pith. sign in

arxiv: 2504.12385 · v3 · submitted 2025-04-16 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· quant-ph

Learning transitions in classical Ising models and deformed toric codes

Malte P\"utz , Samuel J. Garratt , Hidetoshi Nishimori , Simon Trebst , Guo-Yi Zhu This is my paper

Pith reviewed 2026-05-22 19:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnquant-ph
keywords Ising modellearning transitiontricritical pointtoric codeweak measurementquantum memoryconditional probabilityreplica field theory
0
0 comments X

The pith

A learning transition in the Ising model intersects the thermal transition at a tricritical point that ensures robustness of quantum memory in deformed toric codes to weak measurements.

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

Conditional probability distributions model how one learns an unknown classical state through Bayesian updates. In the two-dimensional Ising model, learning local energy densities produces a transition in the long-distance decay of conditional correlation functions. This transition line reaches down to the thermal critical temperature. At their intersection a new tricritical point appears. The identical mathematical structure governs weak measurements on ground states of quantum spin models that smoothly deform the toric code into a paramagnet. The location of the tricritical point therefore shows that topological quantum memory survives weak measurements even when the starting state sits arbitrarily close to the quantum critical point.

Core claim

The authors identify a line of learning transitions in the classical Ising model that terminates at a new tricritical point where it meets the thermal phase transition. This point controls the robustness of quantum memory in a one-parameter family of frustration-free Hamiltonians interpolating between the toric code and a trivial product state, because the classical learning problem exactly reproduces the statistics of weak measurements on those quantum ground states.

What carries the argument

Replica field theory and renormalization-group flow applied to conditional correlation functions generated by Bayesian learning of local energy densities.

If this is right

  • The learning transition persists all the way from infinite temperature to the Ising critical temperature.
  • The intersection defines a tricritical point with distinct scaling properties.
  • Quantum memory in the topological phase remains stable under weak measurements arbitrarily close to the quantum phase transition.
  • The same framework applies to learning effects in more general classical and quantum many-body states.

Where Pith is reading between the lines

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

  • The tricritical point may mark the boundary beyond which measurement-induced transitions destroy topological order in related models.
  • Tensor-network or Monte Carlo methods used here could be adapted to detect similar learning transitions in three-dimensional Ising systems or Potts models.
  • Realizing the classical learning protocol in a quantum simulator would allow direct experimental test of the predicted memory robustness.

Load-bearing premise

The classical model of learning via conditional probabilities exactly captures the statistics of weak measurements performed on the ground states of the interpolated quantum Hamiltonians.

What would settle it

A numerical calculation that finds either no change in the decay of conditional correlations at the intersection or a different scaling exponent from the expected tricritical behavior would falsify the existence of the new tricritical point.

Figures

Figures reproduced from arXiv: 2504.12385 by Guo-Yi Zhu, Hidetoshi Nishimori, Malte P\"utz, Samuel J. Garratt, Simon Trebst.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
read the original abstract

Conditional probability distributions describe the effect of learning an initially unknown classical state through Bayesian inference. Here we demonstrate the existence of a \textit{learning transition}, having signatures in the long distance behavior of conditional correlation functions, in the two-dimensional classical Ising model. This transition, which arises when learning local energy densities, extends all the way from the infinite-temperature paramagnetic state down to the thermal critical state. The intersection of the line of learning transitions and the thermal Ising transition is a new tricritical point. Our model for learning also exactly describes the effects of weak measurements on ground states of frustration-free quantum Hamiltonians, which interpolate between the toric code and a paramagnet. Notably, the location of the above tricritical point implies that the quantum memory defined by the degenerate ground states in the topological phase is robust to weak measurement, even when the initial state is arbitrarily close to the quantum phase transition separating topological and trivial phases. Our analysis uses a replica field theory combined with the renormalization group, and we chart out the phase diagram using a combination of tensor network and Monte Carlo techniques. Our methods can be extended to study the more general effects of learning on both classical and quantum states. The learning induced critical states can be realized in classical or quantum devices.

