A generalized symmetry (defined by the algebra of local symmetric operators) can go beyond group or higher group description. A theory of generalized symmetry (up to holo-equivalence) was developed in terms of symmetry-TO -- a bosonic topological order (TO) with gappable boundary in one higher dimension. We propose a general method to compute the 2+1D symmetry-TO from the local symmetric operators in 1+1D systems. Our theory is based on the commutant patch operators, which are extended operators constructed as products and sums of local symmetric operators. A commutant patch operator commutes with all local symmetric operators away from its boundary. We argue that topological invariants associated with anyon diagrams in 2+1D can be computed as contracted products of commutant patch operators in 1+1D. In particular, we give concrete formulae for several topological invariants in terms of commutant patch operators. Topological invariants computed from patch operators include those beyond modular data, such as the link invariants associated with the Borromean rings and the Whitehead link. These results suggest that the algebra of commutant patch operators is described by 2+1D symmetry-TO. Based on our analysis, we also argue briefly that the commutant patch operators would serve as order parameters for gapped phases with finite symmetries.A generalized symmetry (defined by the algebra of local symmetric operators) can go beyond group or higher group description. A theory of generalized symmetry (up to holo-equivalence) was developed in terms of symmetry-TO -- a bosonic topological order (TO) with gappable boundary in one higher dimension. We propose a general method to compute the 2+1D symmetry-TO from the local symmetric operators in 1+1D systems. Our theory is based on the commutant patch operators, which are extended operators constructed as products and sums of local symmetric operators. A commutant patch operator commutes with all local symmetric operators away from its boundary. We argue that topological invariants associated with anyon diagrams in 2+1D can be computed as contracted products of commutant patch operators in 1+1D. In particular, we give concrete formulae for several topological invariants in terms of commutant patch operators. Topological invariants computed from patch operators include those beyond modular data, such as the link invariants associated with the Borromean rings and the Whitehead link. These results suggest that the algebra of commutant patch operators is described by 2+1D symmetry-TO. Based on our analysis, we also argue briefly that the commutant patch operators would serve as order parameters for gapped phases with finite symmetries.

arXiv:2310.05790Oct 9, 20234 citations

Symmetries and Invariances

Edgard Elbaz

Springer eBooksJan 1, 19981 citation

Commuting-projector Hamiltonians for two-dimensional topological insulators: Edge physics and many-body invariants

Jun Ho Son, Jason Alicea

Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Z. Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy $\mathcal{CP}$ symmetry, leading to intuitive interpretations of the bulk invariant for a $\mathcal{CP}$ -symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Z. Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy $\mathcal{CP}$ symmetry, leading to intuitive interpretations of the bulk invariant for a $\mathcal{CP}$-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.

arXiv:1906.11846Physical review. B./Physical review. BOct 4, 201910 citations

Symmetries and Transformations

Edward B. Manoukian

Springer eBooksJan 16, 200716 citations

Implications of an arithmetical symmetry of the commutant for modular invariants

Philippe Ruelle, E. Thiran, J. Weyers

arXiv:hep-th/9212048Nuclear Physics BAug 1, 199324 citations

Commutators and Anticommutators

WORLD SCIENTIFIC eBooksJul 20, 20161 citation

\mathcal{CP}

symmetry, leading to intuitive interpretations of the bulk invariant for a

\mathcal{CP}

-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.