arXiv:2506.23155 · Physical review. B./Physical review. B

Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders

Yoshiki Fukusumi, Yuma Furuta

Published Mar 4, 2026• 1 citation

Abstract

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 $2+1$ dimensional topologically ordered systems and the corresponding classification of $1+1$ 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.

DOI: 10.1103/1t7d-sjcq

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Classifying fusion rules of anyons or SymTFTs: A general algebraic formula for domain wall problems and quantum phase transitions

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

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Lattice models for phases and transitions with non-invertible symmetries

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Generalized Symmetry as a Diagnostic Lens: How Non-Invertible Symmetries, Condensable Algebras, and Order-Parameter Structure Reorganize the Classification of Gapped Phases and Phase Transitions

Saluca Agentic AI Research Team

Jun 3, 20260 citations

arXiv:2506.23155 · Physical review. B./Physical review. B

Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders

Yoshiki Fukusumi, Yuma Furuta

Published Mar 4, 2026• 1 citation

arXiv HTML PDF

Abstract

2+1

dimensional topologically ordered systems and the corresponding classification of

1+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.

DOI: 10.1103/1t7d-sjcq

Related papers

Classifying fusion rules of anyons or SymTFTs: A general algebraic formula for domain wall problems and quantum phase transitions

Yoshiki Fukusumi

We propose a formula for the transformation law of anyons in topologically ordered phases or topological quantum field theories (TQFTs) through a gapped or symmetry-preserving domain wall. Our formalism is based on the ring homomorphism between the $\mathbb{C}$ -linear commutative fusion rings, also known as symmetry topological field theories (SymTFTs). The fundamental assumption in our formalism is the validity of the Verlinde formula, applicable to commutative fusion rings. By combining it with more specific data of the settings, our formula provides classifications of anyons compatible with developing categorical formulations. It also provides the massless renormalization group (RG) flows between conformal field theories (CFTs), or a series of measurement-induced quantum phase transitions, in the language of SymTFT, through the established correspondence between CFTs and TQFTs. Moreover, by studying the correspondence between the ideal structure in the massless RG and the module in the related massive RG, one can make the Nambu-Goldstone-type arguments for generalized symmetry. By combining our formula with orbifolding, extension, and similarity transformation, one can get a series of classifications for the corresponding extended models, or symmetry-enriched topological orders and quantum criticalities.We propose a formula for the transformation law of anyons in topologically ordered phases or topological quantum field theories (TQFTs) through a gapped or symmetry-preserving domain wall. Our formalism is based on the ring homomorphism between the $\mathbb{C}$-linear commutative fusion rings, also known as symmetry topological field theories (SymTFTs). The fundamental assumption in our formalism is the validity of the Verlinde formula, applicable to commutative fusion rings. By combining it with more specific data of the settings, our formula provides classifications of anyons compatible with developing categorical formulations. It also provides the massless renormalization group (RG) flows between conformal field theories (CFTs), or a series of measurement-induced quantum phase transitions, in the language of SymTFT, through the established correspondence between CFTs and TQFTs. Moreover, by studying the correspondence between the ideal structure in the massless RG and the module in the related massive RG, one can make the Nambu-Goldstone-type arguments for generalized symmetry. By combining our formula with orbifolding, extension, and similarity transformation, one can get a series of classifications for the corresponding extended models, or symmetry-enriched topological orders and quantum criticalities.

arXiv:2512.21687Dec 25, 20250 citations

Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls

Yoshiki Fukusumi, Taishi Kawamoto

We study conformal field theories (CFTs) and their classifications from a modern perspective based on the abstract algebraic formalism of symmetries or conserved charges, known as symmetry topological field theories (SymTFTs). By studying the algebraic structure of the SymTFTs in detail, we found a natural generalization of the quantum dimensions associated with (pseudo-)Hermitian systems and (non)-unitary CFTs. These generalized data of SymTFTs provide classifications of massless and massive renormalization group flows, which will describe the quantum phase transitions of the corresponding pseudo-Hermitian systems. Moreover, our discussions straightforwardly enable one to relate a general class of coset constructions or level-rank dualities to domain wall problems between topological quantum field theories (or a series of corresponding quantum phase transitions related to the Higgs mechanism). Our work provides a systematic reduction and classification of algebraic data, symmetries, for pseudo-Hermitian systems based on ideas from established mathematical fields, linear algebra and ring theory.We study conformal field theories (CFTs) and their classifications from a modern perspective based on the abstract algebraic formalism of symmetries or conserved charges, known as symmetry topological field theories (SymTFTs). By studying the algebraic structure of the SymTFTs in detail, we found a natural generalization of the quantum dimensions associated with (pseudo-)Hermitian systems and (non)-unitary CFTs. These generalized data of SymTFTs provide classifications of massless and massive renormalization group flows, which will describe the quantum phase transitions of the corresponding pseudo-Hermitian systems. Moreover, our discussions straightforwardly enable one to relate a general class of coset constructions or level-rank dualities to domain wall problems between topological quantum field theories (or a series of corresponding quantum phase transitions related to the Higgs mechanism). Our work provides a systematic reduction and classification of algebraic data, symmetries, for pseudo-Hermitian systems based on ideas from established mathematical fields, linear algebra and ring theory.