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

1 major / 1 minor

Summary. The manuscript studies learning transitions in the 2D classical Ising model induced by Bayesian inference through conditional probability distributions when learning local energy densities. Replica field theory and renormalization group analysis, supplemented by tensor network and Monte Carlo simulations, are used to identify a line of learning transitions extending from the infinite-temperature paramagnet to the thermal Ising critical point, with their intersection constituting a new tricritical point. The classical model is asserted to exactly describe the effects of weak measurements on ground states of frustration-free quantum Hamiltonians interpolating between the toric code and a paramagnet, implying that quantum memory encoded in the topological phase remains robust to weak measurements even arbitrarily close to the topological-trivial quantum phase transition.

Significance. If the exact mapping between the classical conditional-probability model and quantum weak measurements holds in the relevant regime and the numerical results confirm the tricritical point, the work would be significant for connecting classical learning transitions to the stability of topological quantum memory. The multi-method approach combining replica field theory, RG analysis, tensor networks, and Monte Carlo simulations is a strength that allows charting the phase diagram comprehensively. The result offers a falsifiable prediction for quantum memory lifetime under weak measurements near criticality.

major comments (1)
  1. [Abstract and quantum mapping section] The assertion that the classical learning model 'exactly describes' the effects of weak measurements on the deformed toric-code ground states (abstract) is load-bearing for the quantum memory robustness claim. Without an explicit derivation of the mapping—including the form of the conditional probabilities, the measurement operators, and the regime of validity near the quantum critical point—the transfer of the classical tricritical point to a statement about quantum memory lifetime does not follow rigorously.
minor comments (1)
  1. The abstract would benefit from a short statement on the specific observables or correlation functions used to detect the learning transition signatures.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting the importance of rigorously establishing the quantum mapping. We agree that clarifying this connection will strengthen the presentation of our results on the robustness of quantum memory. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and quantum mapping section] The assertion that the classical learning model 'exactly describes' the effects of weak measurements on the deformed toric-code ground states (abstract) is load-bearing for the quantum memory robustness claim. Without an explicit derivation of the mapping—including the form of the conditional probabilities, the measurement operators, and the regime of validity near the quantum critical point—the transfer of the classical tricritical point to a statement about quantum memory lifetime does not follow rigorously.

    Authors: We thank the referee for this constructive comment. The quantum mapping section of the manuscript derives the correspondence by showing that weak measurements on the ground states of the interpolated toric-code Hamiltonian generate conditional probability distributions for local spin configurations that are identical to those arising in the Bayesian learning of local energy densities in the classical Ising model. The measurement operators are the deformed projectors whose expectation values yield the conditional probabilities P(observed configuration | local energy), and the weak-measurement regime ensures that the post-measurement state remains within the ground-state manifold. The exactness holds because the frustration-free property allows the quantum state to factorize into independent classical probabilities under these measurements. Nevertheless, we acknowledge that expanding this derivation with explicit operator expressions, the step-by-step reduction to the conditional probabilities, and a dedicated discussion of the validity range approaching the quantum critical point will make the transfer to the tricritical point and memory lifetime fully transparent. We will revise the quantum mapping section to include these details. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained.

full rationale

The paper derives the learning transition and tricritical point in the classical 2D Ising model via replica field theory plus RG, with the phase diagram charted by independent tensor-network and Monte Carlo computations. The classical tricritical point is located by these standard methods rather than by any fit or self-definition. The statement that the same conditional-probability model 'exactly describes' weak measurements on frustration-free deformed toric-code Hamiltonians is an asserted equivalence used to transfer the result; it does not reduce the classical derivation to its own inputs by construction, nor does any load-bearing step rely on a self-citation chain whose validity is internal to the paper. The analysis therefore remains externally falsifiable and is scored as non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the applicability of replica field theory to conditional probabilities and on an exact equivalence between a classical inference problem and quantum weak measurements; no free parameters or new entities are indicated in the abstract.

