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arxiv: 2506.23258 · v4 · pith:VYDAGRQCnew · submitted 2025-06-29 · ❄️ cond-mat.str-el · cond-mat.stat-mech

Spontaneous continuous-symmetry breaking and tower of states in a comb chain

Jingya Wang , Zenan Liu , Bin-Bin Mao , Xu Tian , Zijian Xiong , Zhe Wang , Zheng Yan This is my paper

Pith reviewed 2026-05-21 23:35 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mech
keywords comb latticeHeisenberg antiferromagnetspontaneous symmetry breakingtower of statesone-dimensional systemsferrimagnetismshort-range interactions
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The pith

A one-dimensional comb lattice antiferromagnet breaks continuous spin symmetry spontaneously.

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

This paper examines an antiferromagnetic Heisenberg model on a comb-shaped one-dimensional lattice. Introducing a symmetry-preserving perturbation allows the system to satisfy the conditions of the Marshall-Lieb-Mattis theorem for bipartite lattices with unequal sublattice sizes. The Shen-Qiu-Tian theorem then connects this setup directly to spontaneous breaking of continuous symmetry. The authors also report a tower of states in the low-energy spectrum, a feature they observe in this realistic one-dimensional ferrimagnetic system with short-range interactions. A sympathetic reader would care because the result supplies a concrete case where continuous symmetry breaking occurs in one dimension despite the usual expectations for short-range models.

Core claim

The central claim is that the one-dimensional antiferromagnetic Heisenberg model on a comb lattice, when a symmetry-preserving relevant perturbation is applied, can always be described by the Marshall-Lieb-Mattis theorem for bipartite lattices with unequal sublattice sizes, and the Shen-Qiu-Tian theorem thereby establishes spontaneous continuous symmetry breaking, accompanied by the emergence of a tower of states in the spectrum of this ferrimagnetic lattice system.

What carries the argument

The comb lattice geometry under a symmetry-preserving perturbation, which makes the two sublattices unequal in site number and thereby activates the Marshall-Lieb-Mattis theorem connected to continuous symmetry breaking via the Shen-Qiu-Tian theorem.

If this is right

  • Spontaneous long-range magnetic order develops in this one-dimensional short-range system.
  • A tower of states appears in the spectrum as a diagnostic of the symmetry-breaking tendency.
  • The same description applies to other realistic one-dimensional ferrimagnetic lattices with short-range interactions.
  • Numerical searches for symmetry breaking in one dimension can now target comb-like or similarly perturbed geometries.

Where Pith is reading between the lines

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

  • Other one-dimensional lattices that can be made bipartite with unequal sublattices under perturbation may show analogous spontaneous symmetry breaking.
  • The tower of states could serve as a broader numerical indicator for detecting spontaneous continuous symmetry breaking in one-dimensional quantum magnets beyond the specific comb case.
  • This example opens the possibility of engineering comb-chain geometries in materials or cold-atom setups to realize long-range order where it is usually forbidden.

Load-bearing premise

The comb lattice with the applied symmetry-preserving perturbation satisfies the Marshall-Lieb-Mattis theorem conditions for bipartite lattices with unequal sublattice sizes.

What would settle it

Numerical computation of the low-energy spectrum for the perturbed comb model that shows no tower of states with the expected quantum numbers would indicate that continuous symmetry breaking does not occur.

Figures

Figures reproduced from arXiv: 2506.23258 by Bin-Bin Mao, Jingya Wang, Xu Tian, Zenan Liu, Zheng Yan, Zhe Wang, Zijian Xiong.

Figure 2
Figure 2. Figure 2: FIG. 2. Correlation [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. A 1D comb lattice [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. spectrum calculated by quantum Monte-Carlo-stochastic analytic continuation for L=256, where [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Correlation function [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Based on the study of a one-dimensional (1D) antiferromagnetic Heisenberg model on a comb lattice, this work identifies an example of spontaneous continuous symmetry breaking in a 1D system with short-range interactions. When a symmetry-preserving relevant perturbation is applied to the system, we find that this model can always be described by the Marshall-Lieb-Mattis (MLM) theorem. The Shen-Qiu-Tian theorem establishes a direct connection between the MLM theorem (in the case of bipartite lattices with unequal numbers of sites in the two sublattices) and the breaking of continuous symmetry. Moreover, although authors of previous studies have suggested that the presence of a tower of states (TOS) serves as an important numerical diagnostic of the tendency of a system toward spontaneous symmetry breaking, these investigations have primarily focused on two-dimensional systems. In 1D systems, however, the presence of long-range order does not automatically imply the emergence of a TOS. Here, we observe the existence of a TOS in a 1D realistic ferrimagnetic lattice system with short-range interactions.

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 / 2 minor

Summary. The manuscript studies the antiferromagnetic Heisenberg model on a one-dimensional comb lattice. After introducing a symmetry-preserving relevant perturbation, the authors argue that the model satisfies the Marshall-Lieb-Mattis theorem because of unequal sublattice sizes. They then invoke the Shen-Qiu-Tian theorem to conclude spontaneous breaking of continuous SU(2) symmetry and report numerical evidence for a tower of states in this 1D ferrimagnetic system.

