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arXiv:2501.07511 · Physical Review Letters

Minimal Model Renormalization Group Flows: Noninvertible Symmetries and Nonperturbative Description

Federico Ambrosino, Stefano Negro

Published Jun 18, 2025• 5 citations

Abstract

In this Letter we continue the investigation of RG flows between Virasoro minimal models of two-dimensional conformal field theories that are protected by noninvertible symmetries. RG flows leaving unbroken a subcategory of noninvertible symmetries are associated with anomaly matching conditions that we employ systematically to map the space of flows between minimal models beyond the Z_{2}-symmetric proposed recently in the literature. We introduce a family of nonlinear integral equations that appear to encode the exact finite-size, ground-state energies of these flows, including nonintegrable cases, such as the recently proposed M(kq+I,q)→M(kq-I,q). Our family of NLIEs encompasses and generalizes the integrable flows known in the literature: ϕ_{(1,3)}, ϕ_{(1,5)}, ϕ_{(1,2)} and ϕ_{(2,1)}. This work uncovers a new interplay between exact solvability and noninvertible symmetries. Furthermore, our nonperturbative description provides a nontrivial test for all the flows conjectured by anomaly matching conditions, but so far not observed by other means.

DOI: 10.1103/dg1s-5vp6

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In this Letter we continue the investigation of RG flows between Virasoro minimal models of two-dimensional conformal field theories that are protected by noninvertible symmetries. RG flows leaving unbroken a subcategory of noninvertible symmetries are associated with anomaly matching conditions that we employ systematically to map the space of flows between minimal models beyond the Z_{2}-symmetric proposed recently in the literature. We introduce a family of nonlinear integral equations that appear to encode the exact finite-size, ground-state energies of these flows, including nonintegrable cases, such as the recently proposed M(kq+I,q)→M(kq-I,q). Our family of NLIEs encompasses and generalizes the integrable flows known in the literature: ϕ_{(1,3)}, ϕ_{(1,5)}, ϕ_{(1,2)} and ϕ_{(2,1)}. This work uncovers a new interplay between exact solvability and noninvertible symmetries. Furthermore, our nonperturbative description provides a nontrivial test for all the flows conjectured by anomaly matching conditions, but so far not observed by other means.

A bstract Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models

\mathcal{M}

M ( kq + I , q ) →

\mathcal{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.

\mathcal{M}

M (10, 3) →

\mathcal{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.

RG flows of minimal

\mathcal W

-algebra CFTs via non-invertible symmetries

In this letter we study renormalization group (RG) flows between 2d conformal field theories enjoying extended higher-spin

\mathcal{W}

-symmetry. We propose a new class of RG flows between the diagonal minimal models of

\mathcal{W}_N

-algebra that take the form

\mathcal{W}_N(p,q)\to\mathcal{W}_N(p,kp-q)

. These are obtained by matching the anomalies of the non-invertible symmetry

{\mathrm{Rep}}[SU(N)_{p-N}]

(and its discrete quotients) that is preserved by special relevant primary fields. This large non-invertible symmetry includes the familiar

\mathbb{Z}_N

symmetry of the minimal models. Our new flows furnish a significant generalization of the ones recently found in the case of Virasoro algebra, and include all previously known RG flows of

\mathcal{W}_N

. They have the remarkable property of being uniform in the rank

N

of the

\mathcal{W}

-algebra.

We construct complete solutions of the massless S-matrix bootstrap equations describing a new infinite family of integrable RG flows from the higher-fusion-level minimal models M(3,2n+3;n) to the non-unitary Virasoro minimal models M(2,2n+3), including a flow from M(3,5) to the Yang--Lee CFT M(2,5) when n=1. We provide strong evidence for this identification by matching conformal perturbation theory around the UV fixed points with the small-volume expansion of the corresponding thermodynamic Bethe ansatz equations. We further show that a family of non-invertible Verlinde defect lines is preserved along the flows.

Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models

\mathcal{W}_N

induced by

ϕ_{(1,2k+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

\mathbb{Z}_2

preserving renormalization group flows sporadically known in the literature (e.g.

\mathcal{M}(10,3) \to \mathcal{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.

Conformal interfaces separating two conformal field theories (CFTs) provide maps between different CFTs, and naturally exist in nature as domain walls between different phases. One particularly interesting construction of a conformal interface is the renormalization group (RG) domain wall between CFTs. For a given Virasoro minimal model \mathcal{M}_{k+3,k+2} ℳ k + 3 , k + 2 , an RG domain wall can be generated by a specific deformation which triggers an RG flow towards its adjacent Virasoro minimal model \mathcal{M}_{k+2,k+1} ℳ k + 2 , k + 1 with the deformation turned on over part of the space. An algebraic construction of this domain wall was proposed by Gaiotto in [J. High Energy Phys. 12, 103 (2012)]. In this paper, we will provide a study of this RG domain wall for the minimal case k=2 k = 2 , which can be thought of as a nonperturbative check of the construction. In this case the wall is separating the Tricritical Ising Model (TIM) CFT and the Ising Model (IM) CFT. We will check the analytical results of correlation functions from the RG brane construction with the numerical density matrix renormalization group (DMRG) calculation using a lattice model proposed in [arXiv.1206.1332, Science 344, 280 (2014)], and find a perfect agreement. We comment on possible experimental realizations of this RG domain wall.