axioms (2)
  • standard math Replica field theory combined with the renormalization group accurately captures the long-distance behavior of conditional correlation functions arising from Bayesian learning of local energy densities.
    This is a standard technique in statistical mechanics for disordered or inference problems and is invoked to analyze the learning transition.
  • domain assumption The classical learning model exactly describes the effects of weak measurements on ground states of frustration-free quantum Hamiltonians interpolating between the toric code and a paramagnet.
    Directly asserted in the abstract as the basis for transferring the tricritical point result to quantum memory robustness.

pith-pipeline@v0.9.0 · 5773 in / 1688 out tokens · 90051 ms · 2026-05-22T19:53:35.483868+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 4 Pith papers

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

  1. Higher Nishimori Criticality and Exact Results at the Learning Transition of Deformed Toric Codes

    cond-mat.stat-mech 2026-04 unverdicted novelty 8.0

    The tricritical point at the learning transition of deformed toric codes is a higher Nishimori critical point where the Edwards-Anderson correlation exponent exactly matches the clean Ising spin exponent and c_eff is ...

  2. Bayesian phase transition for the critical Ising model: Enlarged replica symmetry in the epsilon expansion and in 2D

    cond-mat.stat-mech 2026-04 unverdicted novelty 7.0

    Measurement phases in the critical Ising model exhibit an enlarged replica symmetry, analogous to the Nishimori phenomenon, that exactly determines the Edwards-Anderson correlator exponent in 2D and near six dimensions.

  3. Revisiting Nishimori multicriticality through the lens of information measures

    cond-mat.stat-mech 2025-11 unverdicted novelty 6.0

    Generalized coherent information acts as a sharp phase-transition indicator over the entire p-T plane in the 2D ±J random-bond Ising model, yielding a high-precision multicritical point estimate p_c=0.1092212(4) with ...

  4. Born-rule statistical dynamical quantum phase transitions under measurement

    quant-ph 2026-05 unverdicted novelty 5.0

    Introduces statistical dynamical quantum phase transitions via Born-rule sampling of post-measurement states in quenched Ising chains, recovering DQPT features in high moments and proposing a measurement-based simulat...

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages · cited by 4 Pith papers · 2 internal anchors

  1. [1]

    I” and the tricritical point “T

    The pink shade high- lights the critical region with finite fraction of Ic, between the Ising critical point “I” and the tricritical point “T”, atγT ≈0.598(2). It is found that Ic yields a nonzero scaling dimensionη/ν at the tricritical point, in contrast to scaling invariant at the Ising critical point. (b) Ic along tuning temperature at a chosen finite ...

    work page 2004

  2. [2]

    E. T. Jaynes, Information Theory and Statistical Mechanics, Phys. Rev. 106, 620 (1957)

    work page 1957

  3. [3]

    E. T. Jaynes, Information Theory and Statistical Mechanics. II, Phys. Rev. 108, 171 (1957)

    work page 1957

  4. [4]

    Nishimori, Statistical Physics of Spin Glasses and Infor- mation Processing: An Introduction (Oxford University Press, 2001)

    H. Nishimori, Statistical Physics of Spin Glasses and Infor- mation Processing: An Introduction (Oxford University Press, 2001)

    work page 2001

  5. [5]

    Zdeborov ´a and F

    L. Zdeborov ´a and F. Krzakala, Statistical physics of inference: thresholds and algorithms, Adv. Phys. 65, 453 (2016)

    work page 2016

  6. [6]

    M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2010)

    work page 2010

  7. [7]

    M. P. Fisher, V . Khemani, A. Nahum, and S. Vijay, Random Quantum Circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023)

    work page 2023

  8. [8]

    A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, in Entanglement in Spin Chains: From The- ory to Quantum Technology Applications , edited by A. Bayat, S. Bose, and H. Johannesson (Springer International Publish- ing, Cham, 2022) pp. 211–249

    work page 2022

  9. [9]

    Iba, The Nishimori line and Bayesian statistics, J

    Y . Iba, The Nishimori line and Bayesian statistics, J. Phys. A 32, 3875 (1999)

    work page 1999

  10. [10]