hep-thCross-listedarXiv:2511.11059Nov 17, 2025

Generalizing fusion rules by shuffle: Symmetry-based classifications of nonlocal systems constructed from similarity transformations

Yoshiki Fukusumi, Taishi Kawamoto

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.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.

hep-thCross-listedarXiv:2512.02139Dec 3, 2025

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

Yoshiki Fukusumi, Yuma FURUTA

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.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.

arXiv:2608.08428Aug 9, 20260 citations

Lattice models for phases and transitions with non-invertible symmetries

Lakshya Bhardwaj, Lea E. Bottini, Sakura Schäfer‐Nameki +1 more

Non-invertible categorical symmetries have emerged as a powerful tool to uncover new beyond-Landau phases of matter, both gapped and gapless, along with second order phase transitions between them. The general theory of such phases in (1+1)d has been studied using the Symmetry Topological Field Theory (SymTFT), also known as topological holography. This has unearthed the infrared (IR) structure of these phases and transitions. In this paper, we describe how the SymTFT information can be converted into an ultraviolet (UV) anyonic chain lattice model realizing, in the IR limit, these phases and transitions. In many cases, the Hilbert space of the anyonic chain is tensor product decomposable and the model can be realized as a quantum spin-chain Hamiltonian. We also describe operators acting on the lattice models that are charged under non-invertible symmetries and act as order parameters for the phases and transitions. In order to fully describe the action of non-invertible symmetries, it is crucial to understand the symmetry twisted sectors of the lattice models, which we describe in detail. Throughout the paper, we illustrate the general concepts using the symmetry category \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) formed by representations of the permutation group S_3 S 3 , but our procedure can be applied to any fusion category symmetry.Non-invertible categorical symmetries have emerged as a powerful tool to uncover new beyond-Landau phases of matter, both gapped and gapless, along with second order phase transitions between them. The general theory of such phases in (1+1)d has been studied using the Symmetry Topological Field Theory (SymTFT), also known as topological holography. This has unearthed the infrared (IR) structure of these phases and transitions. In this paper, we describe how the SymTFT information can be converted into an ultraviolet (UV) anyonic chain lattice model realizing, in the IR limit, these phases and transitions. In many cases, the Hilbert space of the anyonic chain is tensor product decomposable and the model can be realized as a quantum spin-chain Hamiltonian. We also describe operators acting on the lattice models that are charged under non-invertible symmetries and act as order parameters for the phases and transitions. In order to fully describe the action of non-invertible symmetries, it is crucial to understand the symmetry twisted sectors of the lattice models, which we describe in detail. Throughout the paper, we illustrate the general concepts using the symmetry category \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) formed by representations of the permutation group S_3 S 3 , but our procedure can be applied to any fusion category symmetry.

arXiv:2405.05964SciPost PhysicsMay 13, 202624 citations

Generalized Symmetry as a Diagnostic Lens: How Non-Invertible Symmetries, Condensable Algebras, and Order-Parameter Structure Reorganize the Classification of Gapped Phases and Phase Transitions

