Dimensional crossover of class D real-space topological invariants

TL;DR

This paper introduces a novel framework using real-space topological markers to analyze how topological invariants evolve when a material's spatial dimension changes, exemplified by Shiba lattices.

Contribution

It presents a generalizable method to study topological phase transitions across dimensions, aiding material design by linking geometry to topological protection.

Findings

Framework applied to Shiba lattices shows topology evolution during deformation.

Provides a measure of disorder protection in topological states.

Guides minimum thickness requirements for 3D topological properties.

Abstract

The topological properties of a material depend on its symmetries, parameters, and spatial dimension. Changes in these properties due to parameter and symmetry variations can be understood by computing the corresponding topological invariant. Since topological invariants are typically defined for a fixed spatial dimension, there is no existing framework to understand the effects of changing spatial dimensions via invariants. Here, we introduce a framework to study topological phase transitions as a system's dimensionality is altered using real-space topological markers. Specifically, we consider Shiba lattices, which are class D materials formed by magnetic atoms on the surface of a conventional superconductor, and characterize the evolution of their topology when an initial circular island is deformed into a chain. We also provide a measure of the corresponding protection against disorder. Our framework is generalizable to any symmetry class and spatial dimension, potentially guiding the design of materials by identifying, for example, the minimum thickness of a slab required to maintain three-dimensional topological properties.