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arxiv: 2605.28967 · v1 · pith:FH7TSBIPnew · submitted 2026-05-27 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

Local Strong-to-Weak Spontaneous Symmetry Breaking

Francisco Divi , Leonardo A. Lessa , Chong Wang This is my paper

Pith reviewed 2026-06-29 11:18 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el
keywords strong-to-weak spontaneous symmetry breakinglocal fidelity correlatorthermodynamic limitconformal field theoryfree-fermion metalseigenstate thermalization hypothesisconditional mutual informationsymmetry breaking
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The pith

A local one-point fidelity correlator defines strong-to-weak spontaneous symmetry breaking that stays well-defined in the thermodynamic limit.

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

The paper introduces a local definition of strong-to-weak spontaneous symmetry breaking that replaces the earlier two-point fidelity correlator with a one-point version. This change allows the order to be detected in a system of size N using only polynomial resources up to volume scales of order log N. The local order parameter remains meaningful even when the global density matrix itself ceases to be well-defined in the thermodynamic limit. Stability under finite-depth symmetric channels and the presence of long-range conditional mutual information carry over to the local setting. The construction is illustrated on conformal field theory ground states and on ballistic and diffusive free-fermion metals, where the correlator shows universal scaling.

Core claim

We propose a local notion of strong-to-weak spontaneous symmetry breaking (SW-SSB) through a local one-point fidelity correlator. Compared with the previous definition in terms of a two-point fidelity correlator, our local formulation offers two key advantages: it is easier to detect in large systems and the local SW-SSB order remains well defined in the thermodynamic limit, where the density matrix itself is not well defined. We show that key features of SW-SSB, including stability under finite-depth symmetric channels and long-range conditional mutual information, persist within this local framework.

What carries the argument

The local one-point fidelity correlator, which compares a local reduced density matrix to its image under a symmetry transformation and thereby serves as an order parameter for local SW-SSB.

If this is right

  • Detection of the order becomes feasible with poly(N) resources up to O(log N) volume scales.
  • The order parameter stays well-defined when the global density matrix is no longer defined.
  • Stability under finite-depth symmetric channels holds for the local version.
  • Long-range conditional mutual information remains a signature of the local order.
  • In critical systems the correlator defines a class of defect problems with universal scaling in CFTs and free-fermion metals.

Where Pith is reading between the lines

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

  • Numerical studies of SW-SSB can now target substantially larger system sizes than the two-point definition permitted.
  • The analogy with local thermalization under ETH suggests that local SW-SSB may classify eigenstates in a manner parallel to thermalization diagnostics.
  • The local formulation may extend naturally to open quantum systems or to symmetry-protected phases in quantum circuits.

Load-bearing premise

The local one-point fidelity correlator fully captures the essential physics of the earlier two-point definition, so that stability and long-range mutual information automatically transfer without further conditions.

What would settle it

A concrete system in which the two-point fidelity detects SW-SSB but the local one-point version shows no order, or in which the local version loses stability under a finite-depth symmetric channel that preserves the two-point order.

Figures

Figures reproduced from arXiv: 2605.28967 by Chong Wang, Francisco Divi, Leonardo A. Lessa.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the local fidelity correlator [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Tripartition [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. After applying a finite-depth local channel [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Two coverings [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The tripartitions used in defining the local CMI [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. One-point R´enyi-1 correlator, [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. One-point R´enyi-1 correlator, [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Geometric representation of the R´enyi-1 one-point [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. One-point R´enyi-1 correlator, [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
read the original abstract

We propose a local notion of strong-to-weak spontaneous symmetry breaking (SW-SSB), through a local one-point fidelity correlator. Compared with the previous definition in terms of a two-point fidelity correlator, our local formulation offers two key advantages: (1) it is easier to detect in large systems: for a system of size $N$ and with ${\rm poly}(N)$ amount of resources, one can detect the local fidelity order up to volume scale $O(\log(N))$; and (2) the local SW-SSB order remains well defined in the thermodynamic limit, where the density matrix itself is not well defined. We show that key features of SW-SSB, including stability under finite-depth symmetric channels and long-range conditional mutual information, persist within this local framework. Our definition is conceptually analogous to local thermalization, as exemplified by pure states obeying the eigenstate thermalization hypothesis (ETH). For critical states, the local one-point fidelity correlator defines an interesting class of defect problems. We demonstrate the applicability of the local formulation through several concrete examples, and derive the universal scaling behavior of the local fidelity correlator in a range of critical systems, including ground states of conformal field theories as well as ballistic and diffusive free-fermion metals.

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

0 major / 2 minor

Summary. The manuscript proposes a local definition of strong-to-weak spontaneous symmetry breaking (SW-SSB) using a one-point fidelity correlator. It claims two advantages over the prior two-point version: easier detection in large systems (poly(N) resources up to O(log N) volume) and well-definedness in the thermodynamic limit. The authors assert that core properties—stability under finite-depth symmetric channels and long-range conditional mutual information—persist, demonstrate this via concrete examples, and derive universal scaling of the correlator for CFT ground states as well as ballistic and diffusive free-fermion metals. The local order is presented as analogous to local thermalization under ETH.

