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arXiv:2610.10764

Gauging Modulated Symmetries: Bond Algebras, Higher-Form Symmetries, and Symmetry-Enriched Topological Order

Gustavo M. Yoshitome, Pedro R. S. Gomes, Guilherme Delfino

Published Oct 7, 2026• 0 citations
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Abstract

We develop a constructive framework for characterizing modulated symmetries and their gauging in $d$-dimensional lattice systems. The modulation profiles are organized as a finite-dimensional representation of the space group acting on a finite Abelian internal symmetry group. Since the corresponding automorphism group is finite, the nontrivial spatial action factors through a finite quotient of the space group, leading to periodic modulation patterns and providing a classification framework for discrete Abelian modulated symmetries. The symmetry-allowed local interactions are systematically obtained from a kernel construction, which also yields an algebra of open operators. Upon gauging, its elements are transmuted into extended Wilson and boundary operators while retaining their algebraic relations. In the deconfined regime, the resulting algebra realizes a mixed 't Hooft anomaly, providing a microscopic route to Abelian crystalline symmetry-enriched topological phases (SETs). These SETs do not mix electric and magnetic sectors, which transform in mutually contragredient representations, thereby preserving their braiding pairing. We illustrate the framework through explicit examples and show that several models present in the literature fit naturally within our construction.

DOI
10.48550/arxiv.2610.10764

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We develop a constructive framework for characterizing modulated symmetries and their gauging in ddd-dimensional lattice systems. The modulation profiles are organized as a finite-dimensional representation of the space group acting on a finite Abelian internal symmetry group. Since the corresponding automorphism group is finite, the nontrivial spatial action factors through a finite quotient of the space group, leading to periodic modulation patterns and providing a classification framework for discrete Abelian modulated symmetries. The symmetry-allowed local interactions are systematically obtained from a kernel construction, which also yields an algebra of open operators. Upon gauging, its elements are transmuted into extended Wilson and boundary operators while retaining their algebraic relations. In the deconfined regime, the resulting algebra realizes a mixed 't Hooft anomaly, providing a microscopic route to Abelian crystalline symmetry-enriched topological phases (SETs). These SETs do not mix electric and magnetic sectors, which transform in mutually contragredient representations, thereby preserving their braiding pairing. We illustrate the framework through explicit examples and show that several models present in the literature fit naturally within our construction.
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A bstract We present a systematic framework for constructing exactly-solvable lattice models of symmetry-enriched topological (SET) phases based on an enlarged version of the string-net model. We also gauge the global symmetries of our SET models to obtain string-net models of pure topological phases. Without invoking externally imposed onsite symmetry actions, our approach promotes the string-net model of a pure topological order, specified by an input unitary fusion category F \mathcal{F} F F , to an SET model, specified by a multifusion category together with a set of isomorphisms. Two complementary construction strategies are developed in the main text: (i) promotion via outer automorphisms of F \mathcal{F} F F and (ii) promotion via the Frobenius algebras of F \mathcal{F} F F . The global symmetries derived via these two strategies are intrinsic to topological phases and are thus termed blood symmetries, as opposed to adopted symmetries, which can be arbitrarily imposed on topological phases. We propose the concept of symmetry-gauging family of topological phases, which are related by gauging their blood symmetries. With our approach, we construct the first explicit lattice realization of a nonabelian-symmetry-enriched topological phase — the S 3 symmetry-enriched ℤ 2 × ℤ 2 quantum-double phase. The approach further reveals the role of local excitations in SET phases and establishes their symmetry constraints.
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Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group GGG, the 2D SPT phase protected by GGG is dual to the 2D topological phase exemplified by the twisted quantum double model ${D}^{\ensuremath{\omega}}(G)$ via gauging the global symmetry GGG. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when GGG is a finite, solvable group. Here, we review the general approach to gauging a $G\text{\ensuremath{-}}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a NNN-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the NNN-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data GGG and $[\ensuremath{\omega}]\ensuremath{\in}{H}^{3}(G,U(1))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups Dn{D}_{n}Dn​ and the quaternion group Q8{Q}_{8}Q8​.