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:1202.3983 · New Journal of Physics

A new class of (2 + 1)-dimensional topological superconductors with $\mathbb {Z}_8$ topological classification

Xiao-Liang Qi

Published Jun 3, 2013• 137 citations
arXivHTMLPDF

Abstract

The classification of topological states of matter depends on spatial dimension and symmetry class. For non-interacting topological insulators and superconductors, the topological classification is obtained systematically and non-trivial topological insulators are classified by either integer or Z 2 . The classification of interacting topological states of matter is much more complicated and only special cases are understood. In this paper we study a new class of topological superconductors in (2 + 1) dimensions which has time-reversal symmetry and a spin conservation symmetry. We demonstrate that the superconductors in this class are classified by when electron interaction is considered, while the classification is without interaction.

DOI
10.1088/1367-2630/15/6/065002

Related papers

Fermionic topological phases and bosonization in higher dimensions

Ryohei Kobayashi

Abstract We review a recipe to produce a lattice construction of fermionic phases of matter in the presence of time reversal symmetry by extending the fermionization and bosonization known in (1+1) dimensions to various setups including higher spacetime dimensions in the presence of global symmetries. As an application, we provide a state sum lattice path integral for a (1+1)-dimensional topological superconductor with time reversal symmetry generating the $\mathbb{Z}_8$ classification of the symmetry-protected topological phase. We also illustrate a state sum path integral for a (3+1)-dimensional topological superconductor with time reversal symmetry that generates the $\mathbb{Z}_{16}$ classification.

Progress of Theoretical and Experimental Physics•Aug 17, 2021•1 citation

Inversion symmetry protected topological insulators and superconductors

Yuan-Ming Lu, D. H. Lee

Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry. In this work we systematically study another class of topological phases of weakly interacting electrons protected by spatial inversion symmetry, which generally don't support stable gapless boundary states. We classify these inversion-symmetric topological insulators and superconductors in the framework of K-theory, and construct their lattice models. We also discuss quantized response functions of these inversion-protected topological phases, which serve as their experimental signatures.

arXiv:1403.5558•APS•Mar 21, 2014•8 citations

Classification of topological insulators and superconductors in three spatial dimensions

Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki +1 more

We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced ${\mathbb{Z}}_{2}$ topological insulator in the symplectic (or spin-orbit) symmetry class. We show that there exist precisely four more topological insulators. For these systems, all of which are time-reversal invariant in three dimensions, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. Three of the above five topologically nontrivial phases can be realized as time-reversal invariant superconductors. In these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a two-dimensional surface, they support a number (which may be an arbitrary nonvanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin-rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogs of well-known paired topological phases in two spatial dimensions such as the spinless chiral $({p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y})$-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically nontrivial (analogous to ``weak pairing'') and topologically trivial (analogous to ``strong pairing'') 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with nonvanishing winding number possess nontrivial topological ground-state degeneracies.

arXiv:0803.2786•Physical Review B•Nov 26, 2008•3,322 citations

Topological phases of one-dimensional fermions: An entanglement point of view

Ari M. Turner, Frank Pollmann, Erez Berg

The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by L. Fidkowski and A. Kitaev [Phys. Rev. B 81, 134509 (2010)], we find that in the presence of interactions there are only eight distinct phases which obey a ${\mathbb{Z}}_{8}$ group structure. This is in contrast to the $\mathbb{Z}$ classification in the noninteracting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.

arXiv:1008.4346•Physical Review B•Feb 8, 2011•564 citations

Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond

Andreas W W Ludwig

After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.

arXiv:1512.08882•Physica Scripta•Dec 31, 2015•144 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
A new class of (2 + 1)-dimensional topological superconductors with Z8\mathbb {Z}_8Z8​ topological classification
The classification of topological states of matter depends on spatial dimension and symmetry class. For non-interacting topological insulators and superconductors, the topological classification is obtained systematically and non-trivial topological insulators are classified by either integer or Z 2 . The classification of interacting topological states of matter is much more complicated and only special cases are understood. In this paper we study a new class of topological superconductors in (2 + 1) dimensions which has time-reversal symmetry and a spin conservation symmetry. We demonstrate that the superconductors in this class are classified by when electron interaction is considered, while the classification is without interaction.
Abstract We review a recipe to produce a lattice construction of fermionic phases of matter in the presence of time reversal symmetry by extending the fermionization and bosonization known in (1+1) dimensions to various setups including higher spacetime dimensions in the presence of global symmetries. As an application, we provide a state sum lattice path integral for a (1+1)-dimensional topological superconductor with time reversal symmetry generating the Z8\mathbb{Z}_8Z8​ classification of the symmetry-protected topological phase. We also illustrate a state sum path integral for a (3+1)-dimensional topological superconductor with time reversal symmetry that generates the Z16\mathbb{Z}_{16}Z16​ classification.
Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry. In this work we systematically study another class of topological phases of weakly interacting electrons protected by spatial inversion symmetry, which generally don't support stable gapless boundary states. We classify these inversion-symmetric topological insulators and superconductors in the framework of K-theory, and construct their lattice models. We also discuss quantized response functions of these inversion-protected topological phases, which serve as their experimental signatures.
We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced Z2{\mathbb{Z}}_{2}Z2​ topological insulator in the symplectic (or spin-orbit) symmetry class. We show that there exist precisely four more topological insulators. For these systems, all of which are time-reversal invariant in three dimensions, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. Three of the above five topologically nontrivial phases can be realized as time-reversal invariant superconductors. In these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a two-dimensional surface, they support a number (which may be an arbitrary nonvanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin-rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogs of well-known paired topological phases in two spatial dimensions such as the spinless chiral $({p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y})$-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically nontrivial (analogous to ``weak pairing'') and topologically trivial (analogous to ``strong pairing'') 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with nonvanishing winding number possess nontrivial topological ground-state degeneracies.
The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by L. Fidkowski and A. Kitaev [Phys. Rev. B 81, 134509 (2010)], we find that in the presence of interactions there are only eight distinct phases which obey a Z8{\mathbb{Z}}_{8}Z8​ group structure. This is in contrast to the Z\mathbb{Z}Z classification in the noninteracting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.
After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.
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.