Related papers: Continuous bulk and interface description of topol…
The bulk-edge correspondence predicts that interfaces between topological insulators support robust currents. We prove this principle for PDEs that are periodic away from an interface. Our approach relies on semiclassical methods. It…
In this review, we discuss recent progress in the explorations of topological materials beyond topological insulators; specifically, we focus on topological crystalline insulators and bulk topological superconductors. The basic concepts,…
The unusual electronic states found in topological materials can enable a new generation of devices and technologies, yet a long-standing challenge has been finding materials without deleterious parallel bulk conduction. This can arise…
The relation between bulk topological invariants and experimentally observable physical quantities is a fundamental property of topological insulators and superconductors. In the case of chiral symmetric systems in odd spatial dimensions…
Two-dimensional topological insulators possess conducting edge states at their boundary while being insulating in the bulk. The detection of edge states remains an open question in ultracold atom setups. We propose a configuration to…
We study solutions of $2 \times 2$ systems $(h D_t + \mathcal{D}) \Psi_t = 0$ on $\mathbb{R}^2$ in the semiclassical regime $h \rightarrow 0$. Our Dirac operator $\mathcal{D}$ is a standard model for interfaces between topological…
Topological insulators constitute one of the most intriguing phenomena in modern condensed matter theory. The unique and exotic properties of topological states of matter allow for unidirectional gapless electron transport and extremely…
Topological insulators are noninteracting, gapped fermionic systems which have gapless boundary excitations. They are characterized by topological invariants, which can be written in many different ways, including in terms of Green's…
The main focus of this paper is a nonparametric filtering technique for the estimation of interface geometry in bulk materials obtainable from modern imaging measurements. The filtering methodology relies on an assumed hierarchy of…
Spectral measurements of boundary localized in-gap modes are commonly used to identify topological insulators via the bulk-boundary correspondence. This can be extended to high-order topological insulators for which the most striking…
We consider the analogy between the topological phase transition which occurs as a function of spatial coordinate on a surface of a non-trivial insulator, and the one which occurs in the bulk due to the change of internal parameters (such…
We develop a formalism to extend, simultaneously, the usual definition of bulk and edge indices from topological insulators to the case of a finite sample with open boundary conditions, and provide a physical interpretation of these…
Real and complex Clifford bundles and Dirac operators defined on them are considered. By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford chessboard. Through the relations between…
We discuss the proximate phases of a three-dimensional system with Dirac-like dispersion. Using the cubic lattice with plaquette $\pi$-flux as a model, we find, among others phases, a chiral topological insulator and singlet topological…
Two-dimensional topological insulators are characterized by an insulating bulk and conductive edge states protected by the nontrivial topology of the bulk electronic structure. They remain robust against moderate disorder until Anderson…
We provide an index-theoretic proof of the bulk-boundary correspondence for two- and three-dimensional second-order topological insulators that preserve inversion symmetry, which are modeled as rectangles and rectangular prism-shaped…
We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators…
In this chapter, we report the recent progress in the understanding of the rich mathematical structures of topological insulators in the framework of index theory and noncommutative geometry.
Recent theories and experiments have suggested that strong spin-orbit coupling effects in certain band insulators can give rise to a new phase of quantum matter, the so-called topological insulator, which can show macroscopic entanglement…
Studying deterministic operators, we define an appropriate topology on the space of mobility-gapped insulators such that topological invariants are continuous maps into discrete spaces, we prove that this is indeed the case for the integer…