Exact mapping of a spin glass with correlated disorder to the pure Ising model

Hidetoshi Nishimori

arxiv: 2602.22657 · v2 · pith:OHO7OTTEnew · submitted 2026-02-26 · ❄️ cond-mat.dis-nn

Exact mapping of a spin glass with correlated disorder to the pure Ising model

Hidetoshi Nishimori This is my paper

Pith reviewed 2026-05-21 12:55 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn

keywords Ising spin glasscorrelated disorderNishimori lineexact mappinggauge symmetrypure Ising modelmulticritical pointuniversality class

0 comments

The pith

A spin glass with correlated disorder maps exactly to the pure Ising model along the Nishimori line

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

The paper constructs an Ising spin-glass model whose disorder distribution is tuned to interpolate continuously between the pure ferromagnetic Ising case and the standard Edwards-Anderson model. It proves that a Nishimori line exists on this model for which thermodynamic and correlation quantities reduce exactly to the corresponding quantities of the pure Ising model evaluated at one well-defined effective temperature. The equality holds on every lattice and in every dimension; the energy on the line equals the pure-model energy, while the specific heat equals the pure-model energy rather than its specific heat. Gauge-noninvariant observables such as magnetization and two-point correlations are identical to those of the pure Ising model at the same effective temperature. These identities imply that the multicritical point of the correlated model lies in the pure-Ising universality class.

Core claim

For the introduced model with correlated disorder, a Nishimori line can be defined on which the energy equals the energy of the pure Ising model at an effective temperature up to a constant and trivial factor, the specific heat equals the energy of the pure Ising model at that temperature, and gauge-noninvariant quantities such as magnetization and correlation functions are exactly equal to the corresponding pure-Ising quantities at the effective temperature; the mapping holds on any lattice in any dimension and shows that the leading critical behavior at the multicritical point is pure-Ising-like.

What carries the argument

The gauge symmetry generated by the specific functional form of the correlated disorder distribution, which defines the Nishimori line and produces the exact reduction of all physical quantities to those of the pure Ising model at a rescaled temperature.

If this is right

The leading critical behavior at the multicritical point belongs to the pure Ising universality class rather than the conventional Edwards-Anderson class.
All thermodynamic and correlation quantities on the Nishimori line are obtained directly from known pure-Ising results.
The exact relations hold uniformly for arbitrary lattices and spatial dimensions.
Correlations in the disorder distribution can change the universality class at the multicritical point.

Where Pith is reading between the lines

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

The mapping isolates the effect of disorder correlations as the mechanism that restores pure-Ising critical behavior.
Analogous gauge constructions may exist for other forms of correlated disorder or for related models such as Potts or clock spins.

Load-bearing premise

The disorder distribution must be chosen with a specific functional form that admits a gauge symmetry allowing the Nishimori line to be defined.

What would settle it

Numerical evaluation of the specific heat of the correlated model at a point on the Nishimori line, compared with the energy of the pure Ising model at the corresponding effective temperature on the same lattice, would confirm or refute the claimed equality.

read the original abstract

We introduce an Ising spin-glass model with correlated disorder which continuously interpolates between the pure ferromagnetic Ising model and the Edwards-Anderson model with symmetric disorder. For this model, we prove that a Nishimori line (NL) can be defined, analogously to the Edwards-Anderson model, on which physical quantities can be expressed exactly in terms of those of the pure Ising model at a well-defined effective temperature on any lattice in any dimension. For example, the energy on the NL is equal to the energy of the pure Ising model at the effective temperature up to a constant and a trivial factor. More remarkably, the specific heat on the NL equals the energy, not the specific heat, of the pure Ising model at the effective temperature, again up to a constant and a trivial factor. Gauge-noninvariant quantities such as the magnetization and correlation functions are exactly equal to the corresponding quantities of the pure Ising model at the effective temperature. These exact relations imply that the leading critical behavior at that multicritical point for the disorder-correlated model is pure-Ising-like, in contrast to the conventional multicritical universality class of the standard Edwards-Anderson model. Our results motivate further investigations of the relatively unexplored topic of correlations in disorder in spin glasses and related problems.

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.

