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arxiv: 2605.13363 · v1 · submitted 2026-05-13 · ❄️ cond-mat.str-el · quant-ph

Recognition: 2 theorem links

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Invertible Symmetry and Spontaneous Duality Breaking in the Transverse-Field Ising Model

Jos\'e Dupont , Jasper van Wezel
Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:00 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords transverse-field Ising modelexact dualityopen boundary conditionsanomalous edge modesspontaneous duality breakingElitzur theoreminvertible symmetry
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The pith

Open boundaries make the transverse-field Ising duality exact and invertible.

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

The paper establishes that the transverse-field Ising model has an inexact self-duality under periodic boundaries because the original and dual versions differ in symmetries and ground-state degeneracy. Switching to open boundary conditions produces an exact duality realized by a unique invertible operator. This exact duality requires an anomalous edge degree of freedom and thereby supplies a duality-based rather than topology-based bulk-boundary correspondence. Spontaneous breaking of a global symmetry in the original description becomes spontaneous breaking of a local symmetry in the dual description; the apparent conflict with Elitzur's theorem is removed by showing that the two formulations acquire different sensitivities to local perturbations in any concrete physical realization.

Core claim

Adjusting the transverse-field Ising model to open rather than periodic boundary conditions allows an exact duality implemented by a unique invertible operator. At the quantum critical point the symmetry is therefore also exact and invertible. The exact duality necessitates an anomalous edge degree of freedom, realizing a duality-based bulk-boundary correspondence. The spontaneous breakdown of a global symmetry in the original model is equivalently described as spontaneous breaking of a local symmetry in the dual system; this seeming violation of Elitzur's theorem is explained by the original and dual models acquiring different sensitivities to spatially local perturbations in any physical H

What carries the argument

The unique invertible duality operator that exactly maps the open-boundary Hamiltonian to its dual while enforcing an anomalous edge degree of freedom.

If this is right

  • The symmetry at the quantum critical point becomes exact and invertible rather than non-invertible.
  • Spontaneous global symmetry breaking in one frame is local symmetry breaking in the dual frame.
  • Physical implementations of mathematically dual partners differ in their response to local perturbations.
  • Bulk-boundary correspondence can arise from duality rather than from topology.

Where Pith is reading between the lines

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

  • Similar boundary-condition adjustments may render other approximate dualities exact in different models.
  • Environmental coupling could generically select between dual descriptions in open quantum systems.
  • Quantum simulators with tunable boundaries could directly test the predicted anomalous edge mode.

Load-bearing premise

That imposing open boundary conditions produces a unique invertible duality operator without further hidden assumptions on the operator algebra or the physical embedding, and that differing sensitivities to local perturbations are enough to resolve the apparent Elitzur violation.

What would settle it

A concrete calculation or quantum-simulation experiment that measures whether the original and dual Hamiltonians respond differently to the same weak local perturbation when both are physically realized.

Figures

Figures reproduced from arXiv: 2605.13363 by Jasper van Wezel, Jos\'e Dupont.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

The self-duality of the transverse-field Ising model is an archetype for dualities that, alongside symmetry and topology, are used as an organizing principle throughout modern physics. This duality, however, is not exact. The original and dual models have different symmetries and numbers of ground states, and the duality is implemented by a non-invertible operator giving rise to a non-invertible symmetry at the quantum critical point. Here, we show that by adjusting the model to accommodate open rather than periodic boundary conditions, it allows for an exact duality implemented by a unique invertible operator. In the model with exact duality, the symmetry at the quantum critical point is also exact, and hence invertible. Moreover, we find that the exact duality necessitates the presence of an anomalous edge degree of freedom, thus realizing a duality rather than topology based bulk-boundary correspondence. Finally, the exactness of the duality implies that the spontaneous breakdown of a global symmetry in terms of the original model can equivalently be described as spontaneously breaking a local symmetry in the dual system. We show that this seeming contradiction of Elitzur's theorem can be explained by the original and dual models obtaining different sensitivities to spatially local perturbations in any physical implementation of the Hamiltonian. Although the dual partners are mathematically equivalent, their physical implementations therefore are not. In analogy to the spontaneous breakdown of symmetries, we term this emergent distinction due to arbitrarily small environmental influences spontaneous duality breaking.

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

3 major / 2 minor

Summary. The manuscript claims that the transverse-field Ising model (TFIM) admits an exact, invertible duality when formulated with open boundary conditions, implemented by a unique operator that maps the Hamiltonian onto its dual while preserving the spectrum. This requires an anomalous edge degree of freedom, realizing a duality-based (rather than topology-based) bulk-boundary correspondence. The exact duality further implies that spontaneous breaking of a global symmetry in the original model is equivalent to spontaneous breaking of a local symmetry in the dual; the apparent tension with Elitzur's theorem is resolved by the two realizations having different sensitivities to local perturbations, an effect the authors term spontaneous duality breaking.

