Conference: Mathematical Aspects of Topological Insulators “Bulk-defect” and “Bulk-boundary” correspondences

Welcome

Emil Prodan, Yeshiva University, USA: A Universal Chern Model on Arbitrary Triangulations

Given a triangulation of a closed orientable surface, we place single-mode resonators or single-orbital artificial atoms at its vertices, edges, and facets, and we devise near-neighbor hopping terms derived from the boundary and Poincaré duality maps of the simplicial complex of the triangulation. Regardless of the surface or its triangulation, these terms always lead to tight-binding Hamiltonians with large and clean topological spectral gaps, carrying non-trivial Chern numbers in the limit of infinite refinement of the triangulation. We confirm this via numerical simulations, and demonstrate how these models enable topological edge modes at the surfaces of real-world objects. Furthermore, we describe a metamaterial whose dynamics reproduces that of the proposed model, thus bringing the topological metamaterials closer to real-world applications.

Alexander Cerjan, Sandia National Laboratories, USA: Classifying material topology in real space using matrix homotopy

The classification of topological phases of matter has traditionally relied on assumptions about the underlying material, requiring that the material be an insulator with a well-defined band structure. However, many experimentally relevant systems violate one or more of these assumptions, raising fundamental questions about how topology should be defined and diagnosed in realistic settings. In this talk, I will present an overview of these outstanding challenges and describe how a real-space, operator-based approach called the spectral localizer framework provides a platform-agnostic unifying theory capable of addressing these challenges. In particular, the spectral localizer enables the formulation of local, energy-resolved topological markers that remain well-defined in systems lacking a global spectral gap, translational symmetry, or sharp interfaces. This perspective allows one to meaningfully classify gapless heterostructures, such as photonic systems embedded in air, and can be applied directly to continuum models without first finding a low-energy approximation. Beyond stable topological phases, the spectral localizer also offers new insights into recently identified classes of topology, including fragile topological phases that induce Wannier obstructions resulting in novel forms of quantum materials. I will discuss applications of these ideas to nonlinear polariton systems to achieve reconfigurable topological interfaces and to electronic platforms such as two-dimensional electron gases in semiconductor heterostructures to show the emergence of Hofstadter’s butterfly for intermediate scales of the periodic potential’s strength, demonstrating the versatility of the spectral localizer as a general tool for topological classification in modern materials physics.

Terry Loring, University of New Mexico, USA: $K$-theoretic invariants for $C_2 \mathcal{T}$-symmetric systems via the spectral localizer

The mathematics that shows a Chern insulator is far from any atomic limit is essentially the same as the mathematics that used in a problem involving almost commuting matrices.  If three almost commuting Hermitian matrices have square sum close to the identity then there is a $K$-theory obstruction.  Only when this vanishes can we find nearby Hermitian matrices that commute.
If we make the problem more difficult by asking that two of the matrices are purely imaginary while the third is real, and ask the same of the commuting approximants, then the role of $K$-theory changes.  The two possible values of this invariant can arise from commuting triples.  With these extra conditions, the commuting approximation is always possible.  This modified math problem translates into a tool to study the topology of 2D systems that are invariant under the composition of rotation by 180 degrees with bosonic time reversal.
We will discuss also the Clifford and quadratic pseudospectra as these give good ways to visualize the possible states that are localized in both position and energy.  In particular we look at models of twisted bilayer graphene as well as photonic quasicrystals.

Andres Reyes, U. de los Andes, Colombia: $\mathbb Z_2$ Index and Geometric Phase for Mixed States

In this talk I will discuss a $\mathbb Z_2$ index associated with quadratic gapped Hamiltonians describing fermionic systems in terms of self-dual CAR C*-algebras. A reformulation of the $\mathbb Z_2$ index as a state index will also be presented. Attempts to extend this invariant to the finite-temperature case have led to a generalization of the geometric phase to mixed states. This generalization emerges naturally in the context of the self-dual formalism and leads to a notion of parallel transport for paths of mixed states that differs from the Uhlmann approach and can be defined for general quantum systems.  

Rafael Gonzalez-Hernandez (Universidad del Norte, Colombia): Topological phases in aletermagnets

This talk presents a model Hamiltonian for an altermagnetic topological insulator, protected by combined crystalline and time-reversal symmetry such as C 4z T. Within this framework, the spin Chern number emerges as a robust bulk topological invariant, which can be interpreted as a half-quantized Chern number defined on a symmetry partitioned Brillouin zone. This invariant determines the existence of protected boundary states, including edge, hinge, and surface modes, providing the realization of bulk–boundary correspondence in an altermagnetic system [1,2].

[1] R González-Hernández, H Serrano, B Uribe. Phys. Rev. B 111, 085127 (2025)
[2] R González-Hernández, B Uribe. Phys. Rev. B 112, 184101 (2025)

Wladimir Benalcazar, Emory University, USA: 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.

