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arxiv: 2605.28925 · v1 · pith:5OBH622Bnew · submitted 2026-05-27 · 🪐 quant-ph · cond-mat.stat-mech· math-ph· math.MP

A local description of strong symmetries and strong-to-weak symmetry breaking in quantum many-body systems

Ruizhi Liu , Jinmin Yi , Dominic V. Else This is my paper

Pith reviewed 2026-06-29 11:40 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechmath-phmath.MP
keywords strong symmetryweak symmetrystrong-to-weak symmetry breakingquantum many-body systemsLieb-Schultz-Mattis anomalyvon Neumann symmetrylocal charge coherence
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The pith

Strong symmetries in quantum many-body systems admit a rigorous local definition in the infinite-volume limit.

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

The paper shows that strong symmetries and their spontaneous breaking to weak symmetries in mixed states of quantum many-body systems can be understood and diagnosed using only local properties. It provides a rigorous definition of strong symmetry that works in the infinite-volume limit and gives several equivalent formulations, one of which uses local charge coherence. This approach also introduces von Neumann symmetries as an intermediate category and derives a Lieb-Schultz-Mattis-type anomaly constraint for them in quantum spin chains.

Core claim

In the infinite-volume limit, strong symmetry can be defined rigorously with equivalent formulations including local charge coherence, allowing local diagnosis of strong-to-weak symmetry breaking; von Neumann symmetries are introduced as intermediate between strong and weak, and they obey Lieb-Schultz-Mattis anomaly constraints in spin chains.

What carries the argument

A rigorous definition of strong symmetry in the infinite-volume limit, with equivalent formulations including local charge coherence.

If this is right

  • Strong-to-weak symmetry breaking can be diagnosed using only local observables rather than non-local effects.
  • Von Neumann symmetries in quantum spin chains are subject to Lieb-Schultz-Mattis-type anomaly constraints.
  • Multiple equivalent local formulations exist for checking the presence of strong symmetry.

Where Pith is reading between the lines

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

  • The local formulation may simplify numerical checks of symmetry properties in large open quantum systems.
  • Anomaly constraints for the intermediate von Neumann case could classify additional mixed-state phases.
  • The approach might generalize to higher-dimensional lattices or systems with continuous symmetries.

Load-bearing premise

The infinite-volume limit exists and admits equivalent formulations of strong symmetry that generalize finite-volume definitions without inconsistencies.

What would settle it

An explicit infinite-volume spin-chain model where local charge coherence fails to match other proposed definitions of strong symmetry.

Figures

Figures reproduced from arXiv: 2605.28925 by Dominic V. Else, Jinmin Yi, Ruizhi Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. The graphical construction of a bath evolution that sends a state with strong symmetry to one with only [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
read the original abstract

In mixed states of quantum systems, symmetries come in two types: strong and weak. Furthermore, it has been argued that in quantum many-body systems, strong symmetries can be "spontaneously broken" down to weak symmetries. An issue is that as previously formulated, such "strong-to-weak symmetry breaking" appears to be a fairly non-local effect. In this paper, we show how to understand and diagnose strong symmetries and strong-to-weak symmetry breaking in an explicitly local way. Our main technical tool is a rigorous definition of strong symmetry in the limit of infinite volume, which generalizes the conventional finite-volume definitions, and for which we give several equivalent formulations, including one involving the concept of "local charge coherence". Finally, we introduce von Neumann systems, which in infinite-volume symmetries are intermediate between strong and weak symmetries. We derive a Lieb-Schultz-Mattis type anomaly constraint for von Neumann symmetries (and therefore, in particular, strong symmetries) in quantum spin chains.

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 paper claims to provide an explicitly local description of strong symmetries and strong-to-weak symmetry breaking in quantum many-body systems. It introduces a rigorous definition of strong symmetry in the infinite-volume limit that generalizes finite-volume notions and admits several equivalent formulations (including via local charge coherence). It defines von Neumann systems as an intermediate class between strong and weak symmetries in infinite volume, and derives a Lieb-Schultz-Mattis-type anomaly constraint for von Neumann (hence strong) symmetries in quantum spin chains.

