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arxiv: 2309.16749 · v3 · submitted 2023-09-28 · ✦ hep-th · cond-mat.str-el· math-ph· math.AT· math.MP

A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases

Arun Debray , Sanath K. Devalapurkar , Cameron Krulewski , Yu Leon Liu , Natalia Pacheco-Tallaj , Ryan Thorngren This is my paper

Pith reviewed 2026-05-24 06:26 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elmath-phmath.ATmath.MP
keywords symmetry breakingdefectsanomaly matchinghigher Berry phasesinvertible field theoriessymmetry protected topological phaseslong exact sequencedomain walls
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The pith

The symmetry breaking long exact sequence classifies defect anomalies in symmetry breaking phases and matches them to the anomaly of the broken symmetry.

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

The paper introduces the symmetry breaking long exact sequence as a tool to study defects such as domain walls, vortices, and hedgehogs in symmetry breaking phases. It shows that gapless excitations at defect cores carry 't Hooft anomalies that classify the phases and obey an anomaly-matching relation with the anomaly of the original unbroken symmetry. The sequence also yields obstructions to the existence of symmetry breaking phases that support local defects. These results rely on groups of invertible field theories and extend the bulk-boundary correspondence for higher Berry phases, offering a method to classify symmetry protected topological phases.

Core claim

The symmetry breaking long exact sequence (SBLES) is a long exact sequence of groups of invertible field theories that encodes how the anomaly of a broken symmetry is matched by the anomalies of defects in the resulting phase. This sequence classifies symmetry breaking phases according to the anomalies of their localized gapless modes and derives obstructions to symmetry breaking phases containing local defects. The construction further develops the theory of higher Berry phases.

What carries the argument

The symmetry breaking long exact sequence (SBLES), a long exact sequence of groups of invertible field theories that relates the anomaly of a broken symmetry to the anomalies of its defects.

If this is right

  • Symmetry breaking phases are classified by the 't Hooft anomalies of their defects.
  • An anomaly-matching formula relates defect anomalies directly to the anomaly of the broken symmetry.
  • Obstructions exist to the existence of symmetry breaking phases that admit local defects.
  • The framework supplies a computational tool for classifying symmetry protected topological phases.
  • The bulk-boundary correspondence for higher Berry phases is extended by the sequence.

Where Pith is reading between the lines

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

  • The SBLES may enable explicit calculations of higher Berry phases for additional symmetry groups beyond those treated in the paper.
  • Similar long exact sequences could classify defects under non-invertible or higher-form symmetries.
  • The obstruction results may constrain dynamical symmetry breaking scenarios in quantum field theory models.
  • The classification could be tested in lattice models of condensed matter systems with known defect spectra.

Load-bearing premise

The mathematical exactness of the symmetry breaking long exact sequence in the groups of invertible field theories, as provided by the companion paper.

What would settle it

An explicit computation of the anomaly of a vortex or domain wall in a symmetry breaking phase whose value fails to match the anomaly of the broken symmetry according to the derived formula.

Figures

Figures reproduced from arXiv: 2309.16749 by Arun Debray, Cameron Krulewski, Natalia Pacheco-Tallaj, Ryan Thorngren, Sanath K. Devalapurkar, Yu Leon Liu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a) The phase diagram of a single 2+1D [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) The phase diagram of 3+1D [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an 't Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the "symmetry breaking long exact sequence" (SBLES). The mathematical backbone of the SBLES is studied in a companion paper. Our work further develops the theory of higher Berry phase and its bulk-boundary correspondence, and serves as a new computational tool for classifying symmetry protected topological phases.

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

2 major / 0 minor

Summary. The manuscript claims to classify different symmetry breaking phases in terms of the anomalies of defects (domain walls, vortices, hedgehogs) and to relate these to the anomaly of the broken symmetry via an anomaly-matching formula. It also derives an obstruction to the existence of a symmetry breaking phase with a local defect. These results are obtained by applying a symmetry breaking long exact sequence (SBLES) of groups of invertible field theories; the mathematical construction and properties of the SBLES are deferred to a companion paper. The work further develops the theory of higher Berry phases and bulk-boundary correspondence as a tool for classifying symmetry protected topological phases.

