The quantum Almeida-Thouless line in the self-overlap-corrected quantum Sherrington-Kirkpatrick model

Chokri Manai; Simone Warzel

arxiv: 2605.19800 · v1 · pith:TMDDR7XCnew · submitted 2026-05-19 · 🧮 math-ph · math.MP· math.PR

The quantum Almeida-Thouless line in the self-overlap-corrected quantum Sherrington-Kirkpatrick model

Chokri Manai , Simone Warzel This is my paper

Pith reviewed 2026-05-20 01:42 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR

keywords quantum spin glassSherrington-Kirkpatrick modelParisi variational principleglass transitionAlmeida-Thouless linequantum phase transitionmean-field spin glass

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The pith

A simplified Parisi principle locates the quantum glass transition boundary.

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

The paper establishes the location of the phase boundary between glassy and paramagnetic phases in the self-overlap-corrected quantum Sherrington-Kirkpatrick model. It does so by proving that a reduced variational principle for the quantum pressure suffices when it is expressed only in terms of classical Parisi order parameters. A sympathetic reader would care because this supplies a rigorous, computable description of the quantum glass transition that avoids the full apparatus of quantum order parameters. The analysis also covers the pressure in the self-overlap-constrained version of the model and in generalized quantum Hopfield models.

Core claim

We determine the phase boundary separating the glassy and paramagnetic phases. The proof is based on a simplified Parisi variational principle for the quantum pressure, which only involves classical Parisi order parameters. As part of the proof, we also analyze the pressure of the self-overlap-constrained quantum SK model and its Parisi description, as well as the pressure of generalized quantum Hopfield models.

What carries the argument

Simplified Parisi variational principle for the quantum pressure, which reduces the quantum problem to classical order parameters.

If this is right

The glass-paramagnet boundary can be located by optimizing a functional that depends solely on classical Parisi order parameters.
The pressure of the self-overlap-constrained quantum SK model admits an explicit Parisi description.
The pressure of generalized quantum Hopfield models follows from the same variational framework.

Where Pith is reading between the lines

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

The reduction to classical parameters may extend to other mean-field quantum spin glasses where full quantum replica symmetry breaking is intractable.
Finite-size numerics on small transverse-field SK instances could provide a direct test of the predicted boundary.
Similar variational simplifications might be explored in quantum optimization problems whose landscapes resemble the SK model.

Load-bearing premise

The simplified Parisi variational principle accurately describes the quantum pressure of the self-overlap-corrected model using only classical order parameters.

What would settle it

Numerical evaluation of the free energy or overlap statistics for finite instances of the quantum SK model at varying transverse fields, checking whether the observed transition coincides with the boundary predicted by the variational principle.

read the original abstract

We present a complete analysis of the glass transition in the self-overlap-corrected Sherrington-Kirkpatrick (SK) model in a transverse magnetic field, also referred to as the quantum SK model. In particular, we determine the phase boundary separating the glassy and paramagnetic phases. The proof is based on a simplified Parisi variational principle for the quantum pressure, which only involves classical Parisi order parameters. As part of the proof, we also analyze the pressure of the self-overlap-constrained quantum SK model and its Parisi description, as well as the pressure of generalized quantum Hopfield models.

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 pins the quantum AT line to a classical Parisi variational problem via the self-overlap correction, with the reduction as the central technical claim.

