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arXiv:2608.08428

Organizing transitions and their cascades: Generalized symmetry enforcement in massless flows or Higgs transitions

Yoshiki Fukusumi, Yuma FURUTA

Published Aug 9, 2026• 0 citations
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Abstract

We study the role of generalized symmetry in massless renormalization group flows or Higgs transitions. In particular, we revisit the massless renormalization group flows in unitary minimal models, $\mathbf{M}(p,p+1) \rightarrow \overline{\mathbf{M}(p-1,p)}$ preserving the fusion ring symmetry $\text{FR} (SU(2)_{p-2})\subset \mathbf{M}(p,p+1)$ where $p$ is an integer satisfying $p>3$. In this series of flows, we demonstrate that the unbroken fusion ring symmetry $\text{FR} (SU(2)_{p-2})$ eliminates all relevant perturbations in the $\overline{\mathbf{M}(p-1,p)}$ model. Hence, the infrared theory $\overline{\mathbf{M}(p-1,p)}$ is stable at the level of the scaling analysis and can be interpreted as a (weak-)symmetry-enforced gapless phase in contemporary theoretical physics. Phenomenologically, by the folding trick, the unbroken fusion ring symmetry corresponds to a (half-)integer spin nonsimple current, a variant of the Cooper pair involving nonabelian anyons generated from the coset or level-rank duality structures. Moreover, we demonstrate that the structure of (half-)integer spin nonsimple current plays a fundamental role in causing the resonance effect of relevant and dangerously irrelevant perturbations. This resonance effect may result in the cascade of phase transitions (or the system flows to unexplored fixed points), and the symmetry can be a stopper of such unconventional flows.

