arXiv subjects

Tom Stoiber

Publications and source records attributed to Tom Stoiber.

15 recordsLinked to original sources

arXiv2026

Kubo Formulas and Bulk-Edge Correspondence for Curved Boundaries

Strong topological insulators are classified by integer invariants which admit different real-space expressions. Inspired by recent work studying these invariants for spaces with curved boundaries, we revisit the problem of equivalence of the various expressions, including Fredholm index pairings and Kubo formulas involving half-space projections. Using Roe algebras and a variant of KK-theory suitable for non-separable $C^*$-algebras, we prove a general form of bulk-edge correspondence for the index pairing with a general position-space Dirac operator in the presence of arbitrary boundaries. For spaces coarsely equivalent to $\mathbb{R}^d$, we show that up to multiplicity, these pairings are equal to those obtained with the standard dual Dirac operator, with the multiplicity explicitly given as the topological degree of the symbol function. In addition, we prove a generalized Kubo formula which computes the general index pairing by a real-space formula.

math-ph↗

arXiv2025

The role of self-adjoint extensions in the bulk-edge correspondence

We investigate the role of self-adjoint extensions in the bulk-edge correspondence for topological insulators. While the correspondence is well understood in discrete models with spectral gaps, complications arise in the presence of unbounded Hamiltonians and varying boundary conditions, leading to anomalous behavior that has recently been dubbed violations of bulk-edge correspondence. In this work we use a K-theoretic framework to identify precise conditions needed for unbounded Hamiltonians to be affiliated to the respective observable algebras and define K-theory classes. In special cases we can then exclude anomalous behaviour and obtain the standard bulk-edge correspondence, or, under weaker conditions, obtain a relative bulk-edge correspondence theorem, which compares pairs of Hamiltonians. Applying that relative approach in the bulk we recover among other things the so-called bulk-difference-interface correspondence for Hamiltonians that fail to define a bulk K-theory class in the conventional way. The second main result is that one can define K-theory classes in terms of von Neumann unitaries, which under changes in boundary conditions directly contribute to the number of protected edge states. This approach clarifies apparent violations of the classical bulk-edge paradigm and provides a systematic account of boundary-induced topological corrections.

math-ph↗

arXiv2025

Spectral continuity for \'etale groupoids with the Rapid decay property

We show that the reduced groupoid C*-algebras of continuous fields of \'etale groupoids satisfying the rapid decay property yield continuous fields of C*-algebras. This establishes a new sufficient criterion that applies in the non-amenable case where the full and reduced groupoid algebras may differ. Potential applications include convergence of spectra in inverse systems of finite-index subgroups and magnetic models on hyperbolic lattices.

math.OA↗

arXiv2025

Invariant measures on the transversal hull of cone semigroups and some applications

Let $\LL_{\bf v}\subset \Z^D$ be a suitable cone semigroup and $\A_{\bf v}$ its reduced semigroup $C^*$-algebra. In this paper, we compute the $\LL_{\bf v}$-invariant measures in the transversal hull of the semigroup $\LL_{\bf v}$ that exhibit regularity in the boundaries of $\LL_{\bf v}.$ These measures enable the construction of a trace per-unit hypersurface for observables in $\A_{\bf v}$ supported near the boundaries of $\LL_{\bf v}$, leading to the construction of appropriate Chern cocycles in the "boundary" ideals of $\A_{\bf v}$. Our approach applies to both finitely and non-finitely generated cone semigroups. Applications for the bulk-defect correspondence of lattice models of topological insulators are also provided.

math-ph↗

arXiv2025

On Frustration-Free Quantum Spin Models

The goal of our work is to characterize the landscape of the frustration-free quantum spin models over the Cayley graph of a finitely generated group $G$. This is achieved by establishing $G$-equivariant morphisms from the partially ordered space of frustration-free models to the partially ordered spaces 1) of hereditary $C^\ast$-algebras of the underlying UHF quasi-local algebra of observables, 2) of open projections in its double dual, and 3) of subsets of pure state space. Our main result consists of an intrinsic characterization of the images of these morphisms, which captures the essence of frustration-freeness and enables us to extend the concept to generic AF-algebras. Additionally, using well established facts about AF-algebras, we prove density theorems, provide intrinsic characterizations of frustration-free ground states, and propose a definition of a boundary algebra for models constrained to half-lattices, under the sole assumption of frustration-freeness.

math-ph↗

arXiv2024

A space-adiabatic approach for bulk-defect correspondences in lattice models of topological insulators

