We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions d=2 d = 2 and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed ‘t Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension d=2 d = 2 and 3. In d=2 d = 2 , these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In d=3 d = 3 , among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category.
We use the language of von Neumann subfactors to investigate non-invertible symmetries in two dimensions. A fusion categorical symmetry \mathcal{C} 𝒞 , its module category \mathcal{M} ℳ , and a gauging labeled by an algebra object \mathcal{A} 𝒜 are encoded in the bipartite principal graph of a subfactor. The dual principal graph captures the quantum symmetry \mathcal{C}' 𝒞 ′ obtained by gauging \mathcal{A} 𝒜 in \mathcal{C} 𝒞 , as well as a reverse gauging back to \mathcal{C} 𝒞 . From a given subfactor N \subset M N ⊂ M , we derive a quiver diagram that encodes the representations of the associated non-invertible symmetry. We show how this framework provides necessary conditions for admissible gaugings, enabling the construction of generalized orbifold groupoids. To illustrate this strategy, we present three examples: Rep (D_4) ( D 4 ) as a warm-up, the higher-multiplicity case Rep (A_4) ( A 4 ) with its associated generalized orbifold groupoid and triality symmetry, and Rep (A_5) ( A 5 ) , where A_5 A 5 is the smallest non-solvable finite group. For applications to gapless systems, we embed these generalized gaugings as global manipulations on the conformal manifolds of c=1 c = 1 CFTs and uncover new self-dualities in the exceptional SU(2)_1/A_5 S U ( 2 ) 1 / A
The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.
Spatially modulated symmetries have emerged since the discovery of fractons, which characterize unconventional topological phases with mobility-constrained quasiparticle excitations. On the other hand, noninvertible symmetry operators have attracted substantial attention in communities of high energy and condensed matter physics due to their deep insight into quantum anomalies and exotic phases of matter. However, the connection between these exotic and noninvertible symmetries has not been fully explored. In this paper, we construct concrete lattice models with noninvertible symmetry operators via gauging spatially modulated symmetries and investigate their exotic fusion rules. Specifically, we construct spin models with subsystem symmetries or dipole symmetries on one, two, and three-dimensional lattices. Gauging subsystem symmetries leads to noninvertible symmetry operators whose fusion rules involve 0-form subsystem charges in two dimensions and higher-form operators that correspond to ``lineon'' excitations (excitations which are mobile along one-dimensional line) in three dimensions. Gauging dipole symmetries leads to noninvertible symmetry operators with dipole algebras that describe a hierarchical structure between global and dipole charges. Notably, the hierarchical structure of the dual dipole charges is inverted compared with the original ones. Our work provides a unified and systematic analytical framework for constructing exotic symmetry operators by gauging relevant symmetries.
It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In Ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin- 12 Heisenberg XXZ model. One model has symmetry operators forming the fusion category Rep(S3) of representations of the group S3 . We find that the mapping between its topological sectors and those of the XXZ model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of Rep(S3) . Published by the American Physical Society 2024
The classification of phases using categorical symmetries has greatly expanded the landscape of gapped and gapless phases. So far, however, these developments have largely been restricted to phases with unitary (higher-)categorical symmetries over
C. In this work, we incorporate anti-unitary symmetries, such as time-reversal symmetry
Z2T, and show that the relevant physical structures are naturally described by fusion categories over
R. A class of real fusion categories, which we call Galois-real fusion categories, provides the correct categorical model for anti-unitary symmetries. A simple example is the time-reversal symmetry
Z2T itself. We discuss the basic structures of real fusion categories and present a range of examples, including the group-theoretical categories
(GT)ω and
Rep(GT) associated to anti-linear groups
GT, as well as non-invertible time-reversal symmetries described by a real analogue of Tambara--Yamagami fusion categories. We then classify gapped phases enriched with anti-linear symmetries in terms of module categories over Galois-real fusion categories. We furthermore apply the categorical formulation to prove dualities (i.e. gauge or Morita equivalences) of anti-linear symmetries generated by gauging subgroups. Complementing this, we also develop a Symmetry Topological Field Theory (SymTFT) framework, in which Galois-real fusion categories arise as boundary conditions of a
Z2T-enriched SymTFT. Morita equivalent anti-linear symmetries are shown to arise as different boundaries of the same
Z2T-enriched SymTFT.