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.
Symmetric approximant formalism for statistical topological matter
Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学卡弗里纳米科学研究所）；Freie Universität Berlin, Dahlem Center for Complex Quantum Systems and Fachbereich Physik（柏林自由大学达勒姆复杂量子系统中心及物理系）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）；QuTech, Delft University of Technology（代尔夫特理工大学QuTech）
