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arxiv: 2604.20201 · v1 · submitted 2026-04-22 · ❄️ cond-mat.str-el · hep-th

Symmetry breaking phases and transitions in an Ising fusion category lattice model

Soumil Roychowdhury , Chenjie Wang This is my paper

Pith reviewed 2026-05-09 23:30 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-th
keywords Ising fusion categorylattice modelcategorical symmetry breakingphase diagramnon-invertible symmetryconformal field theorycritical antiferromagnet
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The pith

A lattice model with Ising fusion category symmetry hosts a symmetric Ising critical phase, a categorical ferromagnetic phase with threefold degeneracy, and a critical categorical antiferromagnetic phase.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines an anyon-chain-like lattice model whose symmetries are given by the Ising fusion category. It identifies three phases through numerical and analytical work: a symmetric critical phase that matches the ordinary Ising universality class, a categorical ferromagnetic phase where the full category is broken, and a categorical antiferromagnetic phase that breaks lattice translation together with part of the category while remaining critical. This matters because non-invertible symmetries produce breaking patterns that differ from group symmetries, including antiferromagnetic states whose low-energy manifold grows exponentially with system size due to domain walls of quantum dimension greater than one. The transitions are also mapped: the symmetric-to-ferromagnetic transition follows the tricritical Ising theory with central charge 7/10, while the symmetric-to-antiferromagnetic transition appears continuous with central charge 3/2.

Core claim

In an anyon-chain-like lattice model with symmetry described by the Ising fusion category, numerical and analytical studies reveal a phase diagram with a symmetric critical phase in the usual Ising universality class, a categorical ferromagnetic phase in which the Ising fusion category is fully broken and the ground state is threefold degenerate, and a categorical antiferromagnetic phase that breaks lattice translation symmetry and part of the category while being described by a fourfold degenerate Ising conformal field theory. The transition between the symmetric and categorical ferromagnetic phases is governed by the c=7/10 tricritical Ising CFT, and numerical data indicate that the latter

What carries the argument

The Ising fusion category, which supplies the non-invertible symmetry and whose domain walls carry quantum dimension greater than one, thereby enlarging the low-energy manifold in the antiferromagnetic phase.

If this is right

  • Antiferromagnetic states associated with broken non-invertible symmetries remain critical because domain walls have quantum dimension larger than one.
  • The generalized Landau paradigm correctly predicts the threefold degeneracy of the categorical ferromagnetic phase.
  • The symmetric phase belongs to the Ising universality class even though the microscopic symmetry is non-invertible.
  • Transitions out of the symmetric phase can be captured by known conformal field theories such as the tricritical Ising model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar lattice constructions for other fusion categories may also produce critical antiferromagnetic phases with exponentially large low-energy manifolds.
  • The numerical identification of c=3/2 at one transition invites analytic constructions that match this central charge to known theories.
  • The exponential growth of the low-energy manifold in the categorical antiferromagnetic phase suggests that entanglement entropy will scale differently from conventional gapped antiferromagnets.

Load-bearing premise

The finite-size numerical data correctly identify the central charges, ground-state degeneracies, and phase boundaries in the infinite-volume limit without substantial finite-size effects.

What would settle it

Exact diagonalization or tensor-network calculations on chains of length 30 or larger that yield a central charge clearly different from 3/2 at the symmetric-to-categorical-antiferromagnetic transition or that show a first-order rather than continuous transition.

Figures

Figures reproduced from arXiv: 2604.20201 by Chenjie Wang, Soumil Roychowdhury.

Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagram of the Ising fusion category lattice [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Central charge [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Low energy spectra obtained via exact diagonalization in the different phases. All calculations were performed for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A rather unique behavior is observed: Instead of a single smooth curve, the entanglement entropy S[x] 7 To be specific, the boundary conditions are set as x1 = xL = σ. It is equivalent to set H1|x1⟩ = Uδx1,σ|x1⟩ and HL|xL⟩ = UδxL,σ|xL⟩ in the Hamiltonian (12), with U → ∞. It corre￾sponds to the free boundary conditions for the effective Ising chain HIsing [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Loop-TNR results for the CatFM-symmetric transi [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Entanglement entropy of the critical Ising ground state (obtained from DMRG) before (blue) and after (orange) [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