Tom Stoiber, University of California at Irving, USA: Higher-order Topological Insulators and K-theory

The bulk-edge correspondence of topological insulators is mathematically well-understood through the use of C*-algebras and operator K-theory. In higher-order topological insulators, all faces of a crystal are insulating, while surface states appear only at boundaries of higher codimension, such as hinges or corners. For so-called intrinsic topological insulators the existence of these states is independent of boundary conditions but requires spatial symmetries like mirror, rotational, or inversion symmetry. Since the topological protection emerges only in the infinite volume limit, we need to construct C*-algebras that describe symmetric infinite crystals with various boundary configurations. These algebras naturally admit cofiltrations based on the codimensions of the boundaries, leading to spectral sequences in equivariant K-theory. We demonstrate that the classification and phenomenology of higher-order topological insulators align seamlessly with this formalism. Specifically, the higher differentials in the spectral sequence identify precisely the classes of intrinsic higher-order topological insulators and their associated surface states. This is joint work with Emil Prodan and Danilo Polo.

Hermann Schulz-Baldes, Erlangen, Germany: Stability estimates for the localizer index

The talk is about refinements of estimates on the gap of the spectral localizer which further strengthen its locality properties.

Higinio Serrano, ICMS, Bulgaria: Magnetic Equivariant K-Theory

Magnetic symmetries play a central role in the description of crystalline systems with time–reversal or anti-unitary operations. These symmetries are naturally encoded by magnetic groups, which extend ordinary symmetry groups by allowing elements that act anti-linearly. Such structures appear throughout condensed matter physics, for instance in the study of magnetic crystals and topological phases of matter.
In this talk I will introduce magnetic equivariant K-theory, designed to incorporate these symmetries. The theory is defined using complex vector bundles equipped with compatible actions of magnetic groups. I will explain how this construction generalizes classical equivariant K-theory and how it naturally captures the presence of anti-unitary symmetries. Then I will outline some of the main structural properties of this theory. Finally, I will discuss examples illustrating how these groups provide topological invariants for gapped Hamiltonians with magnetic symmetries, highlighting their relevance for the mathematical description of topological phases of matter.

Carlos Ardila, UIS, Colombia:  Chiral 3D Kagome Lattices: Band Topology Induced by Non-Symmorphic Symmetry

2D Kagome lattices are known as ideal platforms for the coexistence of multiple phenomena, including topological band degeneracies, frustrated magnetism originating from their trihexagonal geometry, and correlated states associated with Van Hove singularities and flat bands. In these symmetries, electronic properties are strongly guided by the ideal s-orbital Kagome geometry, which makes them highly sensitive to perturbations. Motivated by the search for more robust mechanisms, we investigate the novel three-dimensional (2D) Kagome lattices that incorporate crystallographic chirality while preserving the characteristic features of the two-dimensional counterpart.
In this work, we show our results associated with a series of tight-binding models to analyze how non-symmorphic screw symmetries modify the conventional itinerant electron Hamiltonian, and the spin–orbit interaction. For the last one, we develop a generalized Kane-Mele model that consideres the interplay between different screw orders within the same lattice. From these models and using representation-theory, we show the emergence of accordion- and hourglass-like dispersions as a direct consequence of crystallographic chirality. These electronic disperssions leads to enforced band degeneracies with nontrivial topological character. Because these features originate from non-symmorphic screw operations rather than from the underlying trihexagonal symmetry, they are expected to be more robust against perturbations such as multiorbital effects or the introduction of non-Kagome sites. As such, our results provide a materials design rule and symmetry-driven perspective on the role of non-symmorphic operations in stabilizing distinctive bulk band structures, offering a mathematically grounded framework for understanding topological features in chiral 3D Kagome crystals.

Luis Angel de León, Centro de Nanociencias y Nanotecnología UNAM, Ensenada, México: Spin splittting effects in Altermagnetic CoTeMoO6 and MnTeMoO6 materials

Altermagnetism is a magnetic phase that combines features of both ferromagnetism (FM) and antiferromagnetism (AFM). Although altermagnetic materials exhibit zero net magnetization, as in AFM, they display spin splitting in their band structures, similar to FM. This coexistence of properties makes them promising candidates for spintronic applications, especially considering that the phenomenon has a non-relativistic origin. [1,2]
In this first-principles study, we modeled the bulk phases of two isostructural materials—CoTeMoO6 (CTMO) and MnTeMoO6 (MTMO)—composed of layers bound by van der Waals forces, in order to investigate their altermagnetic behavior. The magnetic and electronic structures were analyzed for two different magnetic configurations, without including spin–orbit coupling.
Based on these results, we further modeled 2D phases of the materials with one, two, and three layers, manipulating symmetry through the number of layers. We found that altermagnetism emerges for an odd number of layers, whereas it disappears for an even number.

