Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Tenfold Topology of Crystals

Eyal Cornfeld, Shachar Carmeli

Published Sep 9, 2020• 1 citation
OpenAlex

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.

Related papers

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

Eyal Cornfeld, Shachar Carmeli

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.

arXiv:2009.04486•Physical Review Research•Jan 15, 2021•36 citations

Topology of crystalline insulators and superconductors

Ken Shiozaki, Masatoshi Sato

We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by noninteracting Bloch and Bogoliubov--de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of 10 classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes ${\mathbit{Z}}_{2}$ global symmetry, mirror reflection, twofold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott periodicity in the space dimensions. Various symmetry-protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry-protected Fermi points. The bulk classification and the surface Fermi point classification provide a realization of the bulk-boundary correspondence in terms of the $K$ theory.

arXiv:1403.3331•Physical Review B•Oct 10, 2014•508 citations

Construction and Classification of Crystalline Topological Superconductor and Insulators in Three-Dimensional Interacting Fermion Systems

Jian-Hao Zhang, Shang-Qiang Ning, Yang Qi +1 more

The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological phases, especially for interacting systems. In this paper, we systematically construct and classify all the point-group crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of a topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover several intrinsically interacting topological phases that cannot be realized in any free-fermion systems. Moreover, the crystalline equivalence principle for 3D interacting fermionic systems is also verified, with an additional subtle “twist” on the spin of fermions.

arXiv:2204.13558•Physical Review X•Jul 28, 2025•6 citations

Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems

Jian-Hao Zhang, Shuo Yang, Yang Qi +1 more

The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.

arXiv:2012.15657•Physical Review Research•Jul 29, 2022•37 citations

Topological crystalline insulators and superconductors

Max Geier

Topological phases are quantum phases of matter that appear close to absolute zero temperature. In contrast to commonly known condensed phases of matter, such as a magnet or a superconductor, distinct topological phases may have the same symmetry. Instead, these phases are distinguished by the topology of the electronic wavefunction. Already the single-particle excitations described by band theory in an insulator or a metal or by Bogoliubov-de Gennes theory in a superconductor may have novel anomalous properties associated to their topology. This topological band theory is the subject of this thesis. We aim towards identifying the anomalous boundary and defect states associated to the topology of the band structure in crystalline insulators and superconductors. Furthermore, we determine criteria to easily identify the topological phases from the band structure of a given insulator or superconductor. In Chapter 2 we show that crystalline topological insulators and superconductors may host anomalous gapless or in-gap excitations on corners of a two-dimensional crystal or hinges of a three-dimensional crystal. These topological phases are called "second-order" according to the codimension "2" of their anomalous boundary states. We discuss the precise conditions for their appearance and when they can be linked to the bulk topology. Our discussion includes mirror, twofold-rotation and inversion as crystalline symmetries. In Chapter 3 we discuss the appearance of anomalous gapless or ingap states at defects in a crystalline insulator or superconductor. In particular, we link the existence of anomalous states at disclinations -- lattice defects that violate a rotation symmetry only locally -- to second-order topological crystalline phases. As a side product, we identify the possible contributions of other topological phases to the disclination anomaly. The results of this chapter allow to determine precisely how the disclination anomaly is linked to the bulk topology. In Chapter 4 we present how to construct "symmetry-based indicators" for topological superconductors that admit a description within the Bogoliubov-de Gennes framework. Symmetry-based indicators are necessary criteria for a given Bogoliubov-de Gennes Hamiltonian to realize a topological crystalline phase with anomalous boundary exitations. These criteria are formulated in terms of data from a small set of high-symmetry momenta only. Our approach guarantees to extract the maximal information on the anomalous boundary exitations that can be extracted from a point-wise evaluation of the Bogoliubov-de Gennes Hamiltonian. In the limit of weak superconducting pairing, our criteria can be formulated in terms of data from the normal-state band structure alone.