DOI
10.48550/arxiv.2608.08428

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We study the role of generalized symmetry in massless renormalization group flows or Higgs transitions. In particular, we revisit the massless renormalization group flows in unitary minimal models, M(p,p+1)→M(p−1,p)‾\mathbf{M}(p,p+1) \rightarrow \overline{\mathbf{M}(p-1,p)}M(p,p+1)→M(p−1,p)​ preserving the fusion ring symmetry FR(SU(2)p−2)⊂M(p,p+1)\text{FR} (SU(2)_{p-2})\subset \mathbf{M}(p,p+1)FR(SU(2)p−2​)⊂M(p,p+1) where ppp is an integer satisfying p>3p>3p>3. In this series of flows, we demonstrate that the unbroken fusion ring symmetry FR(SU(2)p−2)\text{FR} (SU(2)_{p-2})FR(SU(2)p−2​) eliminates all relevant perturbations in the M(p−1,p)‾\overline{\mathbf{M}(p-1,p)}M(p−1,p)​ model. Hence, the infrared theory M(p−1,p)‾\overline{\mathbf{M}(p-1,p)}M(p−1,p)​ is stable at the level of the scaling analysis and can be interpreted as a (weak-)symmetry-enforced gapless phase in contemporary theoretical physics. Phenomenologically, by the folding trick, the unbroken fusion ring symmetry corresponds to a (half-)integer spin nonsimple current, a variant of the Cooper pair involving nonabelian anyons generated from the coset or level-rank duality structures. Moreover, we demonstrate that the structure of (half-)integer spin nonsimple current plays a fundamental role in causing the resonance effect of relevant and dangerously irrelevant perturbations. This resonance effect may result in the cascade of phase transitions (or the system flows to unexplored fixed points), and the symmetry can be a stopper of such unconventional flows.
We propose a general quantum Hamiltonian formalism of a renormalization group (RG) flow with an emphasis on generalized symmetry by interpreting the elementary relationship between homomorphism, quotient ring, and projection. In our formalism, the noninvertible nature of the ideal of a fusion ring realizing the generalized symmetry of an ultraviolet (UV) theory plays a fundamental role in determining condensation rules between anyons, resulting in the infrared (IR) theories. Our algebraic method applies to the domain wall problem in 2+12+12+1 dimensional topologically ordered systems and the corresponding classification of 1+11+11+1 dimensional gapped phase, for example. An ideal decomposition of a fusion ring provides a straightforward but strong constraint on the gapped phase with noninvertible symmetry and its symmetry-breaking (or emergent symmetry) patterns. Moreover, even in several specific homomorphisms connected under massless RG flows, less familiar homomorphisms appear, and we conjecture that they correspond to partially solvable models in recent literature. Our work demonstrates the fundamental significance of the abstract algebraic structure, ideal, for the RG in physics.
We study the classification of the gapped phases or massive renormalization group (RG) flows dual to the massless RG flows under changing the sign of the coupling constants. Whereas our formalism is based on combining Higgs- or Nambu-Goldstone-type arguments with Cardy's smeared boundary conformal field theories (SBCFTs), several puzzling structures arise. More specifically, the established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states, which are unphysical in boundary critical phenomena. Hence, the module of the gapped phases can be outside of that of boundary critical phenomena, whereas one can still calculate characterizing quantities by applying SBCFTs to the models. For example, we demonstrate that the massive RG flow dual to the massless RG flow from the tricritical Ising model to the Ising model, one of the simplest massless RG flows, has this unusual structure. This can be regarded as a quantum field-theoretic analogue of order-disorder coexistence in lattice models. More generally, the resultant gapped phases usually spontaneously break non-group-like symmetry (or noninvertible symmetry). Our work provides systematic quantum field theoretic descriptions of such unusual phases with spontaneous symmetry breaking of non-group-like (or noninvertible) symmetries.
We study fusion rings, or symmetry topological field theories (SymTFTs), which lie outside the non-negative integer matrix representation (NIM-rep), by combining knowledge from generalized symmetry and that from pseudo-Hermitian systems. By applying the Galois shuffle operation to the SymTFTs, we reconstruct fusion rings that correspond to nonlocal CFTs constructed from the corresponding local nonunitary CFTs by applying the similarity transformations. The resultant SymTFTs are outside of NIM-rep, whereas they are ring isomorphic to the NIM-rep of the corresponding local nonunitary CFTs. We study the consequences of this correspondence between the nonlocal unitary model and local nonunitary models. We demonstrate the correspondence between their classifications of massive or massless renormalization group flows and the discrepancies between their boundary or domain wall phenomena. Our work reveals a new connection between ring isomorphism and similarity transformations, providing the fundamental implications of ring-theoretic ideas in the context of symmetry in physics.
A bstract Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models M \mathcal{M} M M ( kq + I , q ) → M \mathcal{M} M M ( kq – I , q ) induced by ϕ (1,2 k +1) . They vastly generalize the previously proposed ones k = I = 1 by Zamolodchikov, k = 1, I > 1 by Ahn and Lässig, and k = 2 by Dorey et al. All the other ℤ 2 preserving renormalization group flows sporadically known in the literature (e.g. M \mathcal{M} M M (10, 3) → M \mathcal{M} M M (8, 3) studied by Klebanov et al) fall into our proposal (e.g. k = 3, I = 1). We claim our new flows give a complete understanding of the renormalization group flows between Virasoro minimal models that preserve a modular tensor category with the SU(2) q− 2 fusion ring.
Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models M(kq+I,q)→M(kq−I,q)\mathcal{M}(kq + I, q) \to\mathcal{M}(kq-I, q)M(kq+I,q)→M(kq−I,q) induced by ϕ(1,2k+1)ϕ_{(1,2k+1)}ϕ(1,2k+1)​. They vastly generalize the previously proposed ones k=I=1k=I=1k=I=1 by Zamolodchikov, k=1,I>1k=1, I>1k=1,I>1 by Ahn and Lässig, and k=2k=2k=2 by Dorey et al. All the other Z2\mathbb{Z}_2Z2​ preserving renormalization group flows sporadically known in the literature (e.g. M(10,3)→M(8,3)\mathcal{M}(10,3) \to \mathcal{M}(8,3)M(10,3)→M(8,3) studied by Klebanov et al) fall into our proposal (e.g. k=3,I=1k=3, I=1k=3,I=1). We claim our new flows give a complete understanding of the renormalization group flows between Virasoro minimal models that preserve a modular tensor category with the SU(2)q−2SU(2)_{q-2}SU(2)q−2​ fusion ring.