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1. Fractionalization of coset non-invertible symmetry and exotic Hall conductance

   https://doi.org/10.21468/SciPostPhys.17.3.095  

   Hsin, Po-Shen; Kobayashi, Ryohei; Zhang, Carolyn (January 2024, SciPost Physics)

   We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible0 $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a “fractional charge” under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enrichedS_3 $S_3$andO(2) $O\left(2\right)$gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a “sandwich” construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified. 

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2. Undoing decomposition

   https://doi.org/10.1142/S0217751X19502336  

   Sharpe, Eric (December 2019, International Journal of Modern Physics A)

   In this paper we discuss gauging one-form symmetries in two-dimensional theories. The existence of a global one-form symmetry in two dimensions typically signals a violation of cluster decomposition — an issue resolved by the observation that such theories decompose into disjoint unions, a result that has been applied to, for example, Gromov–Witten theory and gauged linear sigma model phases. In this paper we describe how gauging one-form symmetries in two-dimensional theories can be used to select particular elements of that disjoint union, effectively undoing decomposition. We examine such gaugings explicitly in examples involving orbifolds, nonsupersymmetric pure Yang–Mills theories, and supersymmetric gauge theories in two dimensions. Along the way, we learn explicit concrete details of the topological configurations that path integrals sum over when gauging a one-form symmetry, and we also uncover “hidden” one-form symmetries. 

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3. Spontaneously broken subsystem symmetries

   https://doi.org/10.1007/JHEP03(2022)016  

   Distler, Jacques; Karch, Andreas; Raz, Amir (March 2022, Journal of High Energy Physics)

   A bstract We investigate the spontaneous breaking of subsystem symmetries directly in the context of continuum field theories by calculating the correlation function of charged operators. Our methods confirm the lack of spontaneous symmetry breaking in some of the existing continuum field theories with subsystem symmetries, as had previously been established based on a careful analysis of the spectrum. We present some novel continuum field theory constructions that do exhibit spontaneous symmetry breaking whenever allowed by general principles. These interesting patterns of symmetry breaking occur despite the fact that all the theories we study are non-interacting. 

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4. Notes on gauging noninvertible symmetries. Part II. Higher multiplicity cases

   https://doi.org/10.1007/JHEP05(2025)066  

   Perez-Lona, A; Robbins, D; Sharpe, E; Vandermeulen, T; Yu, X (May 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep(A₄) in detail. We realize Rep(A₄) gaugings for thec= 1 CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep(SL(2, ℤ₃)). We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries. 

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5. Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

   https://doi.org/10.22331/q-2020-02-10-228  

   Daniel, Austin K.; Alexander, Rafael N.; Miyake, Akimasa (February 2020, Quantum)

   What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all states within so-called SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. There are four subsystem symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the former three are fundamentally in one-to-one correspondence with three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices possess 1-form symmetries and have a different capability related to error correction. 

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