A Unified View of Topological Phase Transition in Band Theory

TL;DR

This paper presents a unified framework for understanding topological phase transitions in solids, revealing a universal linear scaling relation at the transition point and its implications for material stability.

Contribution

It introduces a new perspective by integrating topology into classical band theory, identifying a universal scaling law, and establishing bounds for disorder tolerance in topological materials.

Findings

Existence of an intermediate topological insulator phase during band evolution.

Universal linear scaling relation between electron hopping and bond length at TPT.

Upper bound on disorder levels that can destroy topological order.

Abstract

We develop a unified view of topological phase transitions (TPTs) in solids by revising the classical band theory with the inclusion of topology. Re-evaluating the band evolution from an "atomic crystal" [a normal insulator (NI)] to a solid crystal, such as a semiconductor, we demonstrate that there exists ubiquitously an intermediate phase of topological insulator (TI), whose critical transition point displays a linear scaling between electron hopping potential and average bond length, underlined by deformation-potential theory. The validity of the scaling relation is verified in various two-dimensional (2D) lattices regardless of lattice symmetry, periodicity, and form of electron hoppings, based on a generic tight-binding model. Significantly, this linear scaling is shown to set an upper bound for the degree of structural disorder to destroy the topological order in a crystalline solid, as exemplified by formation of vacancies and thermal disorder. Our work formulates a simple framework for understanding the physical nature of TPTs with significant implications in practical applications of topological materials.