Significance. If the persistence of the listed features is established, the local formulation supplies a more experimentally accessible and thermodynamically robust diagnostic for SW-SSB. The explicit scaling results in CFTs and free-fermion systems, together with the defect-problem interpretation, would furnish concrete, falsifiable predictions for critical states. The connection to ETH-style local thermalization is conceptually useful for linking symmetry-breaking diagnostics to eigenstate properties.

minor comments (2)
  1. The abstract states that stability and long-range CMI 'persist within this local framework' and are shown via examples; the main text should include an explicit statement (perhaps in the section introducing the local correlator) of any additional conditions required for the carry-over from the two-point definition.
  2. Figure captions and the scaling derivations for the free-fermion cases would benefit from a brief remark on the system sizes or cutoffs used to extract the universal exponents, to facilitate direct comparison with the CFT results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines a new local one-point fidelity correlator for SW-SSB, distinct from the prior two-point version, and demonstrates persistence of stability and long-range CMI through explicit examples plus independent scaling derivations in CFTs and free-fermion models. These calculations rely on standard techniques (ETH analogy, conformal invariance, ballistic/diffusive transport) rather than reducing to a fitted parameter, self-citation chain, or definitional equivalence. The extension is self-contained; no load-bearing step collapses by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the contribution is a conceptual redefinition rather than a calculation involving fitted constants or new postulated objects.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. From Topological Order to Mixed-State Phases: A Ground-State Probe of Fractionalized Excitations

    cond-mat.str-el 2026-06 unverdicted novelty 6.0

    Reduced density matrices of 2D topological orders at entanglement cuts realize 1D mixed-state phases whose properties detect fractionalized excitations such as anyons and spinons.

  2. Strong-to-Weak Spontaneous Symmetry Breaking

    quant-ph 2026-06 unverdicted novelty 2.0

    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.

Reference graph

Works this paper leans on

76 extracted references · 22 canonical work pages · cited by 2 Pith papers · 7 internal anchors

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    averaged

    For any finite regionAcontainingO x, F(ρ A;O x)2 ≤R (1)(ρA;O x)≤ ∥O x∥∞ F(ρ A;O x).(71) Proof.To establish monotonicity, letC=B\A, and letD C = dimH C, the dimension of the Hilbert space ofC. IfO x is unitary, the monotonicity follows from the standard data processing inequality of the Holevo fidelity Tr(√ρ√σ). For generalO x, we can relate the reduced de...

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    (110), we can relate the SW-SSB properties ofρ p to the RBIM at the Nishimori line

    Local fidelity correlator From Eq. (110), we can relate the SW-SSB properties ofρ p to the RBIM at the Nishimori line. Namely, the 5 This statement can be formalized by noting that the relative homologyH 1(A, ∂Ac) =Z |∂Ac| 2 . 17 local fidelity correlator assumes the following form: F(ρ A;Z x) = X s √psps+ˆex (111) ∝ X s,b Zs,b sP b Zs+ˆex,bP b Zs,b (112)...

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    Replica calculation and boundary magnetization By employing the expression ofp s from Eq. (109), we can relate the one-point R´ enyi-2kmeasure R(2k)(ρp,A;Z x),k∈Z >0, to a 2k-replica calculation: R(2k)(ρp,A;Z x) = X s pk s pk s+ˆex (114) ∝ X s X η(α) eK P2k α=1 HA[η(α)] 2kY α=k+1 η(α) x !Y i∈A 2kY α=1 (η(α) i )si (115) = X η(α) eK P2k α=1 HA[η(α)] 2kY α=k...

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    Two dimensional CFTs Let us consider two-dimensional CFTs (D= 1 + 1). Crucially, we will show that the one-point R´ enyi-1 corre- lator for a primary operatorO, on a finite regionAcan be mapped to a conventional two-point function in the complex plane via the uniformization map. Therefore, the R´ enyi-1 correlator is completely fixed by the normal- izatio...

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    R´ enyi-1 correlators in free fermion systems A useful property of free fermions is that Wick’s the- orem holds for correlation functions restricted to any re- gionA. Consequently, the reduced density matrixρ A of a Gaussian state on regionAis also a Gaussian state, and specifyingρ A is equivalent to specifying the two-point correlation matrix CA(x, y) =⟨...

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    We now show this explicitly; the same structure will later be used to analyze higher-dimensional Fermi gases. We make two simplifications. First, we take the region to be the semi-infinite lineA= [0,∞) and insert the operator atx=ℓ. This is equivalent to considering the finite intervalA= [−ℓ, R], inserting the operator atx= 0, and then takingR→ ∞, withℓfi...

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