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

Classification of equivariant approximately locality-preserving unitaries on spin chains

Alex Bols

Published Jun 3, 2021• 1 citation
OpenAlex

Abstract

We extend the classification of approximately locality-preserving unitaries (ALPUs) on spin chains recently obtained by [1] to take a local symmetry action of a finite symmetry group G into account. We classify such equivariant ALPUs up to strongly continuous deformation and allowing to `stack' with auxiliary spin chains. We find that the equivalence classes are uniquely labelled by a chiral index taking values in a discrete subset of $\mathbb{R}$, and a cohomology index taking values in the second cohomology group $H^2(G, U(1))$. We discuss the relation of this cohomology index to the index for one-dimensional symmetry protected topological (SPT) phases, which also takes values in $H^2(G, U(1))$.

Related papers

Classification of equivariant quasi-local automorphisms on quantum chains

Alex Bols

We classify automorphisms on quantum chains, allowing both spin and fermionic degrees of freedom, that are moreover equivariant with respect to a local symmetry action of a finite symmetry group $G$. The classification is up to equivalence through strongly continuous deformation and stacking with decoupled auxiliary automorphisms. We find that the equivalence classes are uniquely labeled by an index taking values in $\mathbb{Q} \cup \sqrt{2} \mathbb{Q} \times \mathrm{Hom}(G, \mathbb{Z}_2) \times H^2(G, U(1))$. We discuss te relation of this index to the index of one-dimensional symmetry protected topological phases on spin chains, which takes values in $H^2(G, U(1))$.

arXiv:2106.02145•Jun 3, 2021•0 citations

Classification of symmetry protected states of quantum spin chains for continuous symmetry groups

Bruno de O. Carvalho, Wojciech De Roeck, Tijl Jappens

Symmetry protected states (SPT's) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPT's corresponding to finite on-site symmetry groups $G$ are classified by the second cohomology group $H^2(G,U(1))$, as established by Kapustin et al. [J. Math. Phys. (2021)]. We extend this result to the case of compact topological symmetry groups $G$. We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.

arXiv:2409.01112•Sep 2, 2024•0 citations

Classification of symmetry protected states of quantum spin chains for continuous symmetry groups

Bruno de O. Carvalho, Wojciech De Roeck, Tijl Jappens

Symmetry protected states (SPTs) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPTs corresponding to finite on-site symmetry groups G are classified by the second cohomology group H2(G, U(1)), as established in Kapustin, Sopenko, and Yang. J. Math. Phys. 62(8), 081901 (2021). We extend this result to the case of compact and metrizable topological symmetry groups G. We also strengthen the existing results in the sense that our classification results hold within the class of spin chains with locally bounded on-site dimensions.

Journal of Mathematical Physics•Jul 1, 2025•0 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

The equivalence of topological indices for $\mathrm{U}(1)\rtimes\mathbb{Z}_2$-symmetric quantum spin chains

Hal Tasaki

Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires $\mathrm{U}(1)$ symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site $\mathrm{U}(1)\rtimes\mathbb{Z}_2$ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite $S=1$ antiferromagnetic Heisenberg chain with both indices equal to $-1$, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.

math-phCross-listedarXiv:2609.34119•Sep 29, 2026•0 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
We extend the classification of approximately locality-preserving unitaries (ALPUs) on spin chains recently obtained by [1] to take a local symmetry action of a finite symmetry group G into account. We classify such equivariant ALPUs up to strongly continuous deformation and allowing to `stack' with auxiliary spin chains. We find that the equivalence classes are uniquely labelled by a chiral index taking values in a discrete subset of R\mathbb{R}R, and a cohomology index taking values in the second cohomology group H2(G,U(1))H^2(G, U(1))H2(G,U(1)). We discuss the relation of this cohomology index to the index for one-dimensional symmetry protected topological (SPT) phases, which also takes values in H2(G,U(1))H^2(G, U(1))H2(G,U(1)).
We classify automorphisms on quantum chains, allowing both spin and fermionic degrees of freedom, that are moreover equivariant with respect to a local symmetry action of a finite symmetry group GGG. The classification is up to equivalence through strongly continuous deformation and stacking with decoupled auxiliary automorphisms. We find that the equivalence classes are uniquely labeled by an index taking values in Q∪2Q×Hom(G,Z2)×H2(G,U(1))\mathbb{Q} \cup \sqrt{2} \mathbb{Q} \times \mathrm{Hom}(G, \mathbb{Z}_2) \times H^2(G, U(1))Q∪2​Q×Hom(G,Z2​)×H2(G,U(1)). We discuss te relation of this index to the index of one-dimensional symmetry protected topological phases on spin chains, which takes values in H2(G,U(1))H^2(G, U(1))H2(G,U(1)).
Symmetry protected states (SPT's) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPT's corresponding to finite on-site symmetry groups GGG are classified by the second cohomology group H2(G,U(1))H^2(G,U(1))H2(G,U(1)), as established by Kapustin et al. [J. Math. Phys. (2021)]. We extend this result to the case of compact topological symmetry groups GGG. We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.
Symmetry protected states (SPTs) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPTs corresponding to finite on-site symmetry groups G are classified by the second cohomology group H2(G, U(1)), as established in Kapustin, Sopenko, and Yang. J. Math. Phys. 62(8), 081901 (2021). We extend this result to the case of compact and metrizable topological symmetry groups G. We also strengthen the existing results in the sense that our classification results hold within the class of spin chains with locally bounded on-site dimensions.
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.
The equivalence of topological indices for U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​-symmetric quantum spin chains
Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires U(1)\mathrm{U}(1)U(1) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite S=1S=1S=1 antiferromagnetic Heisenberg chain with both indices equal to −1-1−1, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.
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.