Significance. If the central claim is substantiated, the result would be significant: it supplies a concrete, short-range 1D example of spontaneous continuous symmetry breaking, which is generally precluded by the Mermin-Wagner theorem. The work also extends the numerical diagnostic of a tower of states to a realistic 1D ferrimagnet, where the authors correctly note that long-range order does not automatically produce a TOS. The use of established theorems (MLM and Shen-Qiu-Tian) and the focus on a geometrically simple lattice are strengths.

major comments (2)
  1. [Shen-Qiu-Tian application] The section applying the Shen-Qiu-Tian theorem: the manuscript must explicitly verify that the theorem’s assumptions (particularly the suppression of 1D infrared divergences that can produce power-law rather than true long-range order) are satisfied for the perturbed comb geometry. The abstract acknowledges that a TOS does not automatically imply SSB in 1D; therefore the step from finite-S ground-state degeneracy to spontaneous SU(2) breaking in the thermodynamic limit requires a concrete argument or additional check that is currently missing.
  2. [Numerical results] Numerical evidence section (tower-of-states data): finite-size scaling, error bars, and the precise definition of the low-lying states used to identify the TOS are not described in sufficient detail. Without these, it is impossible to assess whether the reported tower survives the thermodynamic limit or is an artifact of the chosen perturbation strength or system sizes.
minor comments (2)
  1. [Introduction] The abstract states that the model “can always be described by the Marshall-Lieb-Mattis theorem”; the precise form of the symmetry-preserving perturbation and the resulting sublattice imbalance should be stated explicitly in the main text.
  2. [Model definition] Notation for the comb lattice (site labeling, coordination numbers) is introduced without a figure or clear diagram; a simple schematic would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and constructive feedback on our manuscript. We address the major comments point by point below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: [Shen-Qiu-Tian application] The section applying the Shen-Qiu-Tian theorem: the manuscript must explicitly verify that the theorem’s assumptions (particularly the suppression of 1D infrared divergences that can produce power-law rather than true long-range order) are satisfied for the perturbed comb geometry. The abstract acknowledges that a TOS does not automatically imply SSB in 1D; therefore the step from finite-S ground-state degeneracy to spontaneous SU(2) breaking in the thermodynamic limit requires a concrete argument or additional check that is currently missing.

    Authors: We thank the referee for this insightful comment. The Shen-Qiu-Tian theorem connects the MLM theorem to spontaneous SU(2) symmetry breaking under specific conditions that ensure the absence of power-law order from infrared divergences. In our perturbed comb-chain model, the relevant perturbation is designed to be symmetry-preserving, which we believe stabilizes the long-range ferrimagnetic order by suppressing such divergences, consistent with the unequal sublattice sizes. To address the referee's concern directly, we will add a new paragraph in the discussion of the theorem application that explicitly verifies these assumptions for the comb geometry, including a brief analysis of the infrared behavior. revision: yes

  2. Referee: [Numerical results] Numerical evidence section (tower-of-states data): finite-size scaling, error bars, and the precise definition of the low-lying states used to identify the TOS are not described in sufficient detail. Without these, it is impossible to assess whether the reported tower survives the thermodynamic limit or is an artifact of the chosen perturbation strength or system sizes.

    Authors: We agree with the referee that additional details on the numerical methods and analysis are necessary for a complete assessment. In the revised manuscript, we will expand the numerical results section to include detailed finite-size scaling of the energy differences in the tower of states, error bars derived from the numerical convergence criteria, and a clear definition of the low-lying states specifying the total spin and other quantum numbers used. This will demonstrate that the TOS persists in the thermodynamic limit and is robust against variations in perturbation strength. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external theorems to a new lattice example

full rationale

The paper's chain starts from the 1D antiferromagnetic Heisenberg model on the comb lattice, applies the Marshall-Lieb-Mattis theorem after adding a symmetry-preserving perturbation to obtain a ferrimagnetic ground state with net spin, invokes the Shen-Qiu-Tian theorem to link this to spontaneous continuous symmetry breaking, and reports numerical observation of a tower of states. These steps cite established external theorems rather than deriving them internally or reducing the conclusion to a self-referential definition, fitted parameter, or self-citation chain. No equations or claims in the abstract or described structure rename a known result, smuggle an ansatz, or present a fitted quantity as a prediction. The work therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on applicability of two named theorems to the comb lattice model after perturbation, with no free parameters or invented entities explicitly stated.

axioms (1)
  • domain assumption The comb lattice antiferromagnetic Heisenberg model with symmetry-preserving perturbation is described by the Marshall-Lieb-Mattis theorem for bipartite lattices with unequal sublattice sizes.
    Directly invoked in the abstract to link the model to continuous symmetry breaking via the Shen-Qiu-Tian theorem.

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