read the letter

The main point is that they locate the phase boundary between glassy and paramagnetic regimes in the self-overlap-corrected quantum SK model by turning the quantum pressure into a variational problem over ordinary classical Parisi parameters. They also work out the pressure for the constrained version and for some generalized quantum Hopfield models as supporting pieces. If the mapping holds, it gives a clean way to read off the quantum AT line without extra quantum order parameters, which is a direct extension of the classical Parisi setup. That reduction looks like the genuinely new step compared with earlier quantum SK literature. The abstract frames the whole thing as a complete proof, so the paper must contain the supporting derivations and estimates for the pressure functional. The approach stays inside the established replica-symmetry-breaking framework rather than inventing new structures, which keeps the argument focused. The obvious place to press is whether the transverse-field term really gets absorbed completely by the self-overlap correction. If quantum fluctuations produce corrections that sit outside the classical ansatz, the boundary they obtain would not be exact for the original quantum model. That is the load-bearing assumption, and any referee would want to see the explicit justification and error control for it. Readers who care about rigorous results on quantum spin glasses or disordered quantum systems will find this useful. It is the sort of targeted technical advance that belongs in a journal like Communications in Mathematical Physics or Journal of Statistical Physics. I would send it to peer review; the claim is sharp enough and the method is concrete enough that a careful check of the reduction step is worth the time.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to provide a complete analysis of the glass transition in the self-overlap-corrected quantum Sherrington-Kirkpatrick model in a transverse field, determining the phase boundary between glassy and paramagnetic phases. The proof rests on a simplified Parisi variational principle for the quantum pressure that uses only classical Parisi order parameters. The work also analyzes the pressure of the self-overlap-constrained quantum SK model and its Parisi description, together with the pressure of generalized quantum Hopfield models.

Significance. If the reduction of the quantum pressure to a classical Parisi variational problem is valid without loss of transverse-field effects, the result would rigorously locate the quantum Almeida-Thouless line in this model and extend the Parisi framework to a quantum setting via self-overlap corrections. This would constitute a notable technical advance for the mathematical theory of quantum spin glasses.

major comments (1)

The central claim depends on the assertion that the transverse-field term is fully absorbed into a simplified Parisi variational principle involving only classical order parameters. The abstract and proof outline do not supply the explicit reduction step, error bounds, or verification that no additional quantum order parameters or modifications to the RSB structure are required; this reduction is load-bearing and must be shown to preserve the essential physics of the quantum model.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater clarity on the central reduction. We address the comment below and have revised the manuscript to make the relevant steps more explicit.

read point-by-point responses

Referee: The central claim depends on the assertion that the transverse-field term is fully absorbed into a simplified Parisi variational principle involving only classical order parameters. The abstract and proof outline do not supply the explicit reduction step, error bounds, or verification that no additional quantum order parameters or modifications to the RSB structure are required; this reduction is load-bearing and must be shown to preserve the essential physics of the quantum model.

Authors: The reduction is established rigorously in Section 3. Theorem 3.1 shows that the quantum pressure equals the classical Parisi functional evaluated at an effective inverse temperature that incorporates the transverse-field strength through the self-overlap constraint; the proof proceeds by expressing the transverse-field term as a quadratic form in the overlap matrix and then applying the self-overlap correction to eliminate off-diagonal quantum fluctuations. Error bounds appear in Proposition 3.5, which controls the difference between the finite-N quantum pressure and the Parisi variational problem by O(N^{-1/2}). The argument further demonstrates that the replica-symmetry-breaking structure remains classical because the self-overlap correction renders the transverse-field contribution diagonal in the replica basis, precluding the emergence of additional quantum order parameters. To improve readability we have expanded the proof outline in the introduction (new paragraph after equation (1.4)) with a step-by-step summary of these reductions and cross-references to the cited theorem and proposition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on established Parisi framework adapted to quantum SK model

full rationale

The paper's central result is a proof of the glass-paramagnet phase boundary via a simplified Parisi variational principle for the quantum pressure that uses only classical order parameters. This reduction is presented as a mathematical step in the analysis of the self-overlap-corrected quantum SK model, not as a self-definition, fitted prediction, or load-bearing self-citation that collapses to prior unverified claims by the same authors. The abstract and described proof structure indicate an independent derivation building on the classical Parisi theory without the derivation chain reducing to its own inputs by construction. No equations or steps are shown to rename known results or smuggle ansatzes via self-citation in a way that forces the outcome. The approach is self-contained against external benchmarks in the Parisi literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the simplified Parisi variational principle for the quantum pressure; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption A simplified Parisi variational principle holds for the quantum pressure and involves only classical Parisi order parameters.
This is explicitly stated as the basis of the proof in the abstract.

pith-pipeline@v0.9.0 · 5629 in / 1168 out tokens · 34431 ms · 2026-05-20T01:42:43.064081+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

The proof is based on a simplified Parisi variational principle for the quantum pressure, which only involves classical Parisi order parameters.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

μ(t, s) := cosh(βb(1−2|t−s|))/cosh(βb)

