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Shin Hayashi

Publications and source records attributed to Shin Hayashi.

9 recordsLinked to original sources
arXiv2025

Index theory and bulk-boundary correspondence for inversion-symmetric second-order topological insulators

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 systems. Our method uses extensions of the symbols of some Toeplitz operators on discrete quarter planes and computations of topological equivariant K-theory groups.

math.KT↗
arXiv2022

An index theorem for quarter-plane Toeplitz operators via extended symbols and gapped Invariants related to corner states

In this paper, we discuss index theory for Toeplitz operators on a discrete quarter-plane of two-variable rational matrix function symbols. By using Gohberg-Krein theory for matrix factorizations, we extend the symbols defined originally on a two-dimensional torus to some three-dimensional sphere and derive a formula to express their Fredholm indices through extended symbols. Variants for families of (self-adjoint) Fredholm quarter-plane Toeplitz operators and those preserving real structures are also included. For some bulk-edge gapped single-particle Hamiltonians of finite hopping range on a discrete lattice with a codimension-two right angle corner, topological invariants related to corner states are provided through extensions of bulk Hamiltonians.

math.KT↗
arXiv2021

Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators

A second-order topological insulator on a two-dimensional square lattice hosts zero-dimensional states inside a band gap. They are localized near $90^{\circ}$ and $270^{\circ}$ corners constituting an edge of the system. When the edge is in a staircase form consisting of these two corners, two families of edge states (i.e., one-dimensional states localized near the edge) appear as a result of the hybridization of zero-dimensional states. We identify symmetry that makes them gapless. We also show that a pair of nontrivial winding numbers associated with this symmetry guarantee a gapless spectrum of edge states, indicating that bulk--boundary correspondence holds in this topological insulator with a staircase edge.

cond-mat.mes-hall↗
arXiv2021

Classification of topological invariants related to corner states

We discuss some bulk-surfaces gapped Hamiltonians on a lattice with corners and propose a periodic table for topological invariants related to corner states aimed at studies of higher-order topological insulators. Our table is based on four things: (1) the definition of topological invariants, (2) a proof of their relation with corner states (3) computations of K-groups and (4) a construction of explicit examples.

math-ph↗
arXiv2019

Second-order topological phases protected by chiral symmetry

We study second-order topological insulators and semimetals characterized by chiral symmetry. We investigate topological phase transitions of a model for construction of the two-dimensional second-order topological insulators protected only by chiral symmetry. By the theory of the phase transitions, we propose a second-order topological semimetal and insulators with flat hinge bands in chiral-symmetric three-dimensional systems. The three-dimensional second-order topological phases can be obtained from the stacked two-dimensional second-order topological insulators with chiral symmetry. Moreover, we show that broken chiral symmetry in the three-dimensional second-order topological phase allows a second-order topological insulator with chiral hinge states. We also demonstrate the second-order topological phases by using a lattice model.

cond-mat.mes-hall↗
arXiv2019

Toeplitz operators on concave corners and topologically protected corner states

We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems with concave corners are studied. Explicit examples clarify that these topological invariants depend on the shape of the system. We discuss the Benalcazar--Bernevig--Hughes' 2-D Hamiltonian and see that there still exists topologically protected corner states even if we break some symmetries as long as the chiral symmetry is preserved.

math-ph↗
arXiv2018

Topological invariants and corner states for Hamiltonians on a three-dimensional lattice

Periodic Hamiltonians on a three-dimensional (3-D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level are considered. By using K-theory applied for the quarter-plane Toeplitz extension, two topological invariants are defined. One is defined for the gapped bulk and edge Hamiltonians, and the non-triviality of the other means that the corner Hamiltonian is gapless. A correspondence between these two invariants is proved. Such gapped Hamiltonians can be constructed from Hamiltonians of 2-D type A and 1-D type AIII topological insulators, and its corner topological invariant is the product of topological invariants of these two phases.

math-ph↗
arXiv2014

Localization of Dirac Operators on 4n+2 Dimensional Open Spin^c Manifolds

An integer valued topological index of a Dirac operator is introduced for a pair of a 4n+2 dimensional open Spin^c manifold and a section of the determinant line bundle satisfying some property. We show a relation between the index and an index of a Dirac operator of its characteristic submanifold, by a localization of K-class.

math.DG↗