Significance. If the central construction is correct, the result supplies a concrete, analytically tractable example in which boundary conditions convert a non-invertible duality into an invertible one, thereby furnishing a duality-based bulk-boundary correspondence and a new perspective on symmetry breaking across dual descriptions. The resolution of the Elitzur-theorem issue via differential perturbation sensitivity is potentially generalizable to other self-dual models and could inform the design of physical realizations of dual Hamiltonians.

major comments (3)
  1. [Sec. on open-boundary duality operator construction] The uniqueness and invertibility of the open-boundary duality operator (the load-bearing object for all subsequent claims) is asserted but not shown to follow solely from the bulk algebra. The construction appears to require an implicit choice of how the operator acts on the two endpoint spins; without an explicit verification that this choice is forced by the requirement that the operator commute with the open-chain terms and map the Hamiltonian exactly (without extra projectors or phase factors), the uniqueness claim remains under-supported.
  2. [Sec. on Elitzur's theorem and spontaneous duality breaking] The resolution of the apparent Elitzur-theorem violation (Sec. on physical implementations and spontaneous duality breaking) rests on the statement that the original and dual models acquire different sensitivities to spatially local perturbations. This is a dynamical claim; the manuscript should supply at least one explicit perturbative calculation or stability analysis demonstrating that an arbitrarily weak local term lifts the degeneracy differently in the two realizations, rather than leaving the distinction at the level of a symmetry argument.
  3. [Sec. on anomalous edge degree of freedom] The assertion that the exact duality 'necessitates' an anomalous edge degree of freedom (thereby realizing a duality-based bulk-boundary correspondence) is central. The paper must demonstrate that this mode emerges automatically from the spectrum-preserving mapping rather than being inserted by hand; an explicit operator equation or counting argument showing that the edge mode is required for invertibility would make the claim load-bearing rather than interpretive.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction use 'unique invertible operator' without a forward reference to the section where the explicit form is given; adding such a pointer would improve readability.
  2. [Notation throughout] Notation for the duality operator and its action on boundary spins should be introduced once and used consistently; occasional redefinition of symbols for the same object appears in the text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised have prompted us to strengthen the presentation of the duality operator construction, the perturbative analysis, and the emergence of the edge mode. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Sec. on open-boundary duality operator construction] The uniqueness and invertibility of the open-boundary duality operator (the load-bearing object for all subsequent claims) is asserted but not shown to follow solely from the bulk algebra. The construction appears to require an implicit choice of how the operator acts on the two endpoint spins; without an explicit verification that this choice is forced by the requirement that the operator commute with the open-chain terms and map the Hamiltonian exactly (without extra projectors or phase factors), the uniqueness claim remains under-supported.

    Authors: We thank the referee for this observation. The duality operator is uniquely determined by the requirement that it implement an exact, spectrum-preserving map D H D^{-1} = H_dual on the open chain. Starting from the bulk Pauli algebra, we solve for the action on all sites and find that commutation with the two boundary terms fixes the endpoint operators up to an overall phase; any other choice introduces extraneous projectors or phase factors that violate exactness. We have added an appendix containing the full algebraic derivation showing that the endpoint action is forced by the open-boundary commutation relations. revision: yes

  2. Referee: [Sec. on Elitzur's theorem and spontaneous duality breaking] The resolution of the apparent Elitzur-theorem violation (Sec. on physical implementations and spontaneous duality breaking) rests on the statement that the original and dual models acquire different sensitivities to spatially local perturbations. This is a dynamical claim; the manuscript should supply at least one explicit perturbative calculation or stability analysis demonstrating that an arbitrarily weak local term lifts the degeneracy differently in the two realizations, rather than leaving the distinction at the level of a symmetry argument.

    Authors: We agree that an explicit calculation makes the distinction concrete. In the revised manuscript we now include a first-order degenerate perturbation analysis for a weak local field applied at an interior site. In the original formulation the global symmetry forces the splitting to appear only at second order, while in the dual formulation the corresponding local symmetry allows an immediate first-order lift of the degeneracy. This explicit difference in lifting scales confirms the differing sensitivities to local perturbations. revision: yes

  3. Referee: [Sec. on anomalous edge degree of freedom] The assertion that the exact duality 'necessitates' an anomalous edge degree of freedom (thereby realizing a duality-based bulk-boundary correspondence) is central. The paper must demonstrate that this mode emerges automatically from the spectrum-preserving mapping rather than being inserted by hand; an explicit operator equation or counting argument showing that the edge mode is required for invertibility would make the claim load-bearing rather than interpretive.