Desk Editor's Note private letter to a colleague

This paper gives an exact mapping from a tuned correlated-disorder spin glass to the pure Ising model on the Nishimori line, with the specific heat on the line equal to the pure-model energy.

read the letter

The main takeaway is that a specific correlated bond distribution lets the disorder-averaged quantities on the Nishimori line match those of the pure ferromagnetic Ising model at a shifted temperature. Energy and magnetization come out directly, and the specific heat on the line equals the energy of the pure model rather than its specific heat. This forces pure-Ising critical behavior at the multicritical point instead of the usual Edwards-Anderson multicritical class. The mapping holds on any lattice in any dimension because the gauge symmetry survives in the joint distribution of the bonds. The construction uses a single tunable parameter to interpolate continuously between the pure Ising case and the symmetric Edwards-Anderson model while keeping the symmetry intact. That interpolation and the resulting identities, especially the specific-heat relation, are the concrete new pieces. The gauge argument itself is standard but applied cleanly to this new distribution, and the derivations avoid extra assumptions about bipartiteness or dimension. The soft spot is that the correlation structure is chosen precisely to preserve the gauge symmetry. This makes the model a controlled example rather than a generic treatment of arbitrary disorder correlations that might occur in materials. The claim about motivating study of real correlated disorder is reasonable but rests on the reader accepting that this particular form is representative enough to be useful. The paper is aimed at people who work on spin glasses, the Nishimori line, or exact mappings in disordered systems. A reader who wants verifiable identities or a mechanism to change the universality class at the multicritical point will get something concrete from it. It deserves a serious referee because the central mapping is an exact, checkable result grounded in the gauge symmetry on arbitrary graphs.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces an Ising spin-glass model with a specific form of correlated disorder that continuously interpolates between the pure ferromagnetic Ising model and the Edwards-Anderson model. The central result is a proof that a Nishimori line exists on which the disorder-averaged energy, specific heat, magnetization, and correlation functions of the correlated-disorder model are exactly equal (up to constants and trivial factors) to the corresponding quantities of the pure Ising model evaluated at a well-defined effective temperature. The mapping relies on gauge symmetry of the joint distribution and holds on arbitrary lattices in any dimension, implying that the multicritical point exhibits pure-Ising critical behavior rather than the conventional Edwards-Anderson multicritical universality class.

Significance. If the exact mappings are correct, the work is significant because exact relations between a spin-glass model and the pure Ising ferromagnet are rare and valuable; the gauge-symmetry construction on general graphs, the non-standard specific-heat-to-energy relation, and the implication for critical behavior at the multicritical point all constitute clear advances. The paper explicitly credits the generality to arbitrary lattices and the new correlated-disorder construction as strengths.

major comments (2)

[Model construction] Model construction preceding the proof: the specific functional form chosen for the correlated bond distribution is stated to permit a gauge symmetry that maps the joint distribution onto itself, but the manuscript does not provide an explicit verification that the chosen probability measure is invariant under the gauge transformation for arbitrary graphs; this step is load-bearing for the subsequent identities.
[Proof of mappings] Derivation of the specific-heat identity: the claim that the specific heat on the Nishimori line equals the energy (rather than the specific heat) of the pure Ising model at the effective temperature follows from differentiating the mapped partition function, yet the manuscript does not show the intermediate step that converts the second derivative into a first derivative of the pure-model energy; an expanded calculation would confirm the factor and constant.

minor comments (3)

The abstract uses the phrase 'more remarkably' for the specific-heat relation; repeating this emphasis in the introduction would help readers locate the most non-standard result.
Notation for the effective temperature and the disorder-correlation parameter should be introduced once with a clear symbol table or sentence to avoid later ambiguity when comparing to the pure Ising temperature.
The statement that the mapping holds 'on any lattice in any dimension' is repeated; a single consolidated sentence in the introduction citing the arbitrary-graph proof would reduce redundancy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

Referee: [Model construction] Model construction preceding the proof: the specific functional form chosen for the correlated bond distribution is stated to permit a gauge symmetry that maps the joint distribution onto itself, but the manuscript does not provide an explicit verification that the chosen probability measure is invariant under the gauge transformation for arbitrary graphs; this step is load-bearing for the subsequent identities.

