Tenfold Topology of Crystals: Unified classification of crystalline topological insulators and superconductors

TL;DR

This paper provides a comprehensive classification framework for crystalline topological insulators and superconductors, extending the tenfold-way to include spatial symmetries and surface states using advanced mathematical tools.

Contribution

It introduces a unified classification scheme for crystalline topological phases across all symmetry classes and space groups, utilizing equivariant spectra for broad applicability.

Findings

Classified topological phases for key space groups and symmetry classes

Developed a unified mathematical framework for topological classification

Provided computational tools for analyzing crystalline topological states

Abstract

The celebrated tenfold-way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of non-spatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly-interacting electrons' non-crystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct groundstates as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold-way of Altland-Zirnbauer non-spatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold-way.