Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2512.02139

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

Yoshiki Fukusumi, Taishi Kawamoto

hep-thcond-mat.stat-mechcond-mat.str-elmath-phquant-phSubmitted Dec 1, 2025• Announced Dec 3, 2025• Revised Dec 20, 2025 (v2)
v1 · Dec 1, 2025v2 · Dec 20, 2025v3 · Feb 14, 2026
arXivHTMLPDF

Abstract

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.

Comments
11 pages, 1 figures (v1). References on chiral fermion and explanations on nonlocal effects of the boundary conditions are added(v2)

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.

arXiv:2512.21687•Dec 25, 2025•0 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.

hep-thCross-listedarXiv:2511.11059•Nov 17, 2025

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

Yoshiki Fukusumi, Yuma Furuta

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.

arXiv:2506.23155•Physical review. B./Physical review. B•Mar 4, 2026•1 citation

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.

arXiv:2608.08428•Aug 9, 2026•0 citations

An operator algebraic approach to fusion category symmetry on the lattice

David Evans, Corey Jones

We propose a framework for fusion category symmetry on the (1+1)D lattice in the infinite-volume limit by giving a formal interpretation of SymTFT decompositions. Our approach is based on axiomatizing physical boundary subalgebra of quasi-local observables, and applying ideas from algebraic quantum field theory to derive the expected categorical structures. We show that given a physical boundary subalgebra $B$ of a quasi-local algebra $A$, there is a canonical fusion category $\mathcal{C}$ that acts on $A$ by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that $B$ is recovered as the fixed point operators. We show that a fusion category can be realized as symmetries on a tensor product quasi-local algebra if and only if all of its objects have integer dimensions, and that it admits an ``on-site" action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove two anomaly enforced gaplessness theorems, one for internal categorical symmetries and one for anomalous duality channels. Using the first, we show that for any fusion category $\mathcal{C}$ with no fiber functor there always exist gapless pure symmetric states on an anyon chain.

arXiv:2507.05185•Jul 7, 2025•0 citations

S-matrix bootstrap and non-invertible symmetries

Christian Copetti, Lucía Córdova, Shota Komatsu

A bstract We initiate the S-matrix bootstrap analysis of theories with non-invertible symmetries in (1+1) dimensions. Our previous work [1] showed that crossing symmetry of S-matrices in such theories is modified, with modification characterized by the fusion category data. By imposing unitarity, symmetry and the modified crossing, we constrain the space of consistent S-matrices, identifying integrable theories with non-invertible symmetries at the cusps of allowed regions. We also extend the modified crossing rules to cases where vacua transform in non-regular representations of fusion category, utilizing a connection to a dual category $$ {\mathcal{C}}_{\mathcal{M}}^{\ast } $$ C M ∗ and Symmetry Topological Field Theory (SymTFT). This highlights the utility of SymTFT in the analysis of scattering amplitudes.

arXiv:2408.13132•Journal of High Energy Physics•Mar 28, 2025•20 citations
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 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 C\mathbb{C}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 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 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 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 framework for fusion category symmetry on the (1+1)D lattice in the infinite-volume limit by giving a formal interpretation of SymTFT decompositions. Our approach is based on axiomatizing physical boundary subalgebra of quasi-local observables, and applying ideas from algebraic quantum field theory to derive the expected categorical structures. We show that given a physical boundary subalgebra BBB of a quasi-local algebra AAA, there is a canonical fusion category C\mathcal{C}C that acts on AAA by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that BBB is recovered as the fixed point operators. We show that a fusion category can be realized as symmetries on a tensor product quasi-local algebra if and only if all of its objects have integer dimensions, and that it admits an ``on-site" action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove two anomaly enforced gaplessness theorems, one for internal categorical symmetries and one for anomalous duality channels. Using the first, we show that for any fusion category C\mathcal{C}C with no fiber functor there always exist gapless pure symmetric states on an anyon chain.
A bstract We initiate the S-matrix bootstrap analysis of theories with non-invertible symmetries in (1+1) dimensions. Our previous work [1] showed that crossing symmetry of S-matrices in such theories is modified, with modification characterized by the fusion category data. By imposing unitarity, symmetry and the modified crossing, we constrain the space of consistent S-matrices, identifying integrable theories with non-invertible symmetries at the cusps of allowed regions. We also extend the modified crossing rules to cases where vacua transform in non-regular representations of fusion category, utilizing a connection to a dual category CM∗ {\mathcal{C}}_{\mathcal{M}}^{\ast } CM∗​ C M ∗ and Symmetry Topological Field Theory (SymTFT). This highlights the utility of SymTFT in the analysis of scattering amplitudes.