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arXiv:2306.02976

Boundary criticality via gauging finite subgroups: a case study on the clock model

Su, Lei

Published Jun 5, 2023• 3 citations
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Abstract

Gauging a finite Abelian normal subgroup $Γ$ of a nonanomalous 0-form symmetry $G$ of a theory in $(d+1)$D spacetime can yield an unconventional critical point if the original theory has a continuous transition where $Γ$ is completely spontaneously broken and if $G$ is a nontrivial extension of $G/Γ$ by $Γ$. The gauged theory has symmetry $G/Γ\times \hatΓ^{(d-1)}$, where $\hatΓ^{(d-1)}$ is the $(d-1)$-form dual symmetry of $Γ$, and a 't Hooft anomaly between them. Thus it can be viewed as a boundary of a topological phase protected by $G/Γ\times \hatΓ^{(d-1)}$. The ordinary critical point, upon gauging, is mapped to a deconfined quantum critical point between two ordinary symmetry-breaking phases ($d =1$) or an unconventional quantum critical point between an ordinary symmetry-breaking phase and a topologically ordered phase ($d\ge 2$) associated with $G/Γ$ and $\hatΓ^{(d-1)}$, respectively. Order parameters and disorder parameters, before and after gauging, can be directly related. As a concrete example, we gauge the $\mathbb{Z}_2$ subgroup of $\mathbb{Z}_4$ symmetry of a 4-state clock model on a 1D lattice and a 2D square lattice. Since the symmetry of the clock model contains $D_8$, the dihedral group of order 8, we also analyze the anomaly structure which is similar to that in the compactified $SU(2)$ gauge theory with $θ=π$ in $(3+1)$D and its mixed gauge theory. The general case is also discussed.

DOI
10.48550/arxiv.2306.02976

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This work taps into the idea of gauging finite subgroups of global symmetries in conventional systems to obtain unconventional phases and phase transitions, greatly expanding the conventional Landau paradigm. Assisted by analytical anomaly computations and numerical density-matrix renormalization group calculations, the authors map the ordinary superfluid-insulator transition of the Bose-Hubbard model to novel phase transitions between exotic topological and/or symmetry-breaking phases. The general framework has potential applications in cold atom experiments and digital quantum simulators.
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A cluster of recent preprints in strongly correlated systems converges on a structural pattern worth investigating: deconfined quantum critical points (DQCPs) and beyond-Landau phase transitions in one and two spatial dimensions can be systematically organized — and in some cases *derived* — by tracking what happens to lattice anomalies and symmetry categories when internal symmetries are gauged. This synthesis draws primarily from cond-mat.str-el sources, with supporting material from cond-mat.mes-hall and cond-mat.dis-nn, and argues the following heuristic reading (not a derivation): the non-invertible symmetries that emerge after gauging anomalous onsite symmetries encode, in their fusion category structure, the full universality data of the associated critical point — including whether the transition is type-I or type-II DQCP. Concretely, we synthesize five findings: (1) the spectral-sequence classification of Lieb-Schultz-Mattis (LSM) anomalies into E∞1,2E_\infty^{1,2}E∞1,2​ versus E∞2,1E_\infty^{2,1}E∞2,1​ types, which predicts whether the dual symmetry after gauging is invertible or non-invertible [corpus:arxiv:2606.06343]; (2) the demonstration that Rep(D8D_8D8​) and Rep(H8H_8H8​) crystalline categorical symmetries, despite sharing fusion rings, differ in FFF-symbols and therefore define inequivalent DQCP universality classes [corpus:arxiv:2606.05856]; (3) the introduction of twin condensable algebras and twin phases — inequivalent phases sharing anyon content but differing in algebra structure — as a source of intrinsically beyond-Landau transitions [corpus:arxiv:2605.31602, corpus:arxiv:2605.31601]; (4) the explicit lattice construction of non-invertible symmetry-breaking phases with local order parameters obeying generalized algebraic structures, and their duality to DQCPs of invertible symmetries [corpus:arxiv:2605.27672]; and (5) the tensor-network derivation that string order parameters in 1D organize into Lagrangian algebras in the Drinfel'd center of the symmetry category [corpus:arxiv:2605.27193]. A concrete falsification path: if two DQCPs with the same fusion ring but different FFF-symbols can be shown numerically (e.g., via DMRG central-charge extraction and operator-content comparison) to lie in identical universality classes, the thesis that FFF-symbols carry physical content distinguishing transitions would be falsified. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.29875, 2605.30421, 2605.31185, 2605.31540, 2605.31601, 2605.31602, 2606.04651, 2606.04748, 2606.04958, 2606.05845, 2606.05856, 2606.06343, 2606.06412, 2606.06432, 2606.06490
We introduce the concept of twin phases for a symmetry S\mathcal{S}S, defined as inequivalent phases, whose order parameters are part of the same generalized charge under S\mathcal{S}S. Stable, direct transitions between such twin phases are never spontaneous-symmetry-breaking transitions, even after (partially) gauging the initial symmetry S\mathcal{S}S: they are phase transitions without hidden symmetry breaking. We illustrate this with an (anomalous) finite group symmetry in 1+1d, which exhibits such intrinsically beyond Landau transition, i.e. an intrinsically Deconfined Quantum Critical Point (DQCP).
We construct an exactly solvable lattice model for a deconfined quantum critical point (DQCP) in (1+11+11+1) dimensions. This DQCP occurs in an unusual setting, namely, at the edge of a (2+12+12+1) dimensional bosonic symmetry protected topological (SPT) phase with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. The DQCP describes a transition between two gapped edges that break different Z2{\mathbb{Z}}_{2}Z2​ subgroups of the full ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. Our construction is based on an exact mapping between the SPT edge theory and a Z4{\mathbb{Z}}_{4}Z4​ spin chain. This mapping reveals that DQCPs in this system are directly related to ordinary Z4{\mathbb{Z}}_{4}Z4​ symmetry breaking critical points.