In space-adiabatic approaches one can approximate Hamiltonians that are modulated slowly in space by phase-space functions that depend on position and momentum. In this paper, we establish a rigorous relation between this approach and the operator-theoretic approach for topological insulators with defects, which employs $C^*$-algebras and operator K-theory. Using such tools, we show that by quantizing phase-space functions one can construct lattice Hamiltonians which are gapped at certain spatial limits and carry protected states at defects such as boundaries, hinges, and corners. Moreover, we show that the topological invariants that protect the latter can be computed in terms of the symbol functions. This enables us to compute boundary maps in K-theory that are relevant for bulk-defect correspondences.

math-ph↗

arXiv2024

A spectral localizer approach to strong topological invariants in the mobility gap regime

Topological phases of gapped one-particle Hamiltonians with (anti)-unitary symmetries are classified by strong topological invariants according to the Altland-Zirnbauer table. Those indices are still well-defined in the regime of strong disorder when the spectral gap is replaced by a mobility gap, however, many questions regarding their robustness and existence of topological boundary states are wide open. We apply the recently developed spectral localizer method to prove results on the stability of strong topological invariants under a notion of continuous homotopy that preserves a mobility gap condition. Using the local computability afforded by the spectral localizer we show that for parametrized random families that satisfy a fractional moments bound the probability distribution of the strong topological invariant changes continuously. In particular, for ergodic families the almost sure index must be constant on any path which preserves the mobility gap. Using similar methods, we also prove a result on the delocalization of interface states between two mobility-gapped systems which have differing strong invariants.

math-ph↗

arXiv2024

C*-framework for higher-order bulk-boundary correspondences

A typical crystal is a finite piece of a material which may be invariant under some point symmetry group. If it is a so-called intrinsic higher-order topological insulator or superconductor, then it displays boundary modes at hinges or corners protected by the crystalline symmetry and the bulk topology. We explain the mechanism behind such phenomena using operator K-theory. Specifically, we derive a groupoid C*-algebra that 1) encodes the dynamics of the electrons in the infinite size limit of a crystal; 2) remembers the boundary conditions at the crystal's boundaries, and 3) admits a natural action by the point symmetries of the atomic lattice. The filtrations of the groupoid's unit space by closed subsets that are invariant under the groupoid and point group actions supply equivariant cofiltrations of the groupoid C*-algebra. We show that specific derivations of the induced spectral sequences in twisted equivariant K-theories enumerate all non-trivial higher-order bulk-boundary correspondences.

math-ph↗

arXiv2024

The generators of the K-groups of the sphere

This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i(C(\SM^d))$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.

math-ph↗

arXiv2022

Topological Spectral Bands with Frieze Groups

Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group $C^\ast$-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style.

cond-mat.mtrl-sci↗

arXiv2022

Harmonic analysis in operator algebras and its applications to index theory and topological solid state systems

This monograph develops the theory of Besov spaces for abelian group actions on semifinite von Neumann algebras and then proves Peller criteria for traceclass properties of associated Hankel operators. This allows to extend known index theorems to symbols lying in Sobolev or Besov spaces. The duality theory for pairings over the smooth Toeplitz extension is developed in detail. Numerous applications to solid state systems are presented. In particular, a bulk-boundary correspondence is obtained for insulators with edges of irrational angles and for chiral semimetals having a pseudogaps. The latter implies the existence of flat bands of edge for tight-binding graphene models and shows how the density of surface states is expressed in terms of weak Chern numbers of the system without boundaries.

math-ph↗

arXiv2022

Spectral localization for semimetals and Callias operators

A semiclassical argument is used to show that the low-lying spectrum of a selfadjoint operator, the so-called spectral localizer, determines the number of Dirac or Weyl points of an ideal semimetal. Apart from the IMS localization procedure, an explicit computation for the local toy models given by a Dirac or Weyl point is the key element of proof. The argument has numerous similarities to Witten's reasoning leading to the strong Morse inequalities. The same techniques allow to prove a spectral localization for the Callias operator in terms of a multi-parameter spectral flow of selfadjoint Fredholm operators.

math-ph↗

arXiv2021

Callias-type operators associated to spectral triples

Callias-type (or Dirac-Schr\"odinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.

math-ph↗

arXiv2021

Invariants of disordered semimetals via the spectral localizer

The spectral localizer consists of placing the Hamiltonian in a Dirac trap. For topological insulators its spectral asymmetry is equal to the topological invariants, providing a highly efficient tool for numerical computation. Here this technique is extended to disordered semimetals and allows to access the number of Dirac or Weyl points as well as weak invariants. These latter invariants imply the existence of surface states.

cond-mat.mes-hall↗

arXiv2020

The spectral localizer for semifinite spectral triples

The notion of spectral localizer is extended to pairings with semifinite spectral triples. By a spectral flow argument, any semifinite index pairing is shown to be equal to the signature of the spectral localizer. As an application, a formula for the weak invariants of topological insulators is derived. This provides a new approach to their numerical evaluation.

math-ph↗