# Computation of topological phase diagram of disordered Pb$_{1-x}$Sn$_{x}$Te using the kernel polynomial method

## TL;DR

This paper introduces a kernel polynomial method-based algorithm to compute topological invariants in large, disordered systems, enabling detailed phase diagram analysis of complex alloys like Pb$_{1-x}$Sn$_{x}$Te.

## Contribution

The paper develops a scalable algorithm for topological invariant calculation in inhomogeneous systems, allowing analysis of large disordered materials with over 10 million degrees of freedom.

## Key findings

- Determined the topological phase diagram of Pb$_{1-x}$Sn$_{x}$Te.
- Identified the critical concentration for the topological phase transition.
- Enabled analysis of large, disordered systems with high computational efficiency.

## Abstract

We present an algorithm to determine topological invariants of inhomogeneous systems, such as alloys, disordered crystals, or amorphous systems. Based on the kernel polynomial method, our algorithm allows us to study samples with more than $10^7$ degrees of freedom. Our method enables the study of large complex compounds, where disorder is inherent to the system. We use it to analyse Pb$_{1-x}$Sn$_{x}$Te and tighten the critical concentration for the phase transition.