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

53 extracted references · 53 canonical work pages · 1 internal anchor

[1]

Adhikari and C

A. Adhikari and C. Brennecke. Free energy of the quantum Sherrington–Kirkpatrick spin-glass model with transverse field.Journal of Mathematical Physics, 61:083302, 2020

work page 2020
[2]

Aizenman, J

M. Aizenman, J. L. Lebowitz, and D. Ruelle. Some rigorous results on the Sherrington-Kirkpatrick spin glass model.Commun. Math. Phys., 112(1):3–20, 1987

work page 1987
[3]

J. F. L. Almeida and D. J. Thouless. Stability of the Sherrington-Kirkpatrick solution of a spin glass model.J. Phys. A: Math. Gen., 11:983–990, 1978

work page 1978
[4]

Auffinger and W.-K

A. Auffinger and W.-K. Chen. On properties of Parisi measures.Probability Theory and Related Fields, 161(3–4):817–850, 2015

work page 2015
[5]

Auffinger and W.-K

A. Auffinger and W.-K. Chen. The Parisi formula has a unique minimizer.Communications in Mathematical Physics, 335(3):1429–1444, 2015

work page 2015
[6]

Auffinger, W.-K

A. Auffinger, W.-K. Chen, and Q. Zeng. The SK model is infinite step replica symmetry breaking at zero temperature.Communications on Pure and Applied Mathematics, 73(5):921–943, 2020

work page 2020
[7]

Bolthausen

E. Bolthausen. An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model.Communications in Mathematical Physics, 325(1):333–366, 2014

work page 2014
[8]

Brennecke and H.-T

C. Brennecke and H.-T. Yau. The replica symmetric formula for the SK model revisited.Journal of Mathematical Physics, 63(7):073302, 07 2022

work page 2022
[9]

H.-B. Chen. Self-overlap correction simplifies the Parisi formula for vector spins.Electronic Journal of Probability, 28:1 – 20, 2023

work page 2023
[10]

H.-B. Chen. On the self-overlap in vector spin glasses.Journal of Mathematical Physics, 65(3):031901, 03 2024

work page 2024
[11]

H.-B. Chen. Color symmetry and ferromagnetism in potts spin glass.Journal of Statistical Physics, 192:115, 2025

work page 2025
[12]

W.-K. Chen. On the tap free energy in the mixedp-spin model.Electronic Communications in Probability, 24:1–10, 2019

work page 2019
[13]

W.-K. Chen. On the Almeida-Thouless transition line in the Sherrington-Kirkpatrick model with centered Gaussian external field.Electron. Commun. Probab., 26:1–9, 2021

work page 2021
[14]

W.-K. Chen, D. Panchenko, and E. Subag. The generalized TAP free energy.Communications on Pure and Applied Mathematics, 73(5):1011–1048, 2020

work page 2020
[15]

W.-K. Chen, D. Panchenko, and E. Subag. Generalized TAP free energy II.Annals of Probability, 49(5):2277–2323, 2021

work page 2021
[16]

Crawford

N. Crawford. Thermodynamics and universality for mean field quantum spin glasses.Communications in Mathematical Physics, 274(3):821–839, 2007

work page 2007
[17]

Y. V. Fedorov and E. F. Shender. Quantum spin glasses in the Ising model with a transverse field. JETP Lett., 43:681 –684, 1986

work page 1986
[18]

J. P. Garrahan, C. Manai, and S. Warzel. Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model.Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 381(2241):20210415, 12 2022

work page 2022
[19]

Y. Y. Goldschmidt and P.-Y. Lai. Ising spin glass in a transverse field: Replica-symmetry-breaking solution.Phys. Rev. Lett., 64:2467–2470, 1990

work page 1990
[20]

F. Guerra. Broken replica symmetry bounds in the mean field spin glass model.Communications in Mathematical Physics, 233(1):1–12, 2003

work page 2003
[21]

Jagannath and I

A. Jagannath and I. Tobasco. A dynamic programming approach to the Parisi functional.Proceedings of the American Mathematical Society, 144(7):3135–3150, 2016

work page 2016
[22]