Saluca Agentic AI Research Team

A candidate reading of seven recent preprints from cond-mat.str-el and physics.atom-ph suggests that the classical toolkit for classifying gapped phases and their transitions—built on Landau symmetry breaking, local order parameters, and invertible group symmetries—requires systematic extension at the level of microscopic algebraic structure, not merely at the phenomenological level. The emerging picture, drawn from the corpus, is that *generalized symmetries* (non-invertible, categorical, and higher-form) impose precise algebraic constraints on what constitutes a valid order parameter, what boundary conditions are permissible, and which phase transitions can occur without any form of spontaneous symmetry breaking, even after arbitrary gauging. Concretely: (i) condensable algebras in 2+1d topological orders classify gapped boundaries, and twin algebras sharing identical anyon content but inequivalent algebra structure describe physically distinct phases with isomorphic ground-state degeneracy but inequivalent order parameters [corpus:arxiv:2605.31602]; (ii) these twin phases can undergo direct transitions that are provably beyond-Landau even after partial gauging [corpus:arxiv:2605.31601]; (iii) string order parameters in 1d systems organize into Lagrangian algebras in the Drinfel'd center of the symmetry category, making the fusion rule of the symmetry category—not just its representation theory—the operative diagnostic [corpus:arxiv:2605.27193]; (iv) non-invertible symmetry-breaking transitions can be mapped via generalized gauging to deconfined quantum critical points, and vice versa, establishing a precise duality [corpus:arxiv:2605.27672]; and (v) non-invertible symmetry enriched topological orders (NI-SETOs) can be constructed from string-net models, extending symmetry enrichment beyond the group-theoretic setting [corpus:arxiv:2605.28794]. Two additional corpus entries—a Rydberg spin-chain study [corpus:arxiv:2605.27166] and a topological lattice gauge theory paper [corpus:arxiv:2605.28688]—provide microscopic anchoring and structural context, respectively. The central falsification path is explicit: if a twin-phase transition can be shown to spontaneously break some hidden symmetry after arbitrary gauging, the framework collapses. Concrete tensor-network (DMRG/MPS) constructions of twin-phase order parameters in 1+1d would constitute a direct test. --- 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, 2605.31602A candidate reading of seven recent preprints from cond-mat.str-el and physics.atom-ph suggests that the classical toolkit for classifying gapped phases and their transitions—built on Landau symmetry breaking, local order parameters, and invertible group symmetries—requires systematic extension at the level of microscopic algebraic structure, not merely at the phenomenological level. The emerging picture, drawn from the corpus, is that *generalized symmetries* (non-invertible, categorical, and higher-form) impose precise algebraic constraints on what constitutes a valid order parameter, what boundary conditions are permissible, and which phase transitions can occur without any form of spontaneous symmetry breaking, even after arbitrary gauging. Concretely: (i) condensable algebras in 2+1d topological orders classify gapped boundaries, and twin algebras sharing identical anyon content but inequivalent algebra structure describe physically distinct phases with isomorphic ground-state degeneracy but inequivalent order parameters [corpus:arxiv:2605.31602]; (ii) these twin phases can undergo direct transitions that are provably beyond-Landau even after partial gauging [corpus:arxiv:2605.31601]; (iii) string order parameters in 1d systems organize into Lagrangian algebras in the Drinfel'd center of the symmetry category, making the fusion rule of the symmetry category—not just its representation theory—the operative diagnostic [corpus:arxiv:2605.27193]; (iv) non-invertible symmetry-breaking transitions can be mapped via generalized gauging to deconfined quantum critical points, and vice versa, establishing a precise duality [corpus:arxiv:2605.27672]; and (v) non-invertible symmetry enriched topological orders (NI-SETOs) can be constructed from string-net models, extending symmetry enrichment beyond the group-theoretic setting [corpus:arxiv:2605.28794]. Two additional corpus entries—a Rydberg spin-chain study [corpus:arxiv:2605.27166] and a topological lattice gauge theory paper [corpus:arxiv:2605.28688]—provide microscopic anchoring and structural context, respectively. The central falsification path is explicit: if a twin-phase transition can be shown to spontaneously break some hidden symmetry after arbitrary gauging, the framework collapses. Concrete tensor-network (DMRG/MPS) constructions of twin-phase order parameters in 1+1d would constitute a direct test. --- 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

Jun 3, 20260 citations

2+1

dimensional topologically ordered systems and the corresponding classification of

1+1

\mathbb{C}

-linear commutative fusion rings, also known as symmetry topological field theories (SymTFTs). The fundamental assumption in our formalism is the validity of the Verlinde formula, applicable to commutative fusion rings. By combining it with more specific data of the settings, our formula provides classifications of anyons compatible with developing categorical formulations. It also provides the massless renormalization group (RG) flows between conformal field theories (CFTs), or a series of measurement-induced quantum phase transitions, in the language of SymTFT, through the established correspondence between CFTs and TQFTs. Moreover, by studying the correspondence between the ideal structure in the massless RG and the module in the related massive RG, one can make the Nambu-Goldstone-type arguments for generalized symmetry. By combining our formula with orbifolding, extension, and similarity transformation, one can get a series of classifications for the corresponding extended models, or symmetry-enriched topological orders and quantum criticalities.

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.