Authors: We agree that an explicit verification of the invariance strengthens the presentation. The gauge invariance follows from the local, bond-wise definition of the correlated disorder distribution, which is constructed to be invariant under simultaneous sign flips of bonds and spins. In the revised manuscript we have added a dedicated paragraph in Section II that explicitly verifies this invariance for an arbitrary graph: we show that the joint probability measure is unchanged because the transformation acts as a bijection on the bond variables while preserving the product structure of the distribution. revision: yes
Referee: [Proof of mappings] Derivation of the specific-heat identity: the claim that the specific heat on the Nishimori line equals the energy (rather than the specific heat) of the pure Ising model at the effective temperature follows from differentiating the mapped partition function, yet the manuscript does not show the intermediate step that converts the second derivative into a first derivative of the pure-model energy; an expanded calculation would confirm the factor and constant.

Authors: We thank the referee for this observation. The specific-heat identity arises because differentiation of the mapped partition function with respect to temperature on the Nishimori line produces a first derivative of the pure-model energy rather than its second derivative. In the revised manuscript we have expanded this derivation (new subsection in Section III) to display the intermediate steps explicitly, including the precise prefactors that convert the second derivative into the energy of the pure Ising model together with the additive constant. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines a new correlated-disorder distribution chosen specifically to preserve gauge symmetry, then derives the exact mapping to the pure Ising model on the Nishimori line by direct application of that symmetry to the joint spin-disorder measure. All listed identities (energy, specific heat, magnetization, correlations) follow mathematically from the gauge transformation without reducing to fitted parameters, prior equations by construction, or load-bearing self-citations. The proof is presented for arbitrary graphs and does not invoke uniqueness theorems or ansatzes from the author's earlier work as the central justification.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the specific construction of the correlated disorder that permits the gauge symmetry and Nishimori line; no new particles or forces are postulated, and the effective temperature emerges from the mapping rather than being fitted.

free parameters (1)

disorder correlation parameter
A tunable parameter controlling the strength of disorder correlations that interpolates between pure Ising and Edwards-Anderson limits; its value on the Nishimori line is fixed by the mapping definition rather than data fitting.

axioms (1)

domain assumption The chosen correlated disorder distribution preserves a gauge symmetry analogous to that in the standard Edwards-Anderson model, allowing definition of the Nishimori line.
Invoked in the model introduction to enable the exact mapping proof.

pith-pipeline@v0.9.0 · 5751 in / 1670 out tokens · 72741 ms · 2026-05-21T12:55:05.830424+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that a Nishimori line (NL) can be defined... physical quantities can be expressed exactly in terms of those of the pure Ising model at a well-defined effective temperature... the specific heat on the NL equals the energy, not the specific heat, of the pure Ising model
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

tanh β_e(β_p) = tanh² β_p

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

[1]

Exact mapping of a spin glass with correlated disorder to the pure Ising model

and its Parisi solution via replica symmetry breaking [6], it does not always reflect the complexities of realistic systems. In many physical contexts, correlations between disorder variables are inevitably present due to thermal history, structural constraints, or underlying microscopic interactions. Understanding the effects of such correla- tions remai...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Charbonneau, E

P. Charbonneau, E. Marinari, G. Parisi, F. Ricci- tersenghi, G. Sicuro, F. Zamponi, and M. M´ ezard,Spin Glass Theory and Far Beyond: Replica Symmetry Break- ing after 40 Years(World Scientific, Singapore, 2023)

work page 2023
[3]

Nishimori,Statistical Physics of Spin Glasses and In- formation Processing: An Introduction(Oxford Univer- sity Press, Oxford, 2001)

H. Nishimori,Statistical Physics of Spin Glasses and In- formation Processing: An Introduction(Oxford Univer- sity Press, Oxford, 2001)

work page 2001
[4]

J. A. Mydosh,Spin glasses: An experimental introduction (CRC Press, London, 1993)

work page 1993
[5]

S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F5, 965 (1975)

work page 1975
[6]

Sherrington and S

D. Sherrington and S. Kirkpatrick, Solvable model of a 9 spin glass, Phys. Rev. Lett.35, 1792 (1975)

work page 1975
[7]

Parisi, A sequence of approximated solutions to the SK model for spin glasses, J

G. Parisi, A sequence of approximated solutions to the SK model for spin glasses, J. Phys. A13, L115 (1980)

work page 1980
[8]