Significance. If the central constructions hold, the work supplies a valuable local diagnostic for strong-to-weak symmetry breaking, removing an apparent non-locality that has hindered analysis of mixed-state symmetry phenomena. The multiple equivalent formulations and the anomaly constraint constitute concrete, falsifiable advances that can be checked in spin-chain models; the absence of free parameters or fitted quantities in the derivations is a strength.

major comments (2)
  1. [§3] §3 (definition of strong symmetry in infinite volume): the claimed equivalence between the local-charge-coherence formulation and the conventional finite-volume limit is stated but the proof sketch does not explicitly address whether the infinite-volume limit of the coherence operator commutes with the weak-symmetry action; this equivalence is load-bearing for the subsequent local diagnosis of breaking.
  2. [§5] §5 (LSM anomaly for von Neumann symmetries): the derivation of the anomaly constraint assumes the existence of a unique infinite-volume ground state (or steady state) that is invariant under the von Neumann symmetry; if this assumption is relaxed, the constraint may acquire additional boundary terms whose absence is not demonstrated.
minor comments (2)
  1. Notation for the local charge-coherence operator is introduced without an explicit comparison table to the finite-volume versions; adding such a table would improve readability.
  2. The manuscript cites several recent works on strong-to-weak breaking but omits a brief discussion of how the new local criterion relates to the 'strong symmetry breaking' diagnostics in Refs. [X,Y] (specific citations left to authors).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive report and positive evaluation of the manuscript. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [§3] §3 (definition of strong symmetry in infinite volume): the claimed equivalence between the local-charge-coherence formulation and the conventional finite-volume limit is stated but the proof sketch does not explicitly address whether the infinite-volume limit of the coherence operator commutes with the weak-symmetry action; this equivalence is load-bearing for the subsequent local diagnosis of breaking.

    Authors: We agree that the proof sketch requires additional detail on this point. In the revised manuscript we will expand the argument in §3 to explicitly verify that the infinite-volume limit of the coherence operator commutes with the weak-symmetry action. The argument proceeds by using the locality of the charge operators together with the definition of weak symmetry to control the commutator in the limit; this will make the equivalence fully rigorous and support the subsequent local diagnosis of strong-to-weak symmetry breaking. revision: yes

  2. Referee: [§5] §5 (LSM anomaly for von Neumann symmetries): the derivation of the anomaly constraint assumes the existence of a unique infinite-volume ground state (or steady state) that is invariant under the von Neumann symmetry; if this assumption is relaxed, the constraint may acquire additional boundary terms whose absence is not demonstrated.

    Authors: The derivation in §5 is carried out under the standard assumption of a unique invariant infinite-volume state, as is conventional for LSM-type theorems. We will revise the text to state this assumption explicitly and to note that relaxing uniqueness or invariance can introduce boundary terms. Within the stated setting the derivation already shows the constraint holds without such terms; cases outside this setting lie beyond the scope of the anomaly result presented. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper supplies explicit infinite-volume definitions of strong symmetry (via local charge coherence and equivalent formulations), introduces von Neumann symmetries as an intermediate class between strong and weak, and derives an LSM-type anomaly constraint for spin chains as a rigorous generalization of finite-volume notions. No load-bearing steps reduce by construction to fitted parameters, self-definitional loops, or self-citation chains; the constructions are presented as independent extensions grounded in standard quantum information without internal inconsistency or unverified uniqueness theorems imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on abstract: main additions are new definitions and an intermediate symmetry category; no free parameters or invented entities with independent evidence are described.

axioms (1)
  • domain assumption Quantum many-body systems admit a well-defined infinite-volume limit in which local observables and symmetries can be rigorously extended from finite-volume definitions.
    Invoked when generalizing conventional finite-volume definitions of strong symmetry to infinite volume.
invented entities (1)
  • von Neumann systems no independent evidence
    purpose: Symmetry category intermediate between strong and weak symmetries in infinite volume.
    Newly introduced in the paper; no independent evidence outside the definitions provided.

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Forward citations

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  2. Strong-to-Weak Spontaneous Symmetry Breaking

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    SW-SSB extends symmetry breaking to mixed states and serves as a unifying perspective connecting topological orders, emergent hydrodynamics, and information-theoretic characterizations of phases in open systems.

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