Significance. If the SBLES exists, is natural with respect to the relevant symmetry groups, and yields the stated anomaly-matching without hidden obstructions, the framework would supply a new algebraic tool for relating defect anomalies to bulk symmetry anomalies and for obstructing certain symmetry-breaking configurations, with potential utility in SPT classification.

major comments (2)
  1. [Abstract] Abstract and main text: The anomaly-matching formula, the classification of symmetry-breaking phases by defect anomalies, and the obstruction to local defects are all obtained by applying the SBLES, yet the existence, exactness, and naturality of this long exact sequence are not derived or verified in the manuscript and are instead referred entirely to a companion paper; this renders the central claims conditional on unexamined results.
  2. [Main text] Main text (applications to defects): Without an explicit statement or check of how the SBLES maps the anomaly of the broken symmetry to the defect anomaly (or produces the obstruction), it is not possible to confirm that the matching is non-trivial rather than tautological with respect to the sequence construction itself.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below. The SBLES construction is indeed in the companion paper, but the present work applies it to new physical questions; we will revise to improve self-containedness of the applications.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main text: The anomaly-matching formula, the classification of symmetry-breaking phases by defect anomalies, and the obstruction to local defects are all obtained by applying the SBLES, yet the existence, exactness, and naturality of this long exact sequence are not derived or verified in the manuscript and are instead referred entirely to a companion paper; this renders the central claims conditional on unexamined results.

    Authors: We agree that the existence, exactness, and naturality of the SBLES are established in the companion paper rather than here. This manuscript applies the sequence to classify defect anomalies and derive obstructions in concrete symmetry-breaking settings. In revision we will insert a concise summary of the SBLES properties actually used (exactness at the relevant terms, naturality with respect to symmetry homomorphisms) together with the explicit connecting maps that implement the anomaly-matching, so that the logical steps in the applications can be followed without first reading the companion. revision: partial

  2. Referee: [Main text] Main text (applications to defects): Without an explicit statement or check of how the SBLES maps the anomaly of the broken symmetry to the defect anomaly (or produces the obstruction), it is not possible to confirm that the matching is non-trivial rather than tautological with respect to the sequence construction itself.

    Authors: The connecting homomorphism of the SBLES is constructed precisely to send the anomaly of the broken symmetry to the anomaly of the defect; this is not tautological but follows from the definition of the sequence (detailed in the companion). In the applications we already compute concrete instances (domain-wall anomalies for discrete groups, vortex anomalies for U(1), hedgehog anomalies for SO(3)) that recover known results or yield new predictions. We will add an explicit diagram of the relevant segment of the long exact sequence in each case and label the maps, making the non-trivial character of the matching visible directly in the text. revision: partial

Circularity Check

1 steps flagged

Central anomaly-matching and obstruction claims rest on SBLES whose backbone is deferred to companion paper

specific steps
  1. self citation load bearing [Abstract]
    "We obtain these results using a long exact sequence of groups of invertible field theories, which we call the 'symmetry breaking long exact sequence' (SBLES). The mathematical backbone of the SBLES is studied in a companion paper."

    The anomaly-matching formula, classification of phases, and obstruction to symmetry-breaking phases with local defects are all derived by applying the SBLES; yet the existence and relevant properties of the SBLES itself are not established in this paper but deferred to the companion, so the central results presuppose the companion's construction without independent derivation or external benchmark here.

full rationale

The paper's strongest claims (classification of symmetry-breaking phases via defect anomalies, anomaly-matching formula relating defect anomalies to the broken symmetry, and obstruction to local defects) are explicitly obtained by applying the SBLES. The abstract states that the mathematical backbone of this sequence is studied in a companion paper, so the derivations here reduce to the existence, naturality, and anomaly-matching properties established elsewhere. This matches the self-citation-load-bearing pattern because the load-bearing exact sequence is not re-derived or independently verified within the present manuscript. No other circular steps are identifiable from the provided text, and the paper does not claim the SBLES is constructed here. The result is therefore partially circular (score 6) rather than fully self-contained or definitionally tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; all such elements are deferred to the companion paper on the SBLES mathematical backbone.

pith-pipeline@v0.9.0 · 5723 in / 1071 out tokens · 17659 ms · 2026-05-24T06:26:58.954136+00:00 · methodology

discussion (0)

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Reference graph

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