Analytical estimations of edge states and extended states in large finite-size lattices

TL;DR

This paper develops an analytical framework using asymptotic estimates to understand how lattice size and boundary conditions affect edge states and bulk boundary correspondence in finite topological lattices, with extensions to nonlinear and higher-dimensional systems.

Contribution

It introduces a novel analytical approach to quantify finite size effects on edge states and bulk boundary correspondence in topological lattices, applicable to various lattice types.

Findings

Edge states depend on lattice size and boundary conditions.

Finite size effects cause deviations from ideal bulk boundary correspondence.

Eigenfrequencies near band edges can be accurately approximated.

Abstract

The bulk boundary correspondence, one of the most significant features of topological matter, theoretically connects the existence of edge modes at the boundary with topological invariants of the bulk spectral bands. However, it remains unspecified in realistic examples how large the size of a lattice should be for the correspondence to take effect. In this work, we employ the diatomic chain model to introduce an analytical framework to characterize the dependence of edge states on the lattice size and boundary conditions. In particular, we apply asymptotic estimates to examine the bulk boundary correspondence in long diatomic chains as well as precisely quantify the deviations from the bulk boundary correspondence in finite lattices due to symmetry breaking and finite size effects. Moreover, under our framework the eigenfrequencies near the band edges can be well approximated where two special patterns are detected. These estimates on edge states and eigenfrequencies in linear diatomic chains can be further extended to nonlinear chains to investigate the emergence of new nonlinear edge states and other nonlinear localized states. In addition to one-dimensional diatomic chains, examples of more complicated and higher dimensional lattices are provided to show the universality of our analytical framework.