# Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -

## TL;DR

This paper develops a comprehensive mathematical framework using twisted equivariant K-theory to classify topological crystalline materials, including insulators, semimetals, and superconductors, based on their symmetry properties and boundary states.

## Contribution

It introduces a systematic classification method for topological crystalline phases using twisted equivariant K-theory, applicable to various gapful and gapless topological materials.

## Key findings

- Classified surface states for 17 wallpaper groups without time-reversal symmetry.
- Identified topological invariants and their symmetry representations.
- Discovered new classes of Weyl semimetals protected by inversion symmetry.

## Abstract

We formulate topological crystalline materials on the basis of the twisted equivariant $K$-theory. Basic ideas of the twisted equivariant $K$-theory is explained with application to topological phases protected by crystalline symmetries in mind, and systematic methods of topological classification for crystalline materials are presented. Our formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states, as well as bulk gapless topological materials such as Weyl and Dirac semimetals, and nodal superconductors. As an application of our formulation, we present a complete classification of topological crystalline surface states, in the absence of time-reversal invariance. The classification works for gapless surface states of three-dimensional insulators, as well as full gapped two-dimensional insulators. Such surface states and two-dimensional insulators are classified in a unified way by 17 wallpaper groups, together with the presence or the absence of (sublattice) chiral symmetry. We identify the topological numbers and their representations under the wallpaper group operation. We also exemplify the usefulness of our formulation in the classification of bulk gapless phases. We present a new class of Weyl semimetals and Weyl superconductors that are topologically protected by inversion symmetry.