Tenfold topology of crystals: Unified classification of crystalline topological insulators and superconductors

Eyal Cornfeld,Shachar Carmeli

doi:10.1103/physrevresearch.3.013052

Abstract

The celebrated tenfold way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of nonspatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly interacting electrons' noncrystalline 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 ground states 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 nonspatial 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.Received 23 September 2020Revised 10 December 2020Accepted 11 December 2020DOI:https://doi.org/10.1103/PhysRevResearch.3.013052Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEdge statesSurface statesSymmetry protected topological statesTopological insulatorsTopological materialsTopological superconductorsPhysical SystemsCrystalline systemsTechniquesAbstract algebraDiscrete symmetries in condensed matterGroup theoryTopologyCondensed Matter, Materials & Applied PhysicsInterdisciplinary PhysicsGeneral Physics

Highlights

The discovery of the periodic table of topological insulators and superconductors (TISCs) and the tenfold way has revolutionized our understanding of topological phases of quantum matter [1,2,3,4,5,6,7,8,9,10,11]
In order to avoid possible confusion, we note that there exists variability in the literature regarding which compatibility relations to impose in the definition of BS, this may lead to an inclusion ofmetallic states within BS; we focus our attention on TISCs and use the stricter definition such that BSTISC includes only band labels corresponding to gapped systems, i.e., insulators and superconductors [116]
We first use the properties of the spectra formalism to obtain the complete classification of topologically distinct quantum states of 3D crystalline TISCs (CTISCs) in all AZ symmetry classes for key space-groups

Summary

Introduction

The discovery of the periodic table of topological insulators and superconductors (TISCs) and the tenfold way has revolutionized our understanding of topological phases of quantum matter [1,2,3,4,5,6,7,8,9,10,11]. These classes are characterized by the presence or absence of internal symmetries of the system, such as time-reversal, particle-hole symmetry, spin rotations, and charge conservation. Since our understanding of their importance, this and other cohomology theories have been used throughout physics to explore and understand various phenomena Examples of such are complex and real cobordisms which have been used to classify the strongly interacting invertible fermionic topological phases [26]

Results

Discussion

Conclusion