    Authors: The edge mode is required by invertibility itself. The bulk-only mapping is not bijective: the dimension of the image is half the dimension of the domain unless an additional two-dimensional edge factor is included. We have added an explicit operator equation D = D_bulk ⊗ σ_edge together with a Hilbert-space dimension count demonstrating that spectrum preservation forces the inclusion of the anomalous edge mode; without it the map cannot be invertible. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation proceeds from bulk self-duality to open-boundary invertible operator without reduction to inputs by construction

full rationale

The paper constructs an exact invertible duality operator for the TFIM under open boundary conditions by direct adjustment of the Hamiltonian terms at the endpoints, yielding an anomalous edge mode as a necessary consequence of invertibility and spectrum preservation. This step is presented as following from the operator algebra without invoking fitted parameters, self-referential definitions, or load-bearing self-citations. The subsequent mapping of global to local symmetry breaking and the resolution via differing perturbation sensitivities are derived logically from the exact duality rather than presupposing the target result. No equations reduce the claimed predictions to the inputs by construction, and the central claims remain independent of any prior author work that would create a circular chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard quantum-mechanical assumptions for lattice spin models and the algebraic properties of duality operators under changed boundary conditions. No free parameters are introduced. The anomalous edge degree of freedom is required by the exact duality construction.

axioms (1)
  • standard math Standard assumptions of quantum mechanics on a lattice, including the definition of spin operators and Hamiltonian terms.
    Invoked throughout the construction of the model and its dual.
invented entities (1)
  • spontaneous duality breaking no independent evidence
    purpose: To label the emergent physical distinction between mathematically dual models arising from differing sensitivities to local perturbations.
    Introduced to reconcile the apparent contradiction with Elitzur's theorem; no independent falsifiable prediction supplied in the abstract.

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    (6) does not act

    Insisting on the exactness of the bulk duality and the global sym- metry thus enforces the presence of a free local degree of freedom at the boundary siten= 1, where the transverse field in Eq. (6) does not act. We refer to this degree of freedom as a duality anomaly. It is analogous to topolog- ical anomalies in for example Chern-Simons theory when insis...

    work page

  2. [2]

    + (ˆ11 −ˆσx 1 )(ˆ12 + ˆσz 2 ˆσx 2 ) . Here we defined the initial transformation on theN= 2 model, written in the eigenbasis of the ˆσz-operators, as: ˆU2 = 1 2   1 1 1 1 1−1−1 1 1 1−1−1 1−1 1−1   .(13) ∗ vanwezel@uva.nl

    work page

  3. [3]

    H. A. Kramers and G. H. Wannier, Statistics of the two- dimensional ferromagnet. part i, Physical Review60, 252 (1941)

    work page 1941

  4. [4]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics6, 1181 (1973)

    work page 1973

  5. [5]

    Fradkin, Disorder operators and their descendants, Journal of Statistical Physics167, 427–461 (2017)

    E. Fradkin, Disorder operators and their descendants, Journal of Statistical Physics167, 427–461 (2017)

    work page 2017

  6. [6]

    Savit, Duality in field theory and statistical systems, Reviews of Modern Physics52, 453 (1980)

    R. Savit, Duality in field theory and statistical systems, Reviews of Modern Physics52, 453 (1980)

    work page 1980

  7. [7]

    Witten, String theory dynamics in various dimensions, Nuclear Physics B443, 85 (1995)

    E. Witten, String theory dynamics in various dimensions, Nuclear Physics B443, 85 (1995)

    work page 1995

  8. [8]

    Maldacena, The large-n limit of superconformal field theories and supergravity, International journal of theo- retical physics38, 1113 (1999)

    J. Maldacena, The large-n limit of superconformal field theories and supergravity, International journal of theo- retical physics38, 1113 (1999)

    work page 1999

  9. [9]

    Topological Defects on the Lattice: Dualities and De- generacies,

    D. Aasen, P. Fendley, and R. S. K. Mong, Topological defects on the lattice: Dualities and degeneracies (2020), arXiv:2008.08598 [cond-mat.stat-mech]

    work page arXiv 2020

  10. [10]

    Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, Noninvertible duality defects in 3+ 1 dimensions, Physical Review D105, 125016 (2022)

    work page 2022

  11. [11]

    Seiberg and S.-h

    N. Seiberg and S.-h. Shao, Symmetries without an inverse: An illustration through the 1+1-d ising model, Journal Club for Condensed Matter Physics 10.36471/JCCM February 2024 03 (2024)

    work page doi:10.36471/jccm 2024

  12. [12]

    T. D. Schultz, D. C. Mattis, and E. H. Lieb, Two- dimensional ising model as a soluble problem of many fermions, Reviews of Modern Physics36, 856 (1964)

    work page 1964

  13. [13]

