb generators.

physics.optics↗

arXiv2024

Quasinormal coupled-mode analysis of dynamic gain in exceptional-point lasers

One of the key features of lasers operating near exceptional points (EPs) is that the gain medium can support an oscillating population inversion above a pump threshold, leading to self-modulated laser dynamics. This unusual behavior opens up new possibilities for frequency comb generation and temporal modulation. However, the dynamic population inversion couples signals with different frequencies and thus cannot be captured by conventional temporal coupled-mode theory (TCMT) based on static saturable gain. In this paper, we develop a perturbative coupled-mode analysis framework to capture the spatial-temporal dynamics of near-EP lasers. By decomposing discrete frequency generation into multiple excitations of resonant modes, our analysis establishes a minimal physical model that translates the local distribution of dynamic population-inversion into a resonant modal interpretation of laser gain. This work enables the exploration of unique properties in this self-time-modulated systems, such as time-varying scattering and non-reciprocal transmission.

physics.optics↗

arXiv2024

Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory

Recently, the spectral localizer framework has emerged as an efficient approach to classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts like topological solitons and topological dynamics, where a system's occupation induces a local change in its topology due to the nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of $C^*$-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches to classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective, and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework's application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell's equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical $K$-theory, two mathematical concepts that form the foundation for the spectral localizer framework.

physics.optics↗