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

arXiv:1412.4808 · Physica Scripta

Bott–Kitaev periodic table and the diagonal map

Ricardo Kennedy, Martin R. Zirnbauer

Published Aug 25, 2015• 8 citations
arXivHTMLPDF

Abstract

Building on the ten-way symmetry classification of disordered fermions, the authors have recently given a homotopy-theoretic proof of Kitaev's ‘periodic table’ for topological insulators and superconductors. The present paper offers an introduction to the physical setting and the mathematical model used. Basic to the proof is the so-called diagonal map, a natural transformation akin to the Bott map of algebraic topology, which increases by one unit both the momentum-space dimension and the symmetry index of translation-invariant ground states of gapped free-fermion systems. This mapping is illustrated here with a few examples of interest. (Based on a talk delivered by the senior author at the Nobel Symposium on ‘New Forms of Matter: Topological Insulators and Superconductors’; Stockholm, 13–15 June, 2014.)

DOI
10.1088/0031-8949/2015/t164/014010

Related papers

Bott Periodicity for $${\mathbb{Z}_2}$$ Z 2 Symmetric Ground States of Gapped Free-Fermion Systems

R. Kennedy, Martin R. Zirnbauer

Building on the symmetry classification of disordered fermions, we give a proof of the proposal by Kitaev, and others, for a “Bott clock” topological classification of free-fermion ground states of gapped systems with symmetries. Our approach differs from previous ones in that (i) we work in the standard framework of Hermitian quantum mechanics over the complex numbers, (ii) we directly formulate a mathematical model for ground states rather than spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools rather than K -theory. Key to our proof is a natural transformation that squares to the standard Bott map and relates the ground state of a d -dimensional system in symmetry class s to the ground state of a ( d + 1)-dimensional system in symmetry class s + 1. This relation gives a new vantage point on topological insulators and superconductors.

arXiv:1409.2537•Communications in Mathematical Physics•Nov 26, 2015•73 citations

Bott periodicity for the topological classification of gapped states of matter with reflection symmetry

Luka Trifunovic, Piet W. Brouwer

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-2 and four period-8 cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme, the classification in arbitrary dimensions $d\ensuremath{\ge}1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys. Rev. B 90, 165114 (2014)]. The use of a scattering-matrix-inspired method allows us to address the second descendant ${\mathbb{Z}}_{2}$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.

arXiv:1707.06306•Physical review. B./Physical review. B•Nov 2, 2017•28 citations

Symmetric Fermi projections and Kitaev’s table: Topological phases of matter in low dimensions

David Gontier, Domenico Monaco, Solal Perrin-Roussel

We review Kitaev’s celebrated “periodic table” for topological phases of condensed matter, which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections that depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for zero- and one-dimensional systems and recover the (weak and strong) topological invariants proposed by Kitaev and others.

arXiv:2201.01576•Journal of Mathematical Physics•Apr 1, 2022•5 citations

Periodic table for topological insulators and superconductors

Alexei Kitaev

Abstract. Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given. Keywords: Topological phase, K-theory, K-homology, Clifford algebra, Bott periodicity

•May 14, 2009•902 citations

Periodic table for Floquet topological insulators

Rahul Roy, Fenner Harper

Periodically driven systems have recently been shown to host topological phases that are inherently dynamical in character, opening up a new arena in which to explore topological physics. One important group of such phases, known as Floquet topological insulators, arise in systems of free fermions and exhibit protected topological edge modes analogous to the edge modes of static topological insulators. In this work, the authors use methods from K theory to provide a complete topological classification of Floquet topological phases of this kind. The main result is a periodic table for Floquet topological insulators, which may be viewed as a time-dependent extension of the periodic table of topological insulators and superconductors originally introduced by Alexei Kitaev.

arXiv:1603.06944•Physical review. B./Physical review. B•Oct 13, 2017•354 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
Building on the ten-way symmetry classification of disordered fermions, the authors have recently given a homotopy-theoretic proof of Kitaev's ‘periodic table’ for topological insulators and superconductors. The present paper offers an introduction to the physical setting and the mathematical model used. Basic to the proof is the so-called diagonal map, a natural transformation akin to the Bott map of algebraic topology, which increases by one unit both the momentum-space dimension and the symmetry index of translation-invariant ground states of gapped free-fermion systems. This mapping is illustrated here with a few examples of interest. (Based on a talk delivered by the senior author at the Nobel Symposium on ‘New Forms of Matter: Topological Insulators and Superconductors’; Stockholm, 13–15 June, 2014.)
Bott Periodicity for Z2{\mathbb{Z}_2}Z2​ Z 2 Symmetric Ground States of Gapped Free-Fermion Systems
Building on the symmetry classification of disordered fermions, we give a proof of the proposal by Kitaev, and others, for a “Bott clock” topological classification of free-fermion ground states of gapped systems with symmetries. Our approach differs from previous ones in that (i) we work in the standard framework of Hermitian quantum mechanics over the complex numbers, (ii) we directly formulate a mathematical model for ground states rather than spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools rather than K -theory. Key to our proof is a natural transformation that squares to the standard Bott map and relates the ground state of a d -dimensional system in symmetry class s to the ground state of a ( d + 1)-dimensional system in symmetry class s + 1. This relation gives a new vantage point on topological insulators and superconductors.
Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-2 and four period-8 cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme, the classification in arbitrary dimensions $d\ensuremath{\ge}1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys. Rev. B 90, 165114 (2014)]. The use of a scattering-matrix-inspired method allows us to address the second descendant Z2{\mathbb{Z}}_{2}Z2​ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.
We review Kitaev’s celebrated “periodic table” for topological phases of condensed matter, which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections that depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for zero- and one-dimensional systems and recover the (weak and strong) topological invariants proposed by Kitaev and others.
Abstract. Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given. Keywords: Topological phase, K-theory, K-homology, Clifford algebra, Bott periodicity
Periodically driven systems have recently been shown to host topological phases that are inherently dynamical in character, opening up a new arena in which to explore topological physics. One important group of such phases, known as Floquet topological insulators, arise in systems of free fermions and exhibit protected topological edge modes analogous to the edge modes of static topological insulators. In this work, the authors use methods from K theory to provide a complete topological classification of Floquet topological phases of this kind. The main result is a periodic table for Floquet topological insulators, which may be viewed as a time-dependent extension of the periodic table of topological insulators and superconductors originally introduced by Alexei Kitaev.
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.