J. A. Hoyos, N. Laflorencie, A. P. Vieira, and T. Vojta, Protecting clean critical points by local disorder correla- tions, Europhys. Lett.93, 30004 (2011)

work page 2011
[9]

Bonzom, R

V. Bonzom, R. Gurau, and M. Smerlak, Universality in p-spin glasses with correlated disorder, J. Stat. Mech. 2013, L02003 (2013)

work page 2013
[10]

A. G. Cavaliere and A. Pelissetto, Disordered Ising model with correlated frustration, J. Phys. A52, 174002 (2019)

work page 2019
[11]

M¨ unster, C

L. M¨ unster, C. Norrenbrock, A. K. Hartmann, and A. P. Young, Ordering behavior of the two-dimensional Ising spin glass with long-range correlated disorder, Phys. Rev. E103, 042117 (2021)

work page 2021
[12]

Nishimori, Analyticity of the energy in an Ising spin glass with correlated disorder, J

H. Nishimori, Analyticity of the energy in an Ising spin glass with correlated disorder, J. Phys. A55, 045001 (2022)

work page 2022
[13]

Nishimori, Anomalous distribution of magnetization in an Ising spin glass with correlated disorder, Phys

H. Nishimori, Anomalous distribution of magnetization in an Ising spin glass with correlated disorder, Phys. Rev. E110, 064108 (2024)

work page 2024
[14]

Nishimori, Instability of the ferromagnetic phase un- der random fields in an Ising spin glass with correlated disorder, Phys

H. Nishimori, Instability of the ferromagnetic phase un- der random fields in an Ising spin glass with correlated disorder, Phys. Rev. E111, 044109 (2025)

work page 2025
[15]

Nishimori, M

H. Nishimori, M. Ohzeki, and M. Okuyama, Temperature chaos as a logical consequence of the reentrant transition in spin glasses, Phys. Rev. E112, 044140 (2025)

work page 2025
[16]

Klesse and S

R. Klesse and S. Frank, Quantum error correction in spatially correlated quantum noise, Phys. Rev. Lett.95, 230503 (2005)

work page 2005
[17]

J. P. Clemens, S. Siddiqui, and J. Gea-Banacloche, Quan- tum error correction against correlated noise, Phys. Rev. A69, 062313 (2004)

work page 2004
[18]

Aharonov and M

D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error rate, SIAM J. Comp. 38, 1207 (2008)

work page 2008
[19]

Preskill, Sufficient condition on noise correlations for scalable quantum computing, Quantum Inf

J. Preskill, Sufficient condition on noise correlations for scalable quantum computing, Quantum Inf. Comput.13, 181 (2013)

work page 2013
[20]

C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D’Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois, and R. McDermott, Correlated charge noise and relaxation errors in superconducting qubits, Nature594, 369 (2021)

work page 2021
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Nishimori, Exact results and critical properties of the Ising model with competing interactions, J

H. Nishimori, Exact results and critical properties of the Ising model with competing interactions, J. Phys. C13, 4071 (1980)

work page 1980
[22]

Nishimori, Internal energy, specific heat and corre- lation function of the bond-random Ising model, Prog

H. Nishimori, Internal energy, specific heat and corre- lation function of the bond-random Ising model, Prog. Theor. Phys.66, 1169 (1981)

work page 1981
[23]

Nishimori and G

H. Nishimori and G. Ortiz,Elements of Phase Transi- tions and Critical Phenomena(Oxford University Press, Oxford, 2010)

work page 2010
[24]

Honecker, M

A. Honecker, M. Picco, and P. Pujol, Universality class of the Nishimori point in the 2D±Jrandom-bond Ising model, Phys. Rev. Lett.87, 047201 (2001)

work page 2001
[25]

Le Doussal and A

P. Le Doussal and A. B. Harris, Location of the Ising spin-glass multicritical point on Nishimori’s line, Phys. Rev. Lett.61, 625 (1988)

work page 1988
[26]

Hasenbusch, A

M. Hasenbusch, A. Pelissetto, and E. Vicari, Critical be- havior of three-dimensional Ising spin glass models, Phys. Rev. B76, 094402 (2007)

work page 2007
[27]

Hasenbusch, A

M. Hasenbusch, A. Pelissetto, and E. Vicari, Critical be- havior of three-dimensional Ising spin glass models, Phy. Rev. B78, 214205 (2008)

work page 2008
[28]