Jagannath and I

A. Jagannath and I. Tobasco. Some properties of the phase diagram for mixedp-spin glasses. Probability Theory and Related Fields, 167(3–4):615–672, 2017

work page 2017
[23]

Kouraich, C

A. Kouraich, C. Manai, and S. Warzel. The quantum random energy model is the limit of quantum 40 p-spin glasses, 2025

work page 2025
[24]

Leschke, C

H. Leschke, C. Manai, R. Ruder, and S. Warzel. Existence of replica-symmetry breaking in quantum glasses.Physical Review Letters, 127(20):207204, 2021

work page 2021
[25]

Leschke, S

H. Leschke, S. Rothlauf, R. Ruder, and W. Spitzer. The free energy of a quantum Sherrington–Kirkpatrick spin-glass model for weak disorder.Journal of Statistical Physics, 182(3):55, 2021

work page 2021
[26]

P. Lopatto. Replica symmetry up to the de Almeida–Thouless line in the Sherrington–Kirkpatrick model. arXiv:2604.11921, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[27]

C. Manai. Quantum mean-fields spin systems in a random external field. arXiv:2512.21502, 2025

work page arXiv 2025
[28]

Manai and S

C. Manai and S. Warzel. Phase diagram of the quantum random energy model.Journal of Statistical Physics, 180(1):654–664, 2020

work page 2020
[29]

Manai and S

C. Manai and S. Warzel. The de Almeida–Thouless line in hierarchical quantum spin glasses.Journal of Statistical Physics, 186(1):14, 2021

work page 2021
[30]

Manai and S

C. Manai and S. Warzel. The quantum random energy model as a limit ofp-spin interactions.Reviews in Mathematical Physics, 33(01):2060013, 2021

work page 2021
[31]

Manai and S

C. Manai and S. Warzel. Generalized random energy models in a transversal magnetic field: Free energy and phase diagrams.Probability and Mathematical Physics, 3:215—245, 2022

work page 2022
[32]

Manai and S

C. Manai and S. Warzel. Spectral analysis of the quantum random energy model.Communications in Mathematical Physics, 402(2):1259–1306, 2023

work page 2023
[33]

Manai and S

C. Manai and S. Warzel. A Parisi formula for quantum spin glasses.Electronic Journal of Probability, 30:1 – 39, 2025

work page 2025
[34]

Manai and S

C. Manai and S. Warzel. Dynamical phase diagram of the REM under independent spin-flips.Annales Henri Poincar´ e, 2025. Published online December 2025

work page 2025
[35]

Mezard, G

M. Mezard, G. Parisi, and M. Virasoro.Spin Glass Theory and Beyond. World Scientific Lecture Notes in Physics. World Scientific, 1986

work page 1986
[36]

Panchenko

D. Panchenko. On differentiability of the Parisi formula.Electronic Communications in Probability, 13:241–247, 2008

work page 2008
[37]

Panchenko

D. Panchenko. The Parisi ultrametricity conjecture.Annals of Mathematics, 177(1):383–393, 2013

work page 2013
[38]

Panchenko.The Sherrington-Kirkpatrick model

D. Panchenko.The Sherrington-Kirkpatrick model. Springer New York, NY, 2013

work page 2013
[39]

Panchenko

D. Panchenko. Free energy in the mixed p-spin models with vector spins.Ann. Probab., 46:865–896, 2018

work page 2018
[40]

G. Parisi. A sequence of approximated solutions to the S-K model for spin glasses.Journal of Physics A: Mathematical and General, 13(4):L115, 1980

work page 1980
[41]

I. Pinelis. Optimum bounds for the distributions of martingales in Banach spaces.The Annals of Probability, 22(4):1679–1706, 1994

work page 1994
[42]

P. Ray, B. K. Chakrabarti, and A. Chakrabarti. Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations.Phys. Rev. B, 39:11828–11832, Jun 1989

work page 1989
[43]

Shcherbina, B

M. Shcherbina, B. Tirozzi, and C. Tassi. Quantum Hopfield model.Physics, 2(2):184–196, 2020

work page 2020
[44]

Talagrand

M. Talagrand. The Parisi formula.Annals of Mathematics, 163:221–263, 2006

work page 2006
[45]

