Topological Weaire–Thorpe models of amorphous matter

Abstract

Amorphous solids remain outside of the classification and systematic discovery of new topological materials, partially due to the lack of realistic models that are analytically tractable. Here we introduce the topological Weaire–Thorpe class of models, which are defined on amorphous lattices with fixed coordination number, a realistic feature of covalently bonded amorphous solids. Their short-range properties allow us to analytically predict spectral gaps. Their symmetry under permutation of orbitals allows us to analytically compute topological phase diagrams, which determine quantized observables like circular dichroism, by introducing symmetry indicators in amorphous systems. These models and our procedures to define invariants are generalizable to higher coordination number and dimensions, opening a route toward a complete classification of amorphous topological states in real space using quasilocal properties.

Amorphous topological phases protected by continuous rotation symmetry

“…Pfaffians are calculated using Pfapack [46]. The numerical density of states, momentum-resolved spectral function, and effective Hamiltonian calculations are performed using the kernel polynomial method [17,23,47,48].…”

“…The spectrum of H eff (k) closely follows the peaks of the spectral function, especially near the gap closing points. The key properties of H eff are that it transforms the same way under symmetries as continuum Hamiltonians discussed before, its gap only closes when the gap in the bulkĤ closes [23], and it is properly regularized in the k → ∞ limit [17]. Hence, the bulk invariants defined for continuum systems are directly applicable to detecting topological phase transitions in amorphous systems.…”

“…In order to apply the construction of bulk invariants to amorphous systems, we introduce the effective k-space Hamiltonian [17,23] H eff (k) = G eff (k) −1 through the projection of the single-particle Green's function onto plane-wave states:…”

“…Unlike crystals, which break continuous rotation symmetry even on average, amorphous systems lack long-range order and are therefore on average compatible with continuous rotations. Strong topological, metallic and insulating phases as well as topological superconductivity have been studied in amorphous systems both theoretically [12][13][14][15][16][17][18] and experimentally [19][20][21].…”

Understanding Topological Insulators in Real Space

“…The methods usually applied to calculate the topological properties have been mainly developed for crystalline cases. However, in some situations, there is no such thing as a crystal lattice [ 34 , 35 ], and the bulk-boundary correspondence—a cornerstone in topological insulators—is not straightforward to follow. For these cases, a local probe of the symmetries giving rise to the topological order could be useful, especially one that provides chemical intuition.…”