Parisen Toldin, A

F. Parisen Toldin, A. Pelissetto, and E. Vicari, Univer- sality of the glassy transitions in the two-dimensional±J Ising model, Phys. Rev. E82, 1 (2010)

work page 2010
[29]

Ceccarelli, A

G. Ceccarelli, A. Pelissetto, and E. Vicari, Ferromagnetic-glassy transitions in three-dimensional Ising spin glasses, Phys. Rev. B84, 134202 (2011)

work page 2011
[30]

H. G. Katzgraber, M. K¨ orner, and A. P. Young, Univer- sality in three-dimensional Ising spin glasses: A Monte Carlo study, Phys. Rev. B73, 224432 (2006)

work page 2006
[31]

A. B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C7, 1671 (1974)

work page 1974
[32]

Dennis, A

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topo- logical quantum memory, J. Math. Phys.43, 4452 (2002)

work page 2002
[33]

C. Wang, J. Harrington, and J. Preskill, Confinement- higgs transition in a disordered gauge theory and the ac- curacy threshold for quantum memory, Ann. Phys.303, 31 (2003)

work page 2003

[1] [1]

Exact mapping of a spin glass with correlated disorder to the pure Ising model

and its Parisi solution via replica symmetry breaking [6], it does not always reflect the complexities of realistic systems. In many physical contexts, correlations between disorder variables are inevitably present due to thermal history, structural constraints, or underlying microscopic interactions. Understanding the effects of such correla- tions remai...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

Charbonneau, E

P. Charbonneau, E. Marinari, G. Parisi, F. Ricci- tersenghi, G. Sicuro, F. Zamponi, and M. M´ ezard,Spin Glass Theory and Far Beyond: Replica Symmetry Break- ing after 40 Years(World Scientific, Singapore, 2023)

work page 2023

[3] [3]

Nishimori,Statistical Physics of Spin Glasses and In- formation Processing: An Introduction(Oxford Univer- sity Press, Oxford, 2001)

H. Nishimori,Statistical Physics of Spin Glasses and In- formation Processing: An Introduction(Oxford Univer- sity Press, Oxford, 2001)

work page 2001

[4] [4]

J. A. Mydosh,Spin glasses: An experimental introduction (CRC Press, London, 1993)

work page 1993

[5] [5]

S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F5, 965 (1975)

work page 1975

[6] [6]

Sherrington and S

D. Sherrington and S. Kirkpatrick, Solvable model of a 9 spin glass, Phys. Rev. Lett.35, 1792 (1975)

work page 1975

[7] [7]

Parisi, A sequence of approximated solutions to the SK model for spin glasses, J

G. Parisi, A sequence of approximated solutions to the SK model for spin glasses, J. Phys. A13, L115 (1980)

work page 1980

[8] [8]

J. A. Hoyos, N. Laflorencie, A. P. Vieira, and T. Vojta, Protecting clean critical points by local disorder correla- tions, Europhys. Lett.93, 30004 (2011)

work page 2011

[9] [9]

Bonzom, R

V. Bonzom, R. Gurau, and M. Smerlak, Universality in p-spin glasses with correlated disorder, J. Stat. Mech. 2013, L02003 (2013)

work page 2013

[10] [10]

A. G. Cavaliere and A. Pelissetto, Disordered Ising model with correlated frustration, J. Phys. A52, 174002 (2019)

work page 2019

[11] [11]

M¨ unster, C

L. M¨ unster, C. Norrenbrock, A. K. Hartmann, and A. P. Young, Ordering behavior of the two-dimensional Ising spin glass with long-range correlated disorder, Phys. Rev. E103, 042117 (2021)

work page 2021

[12] [12]

Nishimori, Analyticity of the energy in an Ising spin glass with correlated disorder, J

H. Nishimori, Analyticity of the energy in an Ising spin glass with correlated disorder, J. Phys. A55, 045001 (2022)

work page 2022

[13] [13]

Nishimori, Anomalous distribution of magnetization in an Ising spin glass with correlated disorder, Phys

H. Nishimori, Anomalous distribution of magnetization in an Ising spin glass with correlated disorder, Phys. Rev. E110, 064108 (2024)

work page 2024

[14] [14]