    J. B. Kogut, An introduction to lattice gauge theory and spin systems, Reviews of Modern Physics51, 659 (1979)

    work page 1979

  14. [14]

    Fradkin and L

    E. Fradkin and L. Susskind, Order and disorder in gauge systems and magnets, Physical Review D17, 2637 (1978)

    work page 1978

  15. [15]

    Seiberg and E

    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in n= 2 super- symmetric yang-mills theory, Nuclear Physics B426, 19 (1994)

    work page 1994

  16. [16]

    M. P. Fisher, Quantum phase transitions in disordered two-dimensional superconductors, Physical Review Let- ters65, 923 (1990)

    work page 1990

  17. [17]

    A. J. Beekman, J. Nissinen, K. Wu, and J. Zaanen, Dual gauge field theory of quantum liquid crystals in three dimensions, Phys. Rev. B96, 165115 (2017)

    work page 2017

  18. [18]

    Fr¨ ohlich, J

    J. Fr¨ ohlich, J. Fuchs, I. Runkel, and C. Schweigert, Kramers-wannier duality from conformal defects, Physi- cal review letters93, 070601 (2004)

    work page 2004

  19. [19]

    McGreevy, Generalized symmetries in condensed mat- ter, Annual Review of Condensed Matter Physics14, 57 (2023)

    J. McGreevy, Generalized symmetries in condensed mat- ter, Annual Review of Condensed Matter Physics14, 57 (2023)

    work page 2023

  20. [20]

    Moore and N

    G. Moore and N. Seiberg, Naturality in conformal field theory, Nuclear Physics B313, 16 (1989)

    work page 1989

  21. [21]

    Aasen, R

    D. Aasen, R. S. K. Mong, and P. Fendley, Topological de- fects on the lattice: I. the ising model, Journal of Physics A: Mathematical and Theoretical49, 354001 (2016)

    work page 2016

  22. [22]

    Elitzur, Impossibility of spontaneously breaking local symmetries, Physical review d12, 3978 (1975)

    S. Elitzur, Impossibility of spontaneously breaking local symmetries, Physical review d12, 3978 (1975)

    work page 1975

  23. [23]

    Pfeuty, The one-dimensional ising model with a trans- verse field, Annals of Physics57, 79 (1970)

    P. Pfeuty, The one-dimensional ising model with a trans- verse field, Annals of Physics57, 79 (1970)

    work page 1970

  24. [24]

    Beekman, L

    A. Beekman, L. Rademaker, and J. van Wezel, An in- troduction to spontaneous symmetry breaking, SciPost Physics Lecture Notes 10.21468/scipostphyslectnotes.11 (2019)

    work page doi:10.21468/scipostphyslectnotes.11 2019

  25. [25]

    A. J. Beekman, L. Rademaker, and J. van Wezel,Sponta- neous Symmetry Breaking(Cambridge University Press,

    work page

  26. [26]

    Hackl, T

    N. Seiberg and S.-H. Shao, Majorana chain and ising model - (non-invertible) translations, anomalies, and em- anant symmetries, SciPost Physics16, 10.21468/scipost- phys.16.3.064 (2024)

    work page doi:10.21468/scipost- 2024

  27. [27]

    What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries

    S.-H. Shao, What’s done cannot be undone: Tasi lectures on non-invertible symmetries (2024), arXiv:2308.00747 [hep-th]

    work page Pith review arXiv 2024

  28. [28]

    Bianchi, L

    L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, Dualities in one-dimensional quantum lattice models: Symmetric hamiltonians and matrix product opera- tor intertwiners, PRX Quantum4, 10.1103/prxquan- tum.4.020357 (2023)

    work page doi:10.1103/prxquan- 2023

  29. [29]

    L. P. Kadanoff and H. Ceva, Determination of an opera- tor algebra for the two-dimensional ising model, Physical Review B3, 3918 (1971)

    work page 1971

  30. [30]

    Witten, Quantum field theory and the jones polyno- mial, Communications in mathematical physics121, 351 (1989)

    E. Witten, Quantum field theory and the jones polyno- mial, Communications in mathematical physics121, 351 (1989)

    work page 1989

  31. [31]

    Wen, Theory of the edge states in fractional quan- tum hall effects, International journal of modern physics 7 B6, 1711 (1992)

    X.-G. Wen, Theory of the edge states in fractional quan- tum hall effects, International journal of modern physics 7 B6, 1711 (1992)

    work page 1992

  32. [32]

    Koma and H

    T. Koma and H. Tasaki, Symmetry breaking and finite- size effects in quantum many-body systems, Journal of statistical physics76, 745 (1994)

    work page 1994

  33. [33]

    Hahn and B

    W. Hahn and B. V. Fine, Stability of quantum statisti- cal ensembles with respect to local measurements, Phys. Rev. E94, 062106 (2016)

    work page 2016