Quantitative mappings between symmetry and topology in solids

Abstract

The study of spatial symmetries was accomplished during the last century and had greatly improved our understanding of the properties of solids. Nowadays, the symmetry data of any crystal can be readily extracted from standard first-principles calculation. On the other hand, the topological data (topological invariants), the defining quantities of nontrivial topological states, are in general considerably difficult to obtain, and this difficulty has critically slowed down the search for topological materials. Here we provide explicit and exhaustive mappings from symmetry data to topological data for arbitrary gapped band structure in the presence of time-reversal symmetry and any one of the 230 space groups. The mappings are completed using the theoretical tools of layer construction and symmetry-based indicators. With these results, finding topological invariants in any given gapped band structure reduces to a simple search in the mapping tables provided.

Topological Semimetals from First Principles

“…These attempts have led to a set of recent ideas centered around the notion of symmetry indicators (162,163,164,165,140). The general philosophy underlying these ideas is to use the symmetry quantum numbers characterizing a band structure (e.g.…”

Topological states from topological crystals

“…In the second and third columns δ(g) is given by its values on the generators of the space group. In the last column, symmetry-based indicators (zw,1, zw,2, zw,3, z4) [48] of centrosymmetric space groups are tabulated.…”

Two-dimensional topological materials discovery by symmetry-indicator method

“…The obtained expansion coefficients from GGA and HSE06 calculations are both q = (4, 0, 0, 1, 2, 1, 2/3). The last coefficient of 2/3 translates into SI = 2 in Z3, which implies WO must have a nonzero MCN [43,44].…”