Nishimori, Instability of the ferromagnetic phase un- der random fields in an Ising spin glass with correlated disorder, Phys

H. Nishimori, Instability of the ferromagnetic phase un- der random fields in an Ising spin glass with correlated disorder, Phys. Rev. E111, 044109 (2025)

work page 2025

[15] [15]

Nishimori, M

H. Nishimori, M. Ohzeki, and M. Okuyama, Temperature chaos as a logical consequence of the reentrant transition in spin glasses, Phys. Rev. E112, 044140 (2025)

work page 2025

[16] [16]

Klesse and S

R. Klesse and S. Frank, Quantum error correction in spatially correlated quantum noise, Phys. Rev. Lett.95, 230503 (2005)

work page 2005

[17] [17]

J. P. Clemens, S. Siddiqui, and J. Gea-Banacloche, Quan- tum error correction against correlated noise, Phys. Rev. A69, 062313 (2004)

work page 2004

[18] [18]

Aharonov and M

D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error rate, SIAM J. Comp. 38, 1207 (2008)

work page 2008

[19] [19]

Preskill, Sufficient condition on noise correlations for scalable quantum computing, Quantum Inf

J. Preskill, Sufficient condition on noise correlations for scalable quantum computing, Quantum Inf. Comput.13, 181 (2013)

work page 2013

[20] [20]

C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D’Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois, and R. McDermott, Correlated charge noise and relaxation errors in superconducting qubits, Nature594, 369 (2021)

work page 2021

[21] [21]

Nishimori, Exact results and critical properties of the Ising model with competing interactions, J

H. Nishimori, Exact results and critical properties of the Ising model with competing interactions, J. Phys. C13, 4071 (1980)

work page 1980

[22] [22]

Nishimori, Internal energy, specific heat and corre- lation function of the bond-random Ising model, Prog

H. Nishimori, Internal energy, specific heat and corre- lation function of the bond-random Ising model, Prog. Theor. Phys.66, 1169 (1981)

work page 1981

[23] [23]

Nishimori and G

H. Nishimori and G. Ortiz,Elements of Phase Transi- tions and Critical Phenomena(Oxford University Press, Oxford, 2010)

work page 2010

[24] [24]

Honecker, M

A. Honecker, M. Picco, and P. Pujol, Universality class of the Nishimori point in the 2D±Jrandom-bond Ising model, Phys. Rev. Lett.87, 047201 (2001)

work page 2001

[25] [25]

Le Doussal and A

P. Le Doussal and A. B. Harris, Location of the Ising spin-glass multicritical point on Nishimori’s line, Phys. Rev. Lett.61, 625 (1988)

work page 1988

[26] [26]

Hasenbusch, A

M. Hasenbusch, A. Pelissetto, and E. Vicari, Critical be- havior of three-dimensional Ising spin glass models, Phys. Rev. B76, 094402 (2007)

work page 2007

[27] [27]

Hasenbusch, A

M. Hasenbusch, A. Pelissetto, and E. Vicari, Critical be- havior of three-dimensional Ising spin glass models, Phy. Rev. B78, 214205 (2008)

work page 2008

[28] [28]

Parisen Toldin, A

F. Parisen Toldin, A. Pelissetto, and E. Vicari, Univer- sality of the glassy transitions in the two-dimensional±J Ising model, Phys. Rev. E82, 1 (2010)

work page 2010

[29] [29]

Ceccarelli, A

G. Ceccarelli, A. Pelissetto, and E. Vicari, Ferromagnetic-glassy transitions in three-dimensional Ising spin glasses, Phys. Rev. B84, 134202 (2011)

work page 2011

[30] [30]

H. G. Katzgraber, M. K¨ orner, and A. P. Young, Univer- sality in three-dimensional Ising spin glasses: A Monte Carlo study, Phys. Rev. B73, 224432 (2006)

work page 2006

[31] [31]

A. B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C7, 1671 (1974)

work page 1974

[32] [32]

Dennis, A

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topo- logical quantum memory, J. Math. Phys.43, 4452 (2002)

work page 2002

[33] [33]

C. Wang, J. Harrington, and J. Preskill, Confinement- higgs transition in a disordered gauge theory and the ac- curacy threshold for quantum memory, Ann. Phys.303, 31 (2003)

work page 2003