arXiv subjects

Tomáš Bzdušek

Publications and source records attributed to Tomáš Bzdušek.

At least 19 recordsLinked to original sources

arXiv2026

Topological characterization of multifold band degeneracies in Altland-Zirnbauer symmetry classes

Topological degeneracies of energy bands in crystalline matter are conventionally characterized by invariants computed on an enclosing sphere over which the spectrum remains gapped, a program completed for minimal degeneracies in all ten Altland-Zirnbauer (AZ) symmetry classes. Higher-order degeneracies have instead been studied almost exclusively under crystalline-symmetry protection. Here, we characterize generic $n$-fold degeneracies stabilized solely by AZ symmetries acting locally in momentum space. Their codimension grows quadratically with $n$, placing multifold nodes in parameter spaces combining momenta with tuning parameters or synthetic dimensions. Crucially, the enclosing-sphere paradigm faces a fundamental obstruction: two $(n\,{-}\,1)$-fold degeneracy loci emanating from the $n$-fold node necessarily pierce every choice of enclosing sphere, leaving no uniform spectral gap and thus no standard homotopy classification. We elevate this obstruction into the diagnostic itself. Namely, on the two nodal manifolds where the loci cross the sphere, complementary spectral gaps are restored, admitting conventional band invariants (Chern numbers, Stiefel-Whitney classes, and winding numbers). This observation establishes a general two-way correspondence: (1) the multifold node is topologically protected whenever the associated nodal manifolds are robustly linked, and (2) invariants on cycles of one manifold encode their linking numbers with cycles of the other. Carrying out this program for minimal models of all ten AZ classes, we recast multifold band topology as the topology of linked nodal manifolds and lay the foundation for characterizing multifold nodes in models with arbitrarily many bands.

cond-mat.mes-hall↗

arXiv2026

Topological non-Abelian gauge structures in Cayley-Schreier lattices

Crystalline constructions known as Cayley-Schreier lattices have been suggested as a platform for realizing arbitrary gauge fields in synthetic crystals with real hopping amplitudes. Here, we reveal that Cayley-Schreier lattices can naturally give rise to implementable lattice systems that incorporate non-Abelian gauge structures transforming into a space-group symmetry. We show that their symmetry sectors can be interpreted as blocks of pseudospin models---some of which correspond to true spinors---that can realize a wealth of different topological invariants in a single setup. We underpin these general results with concrete models and illustrate how they can be implemented in technologically available experimental platforms. Our work sets the stage for a systematic investigation of topological insulators and metals with non-Abelian gauge structures.

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arXiv2026

Universal aspects of bulk density of states in non-Hermitian lattices

Non-Hermitian lattice Hamiltonians generally exhibit strong boundary sensitivity, with periodic and open boundary conditions producing distinct density of states (DOS) in the complex-energy plane. This has led to the view that extended non-Hermitian systems lack a unique bulk DOS, with different prescriptions representing inequivalent bulk physics. Here, we show that this apparent ambiguity is largely illusory. For any finite-range tight-binding Hamiltonian, we establish a universal bulk structure: all DOS definitions arising as thermodynamic limits of finite systems share identical multipole moments and generate identical bulk dynamics at finite times and for observables measured far from boundaries. This universality is intimately tied to the thermodynamic Green's functions, which we show to be independent of the boundary condition for large enough complex frequencies. Among all equivalent descriptions, we identify the Brown measure - obtained via Hermitization and resolvent analysis - as a canonical and convenient representative of the bulk DOS, defined directly from the infinite-volume Hamiltonian. We further show that point-gap topology imposes additional universal constraints: boundary-dependent Green's functions are forced to coincide throughout topologically trivial point gaps. This, in particular, provides a systematic criterion, valid in arbitrary dimension, for determining where and how eigenvalues of different boundary truncations can accumulate in the complex plane, and precisely delineates the regime in which the DOS ambiguity retains physical significance.

