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

Welcome

Emil Prodan, Yeshiva University, USA

Alexander Cerjan, Sandia National Laboratories, USA

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)

Wladimir Benalcazar, Emory University, USA

Tom Stoiber, University of California at Irving, USA

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

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

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

Given a magnetic finite group, we consider the 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 is explicitly determined. Following Karoubi, we argue that the elements of this graded Brauer group parametrize the twistings of the magnetic equivariant K-theory of a point.