    A. Y . Kitaev, Fault-tolerant quantum computation by anyons , Ann. Phys. (N. Y .)303, 2 (2003)

    work page 2003

  11. [11]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43, 4452 (2002)

    work page 2002

  12. [12]

    R. Fan, Y . Bao, E. Altman, and A. Vishwanath, Diagnostics of Mixed-State Topological Order and Breakdown of Quantum Memory, PRX Quantum 5, 020343 (2024)

    work page 2024

  13. [13]

    Chen and T

    Y .-H. Chen and T. Grover, Unconventional topological mixed- state transition and critical phase induced by self-dual coherent errors, Phys. Rev. B 110, 125152 (2024)

    work page 2024

  14. [14]

    Eckstein, B

    F. Eckstein, B. Han, S. Trebst, and G.-Y . Zhu, Robust Telepor- tation of a Surface Code and Cascade of Topological Quantum Phase Transitions, PRX Quantum 5, 040313 (2024)

    work page 2024

  15. [15]

    E. H. Chen, G.-Y . Zhu, R. Verresen, A. Seif, E. B¨aumer, D. Lay- den, N. Tantivasadakarn, G. Zhu, S. Sheldon, A. Vishwanath, S. Trebst, and A. Kandala, Nishimori transition across the error threshold for constant-depth quantum circuits, Nat. Phys. 21, 161 (2025)

    work page 2025

  16. [16]

    Ardonne, P

    E. Ardonne, P. Fendley, and E. Fradkin, Topological order and conformal quantum critical points, Ann. Phys. (N. Y .)310, 493 (2004)

    work page 2004

  17. [17]

    See Supplementary Material for numerical method and supple- mentary data and exactly solved models.,

    work page

  18. [18]

    G.-Y . Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori’s Cat: Stable Long-Range Entangle- ment from Finite-Depth Unitaries and Weak Measurements, Phys. Rev. Lett. 131, 200201 (2023)

    work page 2023

  19. [19]

    J. Y . Lee, W. Ji, Z. Bi, and M. P. A. Fisher, Decoding Measurement-Prepared Quantum Phases and Transitions: from Ising model to gauge theory, and beyond, preprint (2022), arXiv:2208.11699

    work page arXiv 2022

  20. [20]

    S. J. Garratt, Z. Weinstein, and E. Altman, Measurements Con- spire Nonlocally to Restructure Critical Quantum States, Phys. Rev. X 13, 021026 (2023)

    work page 2023

  21. [21]

    J. Y . Lee, C.-M. Jian, and C. Xu, Quantum Criticality Under Decoherence or Weak Measurement, PRX Quantum 4, 030317 (2023)

    work page 2023

  22. [22]

    Y . Bao, R. Fan, A. Vishwanath, and E. Altman, Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions, arXiv 2301.05687

    work page internal anchor Pith review Pith/arXiv arXiv

  23. [23]

    Weinstein, R

    Z. Weinstein, R. Sajith, E. Altman, and S. J. Garratt, Nonlocal- ity and entanglement in measured critical quantum Ising chains, Phys. Rev. B 107, 245132 (2023)

    work page 2023

  24. [24]

    Murciano, P

    S. Murciano, P. Sala, Y . Liu, R. S. K. Mong, and J. Alicea, Measurement-Altered Ising Quantum Criticality, Phys. Rev. X 13, 041042 (2023)

    work page 2023

  25. [25]

    P. Sala, S. Murciano, Y . Liu, and J. Alicea, Quantum Critical- ity Under Imperfect Teleportation, PRX Quantum 5, 030307 (2024)

    work page 2024

  26. [26]

    Z. Yang, D. Mao, and C.-M. Jian, Entanglement in a one- dimensional critical state after measurements, Phys. Rev. B108, 165120 (2023)

    work page 2023

  27. [27]

    R. A. Patil and A. W. W. Ludwig, Highly complex novel critical behavior from the intrinsic randomness of quantum mechanical measurements on critical ground states – a controlled renormal- ization group analysis, preprint (2024), arXiv:2409.02107

    work page arXiv 2024

  28. [28]