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arXiv2026

Topological states and flat bands in exactly solvable decorated Cayley trees

We derive the full spectrum of decorated Cayley trees that constitute tree analogs of selected two-dimensional Euclidean lattices; namely of the Lieb, the double Lieb, the kagome, and the star lattice. The common feature of these Euclidean lattices is that their nearest-neighbor models give rise to flat energy bands interpretable through compact localized states. We find that the tree analogs exhibit similar flat or nearly flat energy bands at the corresponding energies. Interestingly, such flat bands in the decorated Cayley trees acquire an interpretation that is absent in their Euclidean counterparts: as edge states localized to the inner or the outer boundary of the tree branches. In particular, we establish an exact correspondence between the Lieb-Cayley tree and an ensemble of one-dimensional Su-Schrieffer-Heeger chains, which maps topological edge states on one side of the chains to flat-band states localized in the bulk of the tree, furnishing the flat energy band with a topological stability. Similar mapping to topological edge states or to states bound to edge defects in one-dimensional chains is shown for flat-band states in all the considered tree decorations. We finally show that the persistence of exact flat bands on infinite decorated trees (i.e., Bethe lattices) arises naturally from a covering interpretation of tree graphs. Our findings reveal a rich landscape of flat-band and topological phenomena in non-Euclidean systems, where geometry alone can generate and stabilize unconventional quantum states.

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arXiv2026

From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory

Topological principles constitute at present an integral component of condensed matter physics, permeating the modern characterization of electronic states while also guiding materials design. In this brief Perspective, I highlight three research threads in single-particle topological band theory that have recently gained momentum: (i) the rise of the quantum geometric tensor, whose components can at present be directly accessed with optical probes; (ii) the notions of delicate and multigap topology, which fall outside the scope of tenfold way and symmetry-based indicators yet leave robust physical fingerprints; and (iii) the consideration of bundle gerbes, which capture formerly overlooked higher-form topological aspects of energy bands. These distinct directions have been elegantly woven together: delicate and multigap topological insulators have peculiar features in quantum geometry that can be conveniently captured by bundle gerbes. This viewpoint exposes the recently identified quantization of a non-linear optical response, and it invites deeper and systematic investigations into geometric and topological aspects of band structures beyond conventional Berryology.

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arXiv2026

Hopf Exceptional Points

Exceptional points at which eigenvalues and eigenvectors of non-Hermitian matrices coalesce are ubiquitous in the description of a wide range of platforms from photonic or mechanical metamaterials to open quantum systems. Here, we introduce a class of Hopf exceptional points (HEPs) that are protected by the Hopf invariants (including the higher-dimensional generalizations) and which exhibit phenomenology sharply distinct from conventional exceptional points. Saliently, owing to their $\mathbb{Z}_2$ topological invariant related to the Witten anomaly, three-fold HEPs and symmetry-protected five-fold HEPs act as their own ``antiparticles". Furthermore, based on higher homotopy groups of spheres, we predict the existence of multifold HEPs and symmetry-protected HEPs with non-Hermitian topology captured by a range of finite groups (such as $\mathbb{Z}_3$, $\mathbb{Z}_{12}$, or $\mathbb{Z}_{24}$) beyond the periodic table of Bernard-LeClair symmetry classes.

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arXiv2025

Superconductivity in hyperbolic spaces: Cayley trees, hyperbolic continuum, and BCS theory

We investigate $s$-wave superconductivity in negatively curved geometries, focusing on Cayley trees and the hyperbolic plane. Using a self-consistent Bogoliubov-de Gennes approach for trees and a BCS treatment of the hyperbolic continuum, we establish a unified mean-field framework that captures the role of boundaries in hyperbolic spaces. For finite Cayley trees with open boundaries, the superconducting order parameter localizes at the edge while the interior can remain normal, leading to two distinct critical temperatures: $T_\textrm{c}^\textrm{edge} > T_\textrm{c}^\textrm{bulk}$. A corresponding boundary-dominated phase also emerges in hyperbolic annuli and horodisc regions, where radial variations of the local density of states enhance edge pairing. We also demonstrate that the enhancement of the density of states at the boundary is significantly more pronounced for the discrete tree geometry. Our results show that, owing to the macroscopic extent of the boundary, negative curvature can stabilize boundary superconductivity as a phase that persists in the thermodynamic limit on par with the bulk superconductivity. These results highlight fundamental differences between bulk and boundary ordering in hyperbolic matter, and provide a theoretical framework for future studies of correlated phases in negatively curved systems.

