# Topological phase transitions in glassy quantum matter

## TL;DR

This paper develops a scaling theory for topological phase transitions in amorphous two-dimensional systems, revealing unique critical properties and persistence of topological phases down to percolation thresholds.

## Contribution

It introduces a finite-size scaling analysis of topological invariants in amorphous systems, uncovering novel critical behaviors and continuous variation of critical exponents.

## Key findings

- Chern glass phase persists down to the percolation threshold.
- Topological indices follow one-parameter scaling but do not always approach integer values.
- Critical exponents vary continuously along the phase boundary.

## Abstract

Amorphous systems have rapidly gained promise as novel platforms for topological matter. In this work we establish a scaling theory of amorphous topological phase transitions driven by the density of lattice points in two dimensions. By carrying out a finite-size scaling analysis of topological invariants averaged over discrete and continuum random geometries, we discover unique critical properties of Chern and $\mathbb{Z}_2$ glass transitions. Even for short-range hopping models the Chern glass phase may persist down to the fundamental lower bound given by the classical percolation threshold. While the topological indices accurately satisfy the postulated one-parameter scaling, they do not generally flow to the closest integer value in the thermodynamic limit. Furthermore, the value of the critical exponent describing the diverging localization length varies continuously along the phase boundary and is not fixed by the symmetry class of the Hamiltonian. We conclude that the critical behaviour of amorphous topological systems exhibit characteristic features not observed in disordered systems, motivating a wealth of new research directions.