相关论文: Index theory and bulk-boundary correspondence for …
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.
Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry.…
Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five…
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…
In this work, we establish the bulk-edge correspondence principle for finite two-dimensional photonic structures. Specifically, we focus on the divergence-form operator with periodic coefficients and prove the equality between the…
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…
We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of noncommutative index theory of operator algebras. In…
We demonstrate the existence of topological insulators in one dimension protected by mirror and time-reversal symmetries. They are characterized by a nontrivial $\mathbb{Z}_2$ topological invariant defined in terms of the "partial"…
Topological insulators in three spatial dimensions are known to possess a precise bulk/boundary correspondence, in that there is a one-to-one correspondence between the 5 classes characterized by bulk topological invariants and Dirac…
Recently we introduced T-duality in the study of topological insulators. In this paper, we study the bulk-boundary correspondence for three phenomena in condensed matter physics, namely, the quantum Hall effect, the Chern insulator, and…
This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual,…
We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to interpolate between certain Fredholm operators arising in the…
A 2-torsion topological phase exists for Hamiltonians symmetric under the wallpaper group with glide reflection symmetry, corresponding to the unorientable cycle of the Klein bottle fundamental domain. We prove a mod 2 twisted Toeplitz…
Second-order topological insulators and superconductors have a gapped excitation spectrum in bulk and along boundaries, but protected zero modes at corners of a two-dimensional crystal or protected gapless modes at hinges of a…
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…
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 contrast to eigenvalue-based approaches, we formulate the bulk-boundary correspondence for two-dimensional non-Hermitian quadratic lattice Hamiltonians in terms of Toeplitz operators and singular values, which correctly capture the…
We investigate the higher-order bulk-boundary correspondence in a family of chiral-symmetric Bloch Hamiltonians with anticommuting mirror symmetries. These models generalize the $\pi$-flux square lattice, the prototypical topological…
In this work we consider whether nonsymmorphic symmetries such as a glide plane can protect the existence of topological crystalline insulators and superconductors in three dimensions. In analogy to time-reversal symmetric insulators, we…
We prove the existence of higher-order topological insulators in: {\it i}) fourfold rotoinversion invariant bulk crystals, and {\it ii}) inversion-symmetric systems with or without an additional three-fold rotation symmetry. These states of…