An anyon-chain-like lattice model with symmetry described by the Ising fusion category is studied. Combining numerical and analytical studies, we uncover a rich phase diagram that contains three phases: a symmetric critical phase and two categorical symmetry breaking phases. The symmetric phase lies in the same universality class as the usual critical Ising model. The first symmetry-breaking phase, dubbed the \emph{categorical ferromagnetic} phase, has the Ising fusion category fully broken and exhibits a threefold ground-state degeneracy, as expected from the generalized Landau paradigm. The other symmetry-breaking phase is analogous to a conventional antiferromagnet: it breaks lattice translation and part of the Ising fusion category, and therefore is termed the \emph{categorical antiferromagnetic} phase. Unlike ordinary antiferromagnetic states associated with finite invertible symmetry breaking, this phase itself is critical, being described by a fourfold degenerate Ising conformal field theory. We argue more generally that antiferromagnetic states associated with broken non-invertible symmetries have a large low-energy manifold that grows exponentially in system size, due to the greater-than-one quantum dimension of domain walls. We also numerically study the transitions between the three phases. The transition between the symmetric and categorical ferromagnetic phase is described by the $c=7/10$ tricritical Ising CFT, while the transition between the symmetric and categorical antiferromagnetic phases is less understood. Our numerical data suggest that the latter transition is continuous and described by a conformal field theory with central charge $c=3/2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript studies an anyon-chain-like lattice model whose symmetry is described by the Ising fusion category. Combining numerical and analytical methods, the authors map a phase diagram containing a symmetric critical phase in the Ising universality class, a categorical ferromagnetic phase with threefold ground-state degeneracy arising from complete symmetry breaking, and a categorical antiferromagnetic phase that breaks lattice translation together with part of the fusion category. The latter phase is itself critical and described by a fourfold-degenerate Ising CFT. The authors further argue that antiferromagnetic states associated with non-invertible symmetry breaking generically possess an exponentially large low-energy manifold because domain walls carry quantum dimension greater than one. The symmetric-to-categorical-ferromagnetic transition is identified with the tricritical Ising CFT (c=7/10), while the symmetric-to-categorical-antiferromagnetic transition is reported as continuous with central charge c=3/2 on the basis of numerical data.

Significance. If the central claims are substantiated, the work supplies a concrete lattice realization of generalized symmetry breaking for a non-invertible fusion category and illustrates how the greater-than-one quantum dimension of domain walls forces antiferromagnetic phases to remain critical. The general argument concerning the exponential growth of the low-energy manifold is a useful conceptual contribution. The identification of the tricritical Ising point provides a controlled benchmark, while the c=3/2 transition, if confirmed, would furnish an example of a novel universality class arising from non-invertible symmetry breaking.

major comments (1)
  1. [Abstract and numerical results on the symmetric-to-categorical-antiferromagnetic transition] The claim that the symmetric-to-categorical-antiferromagnetic transition is continuous and belongs to a c=3/2 CFT rests on the statement that 'our numerical data suggest' this identification. No system sizes, entanglement-entropy scaling fits, level-spectroscopy data, or controlled extrapolation procedure are supplied in the abstract or the summary of numerical results. Because the categorical antiferromagnetic phase is itself asserted to be critical with fourfold degeneracy, finite-size effects or crossover could produce an apparent central charge near 3/2 without the transition actually being described by a single c=3/2 CFT. This identification is load-bearing for the reported phase diagram and the assertion that the transition is continuous.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for identifying the need for clearer documentation of the numerical evidence supporting the symmetric-to-categorical-antiferromagnetic transition. We address this point directly below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and numerical results on the symmetric-to-categorical-antiferromagnetic transition] The claim that the symmetric-to-categorical-antiferromagnetic transition is continuous and belongs to a c=3/2 CFT rests on the statement that 'our numerical data suggest' this identification. No system sizes, entanglement-entropy scaling fits, level-spectroscopy data, or controlled extrapolation procedure are supplied in the abstract or the summary of numerical results. Because the categorical antiferromagnetic phase is itself asserted to be critical with fourfold degeneracy, finite-size effects or crossover could produce an apparent central charge near 3/2 without the transition actually being described by a single c=3/2 CFT. This identification is load-bearing for the reported phase diagram and the assertion that the transition is continuous.

    Authors: We agree that the abstract and the high-level summary of numerical results are too terse and do not adequately document the supporting data or address possible finite-size artifacts arising from the critical nature of the categorical antiferromagnetic phase. The detailed numerical analysis (system sizes, entanglement-entropy scaling, level spectroscopy, and extrapolation) appears in the main text, but we acknowledge that this is insufficient for a load-bearing claim. We will revise the abstract to include a concise statement of the largest system sizes employed and the extrapolated central charge. We will also expand the summary of numerical results into a dedicated paragraph that (i) reports the entanglement-entropy scaling fits and their stability under extrapolation, (ii) summarizes the level-spectroscopy results confirming the expected fourfold degeneracy without level crossings indicative of a first-order transition, and (iii) explicitly discusses why crossover effects from the adjacent critical phase do not dominate the scaling window. These additions will make the evidence for continuity and the c=3/2 identification transparent while preserving the original interpretation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper identifies phases and transitions via a combination of direct numerical measurements (central charges, degeneracies, entanglement scaling) and analytical arguments grounded in standard fusion-category properties and known CFT universality classes (Ising, tricritical Ising). No derivation step reduces by construction to its own inputs: the threefold degeneracy follows from the generalized Landau paradigm applied to the Ising category, the fourfold degeneracy of the categorical antiferromagnet is tied to the quantum dimension of domain walls (a pre-existing anyon property), and the c=3/2 suggestion is explicitly labeled as a numerical observation rather than a fitted or self-defined prediction. No load-bearing self-citations or ansatz smuggling appear in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the model implements the fusion category symmetry correctly and that the numerical simulations converge to the thermodynamic limit.

free parameters (1)
  • Hamiltonian coupling constants
    Parameters in the lattice model Hamiltonian are chosen to realize the phases but not specified in the abstract.
axioms (1)
  • domain assumption The lattice model realizes the Ising fusion category symmetry
    This is the foundational assumption for constructing the anyon-chain-like model.

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Reference graph

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