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:2006.15232 · Forum of Mathematics Sigma

The classification of symmetry protected topological phases of one-dimensional fermion systems

Chris Bourne, Yoshiko Ogata

Published Jan 1, 2021• 25 citations
arXivHTMLPDF

Abstract

Abstract We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in $\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})$ with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.

DOI
10.1017/fms.2021.19

Related papers

Towards a Complete Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions

Meng Cheng, Zhen Bi, Yi‐Zhuang You +1 more

Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology class $H^3(G, \mathrm{U}(1))$. Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of $G$-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional $H^1(G, \mathbb{Z}_2)$ structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group $G\times\mathbb{Z}_2^f$ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.

•Jan 6, 2015•9 citations

Classification of symmetry-protected topological many-body localized phases in one dimension

Amos Chan, Thorsten B Wahl

We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians.

arXiv:1808.05656•Journal of Physics Condensed Matter•Mar 11, 2020•12 citations

Classification of symmetry-protected topological many-body localized phases in one dimension.

Amos Chan, Thorsten B. Wahl

We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians.

•May 18, 2020•8 citations

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory

Zheng‐Cheng Gu, Xiao-Gang Wen

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have ${T}^{2}=1$ time-reversal ${Z}_{2}^{T}$ and fermion-number-parity ${Z}_{2}^{f}$ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the $K$-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.

arXiv:1201.2648•Physical Review B•Sep 23, 2014•306 citations

Classification of symmetry protected topological phases in quantum spin chains

Yoshiko Ogata

We consider the classification problem of symmetry protected topological (SPT) phases on quantum spin systems.SPT phases are gapped short-range-entangled quantum phases with a symmetry G.We explain that in one and two-dimensional quantum spin systems, there are H 2 (G, U (1))/ H 3 (G, U (1))-valued invariant, confirming a physicists conjecture.Contents 1. Symmetry protected topological phases 41 2. SPT-phases in 1-dimensional systems 63 3. SPT-phases in 2-dimensional systems 78

arXiv:2110.04671•Current Developments in Mathematics•Jan 1, 2020•15 citations

Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants*

Yunqing Ouyang, Qing-Rui Wang, Zheng‐Cheng Gu +1 more

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.

arXiv:2005.06572•Chinese Physics Letters•Dec 1, 2021•11 citations
Abstract We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in Z2×H1(G,Z2)×H2(G,U(1)p)\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})Z2​×H1(G,Z2​)×H2(G,U(1)p​) with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.
Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry GGG can be completely classified by the group cohomology class H3(G,U(1))H^3(G, \mathrm{U}(1))H3(G,U(1)). Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of GGG-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional H1(G,Z2)H^1(G, \mathbb{Z}_2)H1(G,Z2​) structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group G×Z2fG\times\mathbb{Z}_2^fG×Z2f​ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.
We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians.
We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the KKK-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1c=1c=1 gapless edge excitations, if the symmetry is not broken.
In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.