cond-mat.supr-con↗

arXiv2025

Superconductivity in hyperbolic spaces: Regular hyperbolic lattices and Ginzburg-Landau theory

We study $s$-wave superconductivity in hyperbolic spaces using the Bogoliubov-de Gennes theory for discrete hyperbolic lattices and the Ginzburg-Landau theory for the continuous hyperbolic plane. Hyperbolic lattices maintain a finite fraction of boundary sites regardless of system size, thus fundamentally altering superconductivity through enhanced boundary effects absent in flat space. Within the BCS framework for hyperbolic lattices, uniform systems reproduce standard bulk behavior, whereas finite systems with open boundaries, studied through exact diagonalization and Cayley-tree approximations, exhibit boundary-enhanced superconductivity and boundary-only superconducting states that persist above the bulk critical temperature. Numerical studies further reveal that boundary termination critically determines superconducting properties; in particular, rough boundaries with dangling bonds generate zero-energy modes that raise critical temperatures by several times relative to smooth boundaries. Turning to the complementary Ginzburg-Landau analysis of the hyperbolic plane, we find that finite geometries permit radial variations of the condensate absent in infinite space. Owing to the interplay between coherence length and curvature radius, the theory exhibits two types of superconductivity even without magnetic fields, with vortices replaced by lines of vanishing order parameter in the nontrivial type. Our findings establish hyperbolic geometry as a platform for engineering boundary-controlled superconductivity, opening new directions for physics in curved spaces in condensed matter and holography.

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arXiv2025

Hyperbolic Spin Liquids

Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev's spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group symmetries to solve the model exactly. We elucidate the ground-state phase diagram on the $\{8,3\}$ lattice and find a gapped $\mathbb{Z}_2$ spin liquid with Abelian anyons, a gapped chiral spin liquid with non-Abelian anyons and chiral edge states, and a Majorana metal whose finite low-energy density of states is dominated by non-Abelian Bloch states.

cond-mat.str-el↗

arXiv2025

Delicate Wannier insulators

The defining feature of topological insulators is that their valence states are not continuously deformable to a suitably defined atomic limit without breaking the symmetry or closing the energy gap. When the atomic limit is given by symmetric exponentially-localized Wannier orbitals, one finds stable and fragile topological insulators characterized by robust bulk-boundary correspondence. More recently, delicate topological insulators (DIs) have been introduced, whose metallic states are guaranteed only at sharply terminated edges and surfaces. Although Wannierizable, their Wannier orbitals necessarily span multiple unit cells, thus refining the notion of the atomic limit. In this work, we extend delicate topological invariants from Bloch states to hybrid Wannier functions. The resulting models, dubbed delicate Wannier insulators (DWIs), are deformable to unicellular atomic limit in the absence of edges and surfaces; nevertheless, they exhibit obstructions to such deformations as well as topological boundary states in the presence of sharply terminated hinges and corners. We present a layering construction that allows us to elevate a DI in $d$ dimensions into a DWI in $(d\,{+}\,1)$ dimensions. We illustrate the phenomenology of DWIs by deploying the layering construction on three concrete models.

cond-mat.mes-hall↗

arXiv2025

Multifractal statistics of non-Hermitian skin effect on the Cayley tree

Multifractal analysis is a powerful tool for characterizing the localization properties of wave functions. Despite its utility, this tool has been predominantly applied to disordered Hermitian systems. Multifractal statistics associated with the non-Hermitian skin effect remain largely unexplored. Here, we demonstrate that the tree geometry induces multifractal statistics for the single-particle skin states on the Cayley tree by deriving the analytical expression of multifractal dimensions. This sharply contrasts with the absence of multifractal properties for conventional single-particle skin effects in crystalline lattices. Our work uncovers the unique feature of the skin effect on the Cayley tree and provides a novel mechanism for inducing multifractality in open quantum systems without disorder.