    Y . Liu, S. Murciano, D. F. Mross, and J. Alicea, Boundary tran- sitions from a single round of measurements on gapless quan- tum states, preprint (2024), arXiv:2412.07830

    work page arXiv 2024

  29. [29]

    Post-measurement Quantum Monte Carlo

    K. Baweja, D. J. Luitz, and S. J. Garratt, Post-measurement Quantum Monte Carlo, preprint (2024), arXiv:2410.13844

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  30. [30]

    Hoshino, M

    M. Hoshino, M. Oshikawa, and Y . Ashida, Entanglement swapping in critical quantum spin chains, preprint (2024), arXiv:2406.12377

    work page arXiv 2024

  31. [31]

    Tang and X

    Q. Tang and X. Wen, A critical state under weak measurement is not critical, preprint (2024), arXiv:2411.13705

    work page arXiv 2024

  32. [32]

    C. L. Henley, From classical to quantum dynamics at Rokhsar–Kivelson points, Journal of Physics: Condensed Mat- ter 16, S891 (2004)

    work page 2004

  33. [33]

    Gurarie and A

    V . Gurarie and A. W. W. Ludwig, Conformal field theory at central charge c=0 and two-dimensional critical systems with quenched disorder, in From Fields to Strings: Circumnavi- gating Theoretical Physics (World Scientific, 2005) pp. 1384– 1440

    work page 2005

  34. [34]

    Q. Wang, R. Vasseur, S. Trebst, A. W. W. Ludwig, and G.-Y . Zhu, Decoherence-induced self-dual criticality in topological states of matter, preprint (2025), arXiv:2502.14034

    work page arXiv 2025

  35. [35]

    Lang and H

    N. Lang and H. P. B ¨uchler, Entanglement transition in the pro- jective transverse field Ising model, Phys. Rev. B 102, 094204 (2020)

    work page 2020

  36. [36]

    X. Chen, Y . Li, M. P. A. Fisher, and A. Lucas, Emergent confor- mal symmetry in nonunitary random dynamics of free fermions, Phys. Rev. Res. 2, 033017 (2020)

    work page 2020

  37. [37]

    Y . Li, X. Chen, A. W. W. Ludwig, and M. P. A. Fisher, Confor- mal invariance and quantum nonlocality in critical hybrid cir- cuits, Phys. Rev. B 104, 104305 (2021)

    work page 2021

  38. [38]

    Zabalo, M

    A. Zabalo, M. J. Gullans, J. H. Wilson, R. Vasseur, A. W. W. Ludwig, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley, Op- erator Scaling Dimensions and Multifractality at Measurement- Induced Transitions, Phys. Rev. Lett.128, 050602 (2022)

    work page 2022

  39. [39]

    Kumar, K

    A. Kumar, K. Aziz, A. Chakraborty, A. W. W. Ludwig, S. Gopalakrishnan, J. H. Pixley, and R. Vasseur, Boundary transfer matrix spectrum of measurement-induced transitions, Phys. Rev. B 109, 014303 (2024)

    work page 2024

  40. [40]

    Barratt, U

    F. Barratt, U. Agrawal, A. C. Potter, S. Gopalakrishnan, and R. Vasseur, Transitions in the Learnability of Global Charges from Local Measurements, Phys. Rev. Lett. 129, 200602 (2022)

    work page 2022

  41. [41]

    Y . Bao, S. Choi, and E. Altman, Theory of the phase transi- tion in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020)

    work page 2020

  42. [42]

    Agrawal, J

    U. Agrawal, J. Lopez-Piqueres, R. Vasseur, S. Gopalakrishnan, 7 and A. C. Potter, Observing quantum measurement collapse as a learnability phase transition, Phys. Rev. X14, 041012 (2024)

    work page 2024

  43. [43]

    Ippoliti and V

    M. Ippoliti and V . Khemani, Learnability Transitions in Mon- itored Quantum Dynamics via Eavesdropper’s Classical Shad- ows, PRX Quantum 5, 020304 (2024)

    work page 2024

  44. [44]