•Jan 1, 2021•0 citations

The weakly interacting tenfold way

Lucas C. P. A. M. Müssnich, Renato Vasconcellos Vieira

The tenfold way is a classification scheme for the building blocks of free fermion systems. More precisely, it classifies the isomorphism classes of spaces of equivariant free Hamiltonians in irreducible fermion systems with symmetries. This classification scheme naturally leads to the K-theoretical classification of topological phases of matter, known as the periodic table of topological insulators and superconductors. Topological K-theory is represented by spectra $KU$ and $KO$, and in this article we present realizations of these spectra in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We introduce a geometric definition of the space of weakly interacting time evolution operators, as the complement of the cut locus of the subspace of free operators. Our main result is that spectra $KU^{wi}$ and $KO^{wi}$ of weakly interacting time evolution operators deformation retract to $KU$ and $KO$. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.

math-phCross-listedarXiv:2603.16799•Mar 18, 2026
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.
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.
We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by noninteracting Bloch and Bogoliubov--de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of 10 classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes ${\mathbit{Z}}_{2}$ global symmetry, mirror reflection, twofold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott periodicity in the space dimensions. Various symmetry-protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry-protected Fermi points. The bulk classification and the surface Fermi point classification provide a realization of the bulk-boundary correspondence in terms of the KKK theory.
The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological phases, especially for interacting systems. In this paper, we systematically construct and classify all the point-group crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of a topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover several intrinsically interacting topological phases that cannot be realized in any free-fermion systems. Moreover, the crystalline equivalence principle for 3D interacting fermionic systems is also verified, with an additional subtle “twist” on the spin of fermions.
The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.
Topological phases are quantum phases of matter that appear close to absolute zero temperature. In contrast to commonly known condensed phases of matter, such as a magnet or a superconductor, distinct topological phases may have the same symmetry. Instead, these phases are distinguished by the topology of the electronic wavefunction. Already the single-particle excitations described by band theory in an insulator or a metal or by Bogoliubov-de Gennes theory in a superconductor may have novel anomalous properties associated to their topology. This topological band theory is the subject of this thesis. We aim towards identifying the anomalous boundary and defect states associated to the topology of the band structure in crystalline insulators and superconductors. Furthermore, we determine criteria to easily identify the topological phases from the band structure of a given insulator or superconductor. In Chapter 2 we show that crystalline topological insulators and superconductors may host anomalous gapless or in-gap excitations on corners of a two-dimensional crystal or hinges of a three-dimensional crystal. These topological phases are called "second-order" according to the codimension "2" of their anomalous boundary states. We discuss the precise conditions for their appearance and when they can be linked to the bulk topology. Our discussion includes mirror, twofold-rotation and inversion as crystalline symmetries. In Chapter 3 we discuss the appearance of anomalous gapless or ingap states at defects in a crystalline insulator or superconductor. In particular, we link the existence of anomalous states at disclinations -- lattice defects that violate a rotation symmetry only locally -- to second-order topological crystalline phases. As a side product, we identify the possible contributions of other topological phases to the disclination anomaly. The results of this chapter allow to determine precisely how the disclination anomaly is linked to the bulk topology. In Chapter 4 we present how to construct "symmetry-based indicators" for topological superconductors that admit a description within the Bogoliubov-de Gennes framework. Symmetry-based indicators are necessary criteria for a given Bogoliubov-de Gennes Hamiltonian to realize a topological crystalline phase with anomalous boundary exitations. These criteria are formulated in terms of data from a small set of high-symmetry momenta only. Our approach guarantees to extract the maximal information on the anomalous boundary exitations that can be extracted from a point-wise evaluation of the Bogoliubov-de Gennes Hamiltonian. In the limit of weak superconducting pairing, our criteria can be formulated in terms of data from the normal-state band structure alone.
The tenfold way is a classification scheme for the building blocks of free fermion systems. More precisely, it classifies the isomorphism classes of spaces of equivariant free Hamiltonians in irreducible fermion systems with symmetries. This classification scheme naturally leads to the K-theoretical classification of topological phases of matter, known as the periodic table of topological insulators and superconductors. Topological K-theory is represented by spectra KUKUKU and KOKOKO, and in this article we present realizations of these spectra in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We introduce a geometric definition of the space of weakly interacting time evolution operators, as the complement of the cut locus of the subspace of free operators. Our main result is that spectra KUwiKU^{wi}KUwi and KOwiKO^{wi}KOwi of weakly interacting time evolution operators deformation retract to KUKUKU and KOKOKO. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.