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arXiv2024

Hyperbolic non-Abelian semimetal

We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in a negatively curved plane. Because of their distinct translation group structure, such lattices are associated with a high-dimensional reciprocal space. In addition, they support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations. Combining diverse numerical and analytical approaches, we uncover an unconventional scaling in the density of states at low energies, and illuminate a nodal manifold of codimension five in the reciprocal space. The nodal manifold is topologically protected by a nonzero second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number.

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arXiv2024

Topological linear response of hyperbolic Chern insulators

We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to $-e^2C_{ij}/h$, where $C_{ij}$ is the first Chern number. Through a flux-threading argument, we provide an interpretation of the Chern number as a topological invariant in hyperbolic band theory. We demonstrate that, although it receives contributions from both Abelian and non-Abelian Bloch states, the Chern number can be calculated solely from Abelian states, resulting in a tremendous simplification of the topological band theory. Finally, we verify our results numerically by computing various Chern numbers in the hyperbolic Haldane model.

cond-mat.mes-hall↗

arXiv2024

Non-Abelian hyperbolic band theory from supercells

Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states. The method applies Abelian band theory to sequences of supercells, recursively built as symmetric aggregates of smaller cells, and enables a rapidly convergent computation of bulk spectra and eigenstates for both gapless and gapped tight-binding models. Our supercell method provides an efficient means of approximating the thermodynamic limit and marks a pivotal step towards a complete band-theoretic characterization of hyperbolic lattices.

cond-mat.mes-hall↗

arXiv2023

Symmetry and topology of hyperbolic Haldane models

Particles hopping on a two-dimensional hyperbolic lattice feature unconventional energy spectra and wave functions that provide a largely uncharted platform for topological phases of matter beyond the Euclidean paradigm. Using real-space topological markers as well as Chern numbers defined in the higher-dimensional momentum space of hyperbolic band theory, we construct and investigate hyperbolic Haldane models, which are generalizations of Haldane's honeycomb-lattice model to various hyperbolic lattices. We present a general framework to characterize point-group symmetries in hyperbolic tight-binding models, and use this framework to constrain the multiple first and second Chern numbers in momentum space. We observe several topological gaps characterized by first Chern numbers of value $1$ and $2$. The momentum-space Chern numbers respect the predicted symmetry constraints and agree with real-space topological markers, indicating a direct connection to observables such as the number of chiral edge modes. With our large repertoire of models, we further demonstrate that the topology of hyperbolic Haldane models is trivialized for lattices with strong negative curvature.

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arXiv2023

Hyperbolic Matter in Electrical Circuits with Tunable Complex Phases

Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We implement hyperbolic graphene as an example of topologically nontrivial hyperbolic matter. Our work sets the stage to realize more complex forms of hyperbolic matter to challenge our established theories of physics in curved space, while the tunable complex-phase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.

cond-mat.mes-hall↗

arXiv2023

Hyperbolic Topological Band Insulators

Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-)dimensional momentum space. To explore the uncharted topological aspects arising in hyperbolic band theory, we here introduce elementary models of hyperbolic topological band insulators: the hyperbolic Haldane model and the hyperbolic Kane-Mele model; both obtained by replacing the hexagonal cells of their Euclidean counterparts by octagons. Their non-trivial topology is revealed by computing topological invariants in both position and momentum space. The bulk-boundary correspondence is evidenced by comparing bulk and boundary density of states, by modelling propagation of edge excitations, and by their robustness against disorder.

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arXiv2022

Flat bands and band touching from real-space topology in hyperbolic lattices

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and octagonal symmetry. We show that two characteristic features of the energy spectrum of those models, namely the fraction of states in the flat band as well as the number of touching points between the flat band and the dispersive bands, can both be captured exactly by a combination of real-space topology arguments and a reciprocal-space description via the formalism of hyperbolic band theory. Furthermore, using real-space numerical diagonalization on finite lattices with periodic boundary conditions, we obtain new insights into higher-dimensional irreducible representations of the non-Euclidean (Fuchsian) translation group of hyperbolic lattices. First, we find that the fraction of states in the flat band is the same for Abelian and non-Abelian hyperbolic Bloch states. Second, we find that only Abelian states participate in the formation of touching points between the flat and dispersive bands.

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