    Singh, R

    H. Singh, R. Vasseur, A. C. Potter, and S. Gopalakrishnan, Mixed-state learnability transitions in monitored noisy quantum dynamics, preprint (2025), arXiv:2503.10308

    work page arXiv 2025

  45. [45]

    Nishimori, Internal Energy, Specific Heat and Correlation Function of the Bond-Random Ising Model, Prog

    H. Nishimori, Internal Energy, Specific Heat and Correlation Function of the Bond-Random Ising Model, Prog. Theor. Phys. 66, 1169 (1981)

    work page 1981

  46. [46]

    Honecker, M

    A. Honecker, M. Picco, and P. Pujol, Universality Class of the Nishimori Point in the 2D ±J Random-Bond Ising Model, Phys. Rev. Lett. 87, 047201 (2001)

    work page 2001

  47. [47]

    Merz and J

    F. Merz and J. T. Chalker, Two-dimensional random-bond Ising model, free fermions, and the network model, Phys. Rev. B 65, 054425 (2002)

    work page 2002

  48. [48]

    Ohzeki and J

    M. Ohzeki and J. L. Jacobsen, High-precision phase diagram of spin glasses from duality analysis with real-space renormaliza- tion and graph polynomials, J. Phys. A 48, 095001 (2015)

    work page 2015

  49. [49]

    T. Chen, E. Guo, W. Zhang, P. Zhang, and Y . Deng, Ten- sor network Monte Carlo simulations for the two-dimensional random-bond Ising model, Phys. Rev. B 111, 094201 (2025)

    work page 2025

  50. [50]

    P ¨utz, R

    M. P ¨utz, R. Vasseur, A. W. W. Ludwig, S. Trebst, and G.-Y . Zhu, to appear, (2025)

    work page 2025

  51. [51]

    Le Doussal and A

    P. Le Doussal and A. B. Harris, ϵ expansion for the Nishimori multicritical point of spin glasses, Phys. Rev. B40, 9249 (1989)

    work page 1989

  52. [52]

    Schumacher and M

    B. Schumacher and M. A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54, 2629 (1996)

    work page 1996

  53. [53]

    M. J. Gullans and D. A. Huse, Scalable Probes of Measurement- Induced Criticality, Phys. Rev. Lett. 125, 070606 (2020)

    work page 2020

  54. [54]

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y . You, Sponta- neous strong symmetry breaking in open systems: Purification perspective, Phys. Rev. B110, 155150 (2024)

    work page 2024

  55. [55]

    L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States, PRX Quantum 6, 010344 (2025)

    work page 2025

  56. [56]

    Zhang, Y

    C. Zhang, Y . Xu, J.-H. Zhang, C. Xu, Z. Bi, and Z.-X. Luo, Strong-to-weak spontaneous breaking of 1-form symme- try and intrinsically mixed topological order, preprint (2024), arXiv:2409.17530

    work page arXiv 2024

  57. [57]

    Nahum and J

    A. Nahum and J. Lykke Jacobsen, Bayesian critical points in classical lattice models, preprint (2025), arXiv:2504.01264

    work page arXiv 2025

  58. [58]

    Learning transitions in classical Ising models and deformed toric codes

    M. P ¨utz, S. J. Garratt, H. Nishimori, S. Trebst, and G.-Y . Zhu, Data for “Learning transitions in classical Ising models and deformed toric codes”, Zenodo 10.5281/zenodo.15227834 (2025). 8 Supplemental Material Numerical method: sampling and random tensor network In our numerical simulation, we first generate the sampless for the precise learning limit...

    work page doi:10.5281/zenodo.15227834 2025

  59. [59]

    partial order

    Increasing γ corresponds to increasing the correlation be- tween sij and σiσj. To calculate averages of (nonlinear) correlations over s, we use a replica trick. The partition function for the n replica theory is Zn = ∫dsP n(s) and ds= ∏i<j dsij. Integrating out s we have Zn ∼ ∑ σ1⋯σn e β N ∑α ∑j<k σα j σα k+ γ2 nN ∑α<β ∑j<k σα j σα k σβ j σβ k , (14) wher...

    work page