Structural spillage: an efficient method to identify non-crystalline topological materials

Abstract

While topological materials are not restricted to crystals, there is no efficient method to diagnose topology in non-crystalline solids such as amorphous materials. Here we introduce the structural spillage, a new indicator that predicts the unknown topological phase of a non-crystalline solid, which is compatible with first-principles calculations. We illustrate its potential with tight-binding and first-principles calculations of amorphous bismuth, predicting a bilayer to be a new topologically nontrivial material. Our work opens up the efficient prediction of non-crystalline solids via first-principles and high-throughput searches.

Amorphous topological matter: Theory and experiment

“…Neural networks can also detect nontrivial topology, by efficiently learning features associated with topology from for example the wave functions [35], and the flow of the entanglement spectrum [76]. Other approaches include the effective Hamiltonian [32,77], symmetry indicators [32], and the structural spillage [78], which take advantage of the gap closing and band inversion in a topological phase transition.…”

“…Based on both tight-binding and density functional theory calculations, ref. [78] showed that the amorphous Bismuth bilayer remains topological, as indicated by the structural spillage and the conductance. Reference [38] demonstrated a structural-disorderinduced quantum spin Hall phase, constructing a phase diagram as a function of spin-orbit coupling and disorder strength, by modelling the disorder by Gaussian deviations from an initial triangular lattice.…”

Single-point spin Chern number in a supercell framework

“…An alternative non-commmutative approach was proposed by Loring and Hastings [40,41] and relates the Z 2 index to the topological obstruction to approximating almost commuting matrices by exactly commuting matrices; its robustness with respect to the introduction of disorder has been investigated in [42]. Very recently, a novel method based on the concept of spillage [43] has been proposed [44], where the identification of topological phases in a non-crystalline system is obtained by calculating the spillage with respect to a crystalline reference structure, whose topological characterization can be performed with standard methods. The most practical approach from the point of electronic structure simulations has been arguably put forward by Huang and Liu [45,46], who addressed the problem of calculating the Z 2 invariant for non-periodic system in the context of quantum spin Hall quasicrystals, and introduced the spin Bott index, which measures the commutativity of the projected position operators.…”