Forum of Mathematics Pi

An -valued index of symmetry-protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems

Yoshiko Ogata

Published Jan 1, 2021• 15 citations

Abstract

Abstract We consider symmetry-protected topological phases with on-site finite groupGsymmetry $\beta $ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{\mathbb T})$ -valued invariant.

DOI: 10.1017/fmp.2021.17

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Forum of Mathematics Pi

An -valued index of symmetry-protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems

Yoshiko Ogata

Published Jan 1, 2021• 15 citations

Publisher

Abstract

Abstract We consider symmetry-protected topological phases with on-site finite groupGsymmetry

\beta

for two-dimensional quantum spin systems. We show that they have

H^{3}(G,{\mathbb T})

-valued invariant.

DOI: 10.1017/fmp.2021.17

Related papers

A $H^{3}(G,{\mathbb T})$ -valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systemsA $H^{3}(G,{\mathbb T})$-valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems

Yoshiko Ogata

We consider SPT-phases with on-site finite group $G$ symmetry $β$ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{\mathbb T})$ -valued invariant.We consider SPT-phases with on-site finite group $G$ symmetry $β$ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{\mathbb T})$-valued invariant.

arXiv:2101.00426Jan 2, 20216 citations

Classification of symmetry protected topological phases in quantum spin chains

Yoshiko Ogata

arXiv:2110.04671Current Developments in MathematicsJan 1, 202015 citations

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

Chris Bourne, Yoshiko Ogata

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.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.

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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.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:2409.01112Sep 2, 20240 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.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 PhysicsJul 1, 20250 citations

An index for two-dimensional SPT states

Nikita Sopenko

arXiv:2101.00801Journal of Mathematical PhysicsNov 1, 202113 citations

Abstract We consider symmetry-protected topological phases with on-site finite groupGsymmetry

\beta

for two-dimensional quantum spin systems. We show that they have

H^{3}(G,{\mathbb T})

-valued invariant.

H^{3}(G,{\mathbb T})

-valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems

We consider SPT-phases with on-site finite group

G

symmetry

β

for two-dimensional quantum spin systems. We show that they have

H^{3}(G,{\mathbb T})

-valued invariant.

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.

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.