[1] L. Šmejkal, J. Sinova, and T. Jungwirth, “Emerging Research Landscape of Altermagnetism,” Phys Rev X, vol. 12, no. 4, p. 040501, Dec. 2022, doi: 10.1103/PhysRevX.12.040501.
[2] L. Šmejkal, J. Sinova, and T. Jungwirth, “Beyond Conventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry,” Phys Rev X, vol. 12, no. 3, p. 031042, Sep. 2022, doi: 10.1103/PhysRevX.12.031042.

Danilo Polo, Universidad de los Andes, Colombia: A general principle for the bulk-edge correspondence for incommensurable defects

In a crystalline material, a defect may arise either when a potential creates an interface or when the material is physically cut. When the resulting interface forms an irrational angle with respect to the underlying lattice directions (incommensurable defect), translational symmetry along the interface is broken, making the computation of expectation values of observables near the defect more subtle.
In this talk, using the framework of C*-algebras, I will construct a trace per-unit-volume that allows for the computation of expectation values of observables localized near the defect. This construction applies to systems with defects that may be either incommensurable or not. I will further show that this trace is unique and therefore provides a canonical method for performing such computations. Finally, using K-theory for C*-algebras, I will establish a general bulk–edge correspondence for such systems and demonstrate that it is independent of the angle formed by the defect with respect to the underlying lattice directions..

Fredy Vides, Universidad Nacional Autonoma de Honduras: A Basis Algorithm for Centralizers of Finitely Generated Unitary Groups with Applications to Floquet Hamiltonian Estimation

We introduce an algorithm for computing a basis of the centralizer algebra C(ρ(G)) of a finitely generated group G acting through a unitary matrix representation ρ, without requiring enumeration of group elements. The algorithm solves a finite linear system derived from the commutation conditions with the generators of G, [M, ρ(gi)] = 0, i = 1, . . . , k, and remains effective regardless of whether the order of G is large or infinite. This extends the structured matrix approximation framework developed in [1] and [2], which addressed symmetry-preserving computation of Floquet Hamiltonians and equivariant system identification for specific symmetry classes.
The centralizer basis provides an explicit parametrization of all Hermitian operators compatible with a given symmetry representation, enabling refined estimation of the effective Floquet Hamiltonian HF by restricting the logarithm computation to the physically admissible subspace.
By Schur’s lemma, if ρ = ⊕m_k (ρ_k) decomposes into irreducibles of multiplicity m_k, then C(ρ(G)) =⊕M_{m_k}(C), and HF is parametrized by the blocks M_{m_k}(C) alone, reducing the estimation problem to a lower-dimensional and symmetry-consistent subspace.
We illustrate the method with symmetry groups relevant to Floquet topological insulators, including representations of Z/2Z by the unitaries −I and [[0,1],[1,0]] in M_2(C), and discrete space-time symmetries arising in periodically driven crystalline systems. Potential applications include symmetry-constrained Hamiltonian tomography, reduction of numerical drift in stroboscopic evolution, and the classification of symmetry-protected Floquet phases in disordered and quasicrystalline systems.
Keywords: centralizer algebra, Floquet Hamiltonian, symmetry-preserving matrix logarithm,topological insulators, finitely generated groups, structured matrix approximation.

[1] T. A. Loring and F. Vides, Computing Floquet Hamiltonians with symmetries, J. Math. Phys. 61, 113501 (2020). https://doi.org/10.1063/5.0023028
[2] F. Vides, Identifying systems with symmetries using equivariant autoregressive reservoir computers, arXiv:2311.09511 (2023). https://arxiv.org/abs/2311.09511
[3] J. H. Shirley, Solution of the Schr ̈odinger equation with a Hamiltonian periodic in time, Phys. Rev. 138, B979 (1965).
[4] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X 3, 031005 (2013).
[5] J. Cayssol, B. D ́ora, F. Simon, and R. Moessner, Floquet topological insulators, Phys. Status Solidi RRL 7, 101 (2013).
[6] T. A. Loring, K-theory and pseudospectra for topological insulators, Ann. Phys. 356, 383 (2015).

Bernardo Uribe, Universidad del Norte, Colombia: Magnetic Equivariant Graded Brauer Group

It is well known that the real K-theory is 8-periodic. This fact, together with the 2-periodicity of complex K-theory, are the pieces underlying the famous Kitaev periodic table for topological insulators. Now, whenever time reversal is broken, say in magnetic materials, Kitaev’s periodic table needs to be recalculted having in mind the different magnetic groups underlying the materials.
In this talk I will present an approach on how to understand Kitaev’s periodic table in the magnetic setting. For this I will consider similarity classes of magnetic equivariant central simple graded algebras over the complex numbers. We call this set the magnetic equivariant graded Brauer group and its structure as an abelian group will be explicitly determined. Following Karoubi, we will argue that the elements of this graded Brauer group parametrize the twistings of the magnetic equivariant K-theory of a point. This twisted magnetic equivariant K-theory groups will provide the information for the enhanced periodic tables for topological insulators.