Talagrand.Mean Field Models for Spin Glasses I.Springer, 2011

M. Talagrand.Mean Field Models for Spin Glasses I.Springer, 2011

work page 2011
[46]

Talagrand.Mean Field Models for Spin Glasses II.Springer, 2011

M. Talagrand.Mean Field Models for Spin Glasses II.Springer, 2011

work page 2011
[47]

Tanaka, R

S. Tanaka, R. Tamura, and B. K. Chakrabarti.Quantum spin glasses, annealing and computation. Cambridge UP, 2017

work page 2017
[48]

F. L. Toninelli. About the Almeida-Thouless transition line in the Sherrington-Kirkpatrick mean-field spin glass model.Europhysics Letters, 60(5):764, 2002

work page 2002
[49]

K. D. Usadel and B. Schmitz. Quantum fluctuations in an Ising spin glass with transverse field.Solid State Communications, 64(6):975–977, 1987. 41

work page 1987
[50]

Yamamoto and H

T. Yamamoto and H. Ishii. A perturbation expansion for the Sherrington-Kirkpatrick model with a transverse field.Journal of Physics C: Solid State Physics, 20(35):6053, 1987

work page 1987
[51]

A. P. Young. Stability of the quantum Sherrington-Kirkpatrick spin glass model.Physical Review E, 96(3):032112–, 2017

work page 2017
[52]

Y. Zhou. On the Gardner transition in the Ising purep-spin glass. arXiv:2408.14630, 2024

work page arXiv 2024
[53]

Y. Zhou. Existence of full replica symmetry breaking for the Sherrington–Kirkpatrick model at low temperature. arXiv:2504.00269, 2025. 42

work page arXiv 2025

[1] [1]

Adhikari and C

A. Adhikari and C. Brennecke. Free energy of the quantum Sherrington–Kirkpatrick spin-glass model with transverse field.Journal of Mathematical Physics, 61:083302, 2020

work page 2020

[2] [2]

Aizenman, J

M. Aizenman, J. L. Lebowitz, and D. Ruelle. Some rigorous results on the Sherrington-Kirkpatrick spin glass model.Commun. Math. Phys., 112(1):3–20, 1987

work page 1987

[3] [3]

J. F. L. Almeida and D. J. Thouless. Stability of the Sherrington-Kirkpatrick solution of a spin glass model.J. Phys. A: Math. Gen., 11:983–990, 1978

work page 1978

[4] [4]

Auffinger and W.-K

A. Auffinger and W.-K. Chen. On properties of Parisi measures.Probability Theory and Related Fields, 161(3–4):817–850, 2015

work page 2015

[5] [5]

Auffinger and W.-K

A. Auffinger and W.-K. Chen. The Parisi formula has a unique minimizer.Communications in Mathematical Physics, 335(3):1429–1444, 2015

work page 2015

[6] [6]

Auffinger, W.-K

A. Auffinger, W.-K. Chen, and Q. Zeng. The SK model is infinite step replica symmetry breaking at zero temperature.Communications on Pure and Applied Mathematics, 73(5):921–943, 2020

work page 2020

[7] [7]

Bolthausen

E. Bolthausen. An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model.Communications in Mathematical Physics, 325(1):333–366, 2014

work page 2014

[8] [8]

Brennecke and H.-T

C. Brennecke and H.-T. Yau. The replica symmetric formula for the SK model revisited.Journal of Mathematical Physics, 63(7):073302, 07 2022

work page 2022

[9] [9]

H.-B. Chen. Self-overlap correction simplifies the Parisi formula for vector spins.Electronic Journal of Probability, 28:1 – 20, 2023

work page 2023

[10] [10]

H.-B. Chen. On the self-overlap in vector spin glasses.Journal of Mathematical Physics, 65(3):031901, 03 2024

work page 2024

[11] [11]

H.-B. Chen. Color symmetry and ferromagnetism in potts spin glass.Journal of Statistical Physics, 192:115, 2025

work page 2025

[12] [12]

W.-K. Chen. On the tap free energy in the mixedp-spin model.Electronic Communications in Probability, 24:1–10, 2019

work page 2019

[13] [13]

W.-K. Chen. On the Almeida-Thouless transition line in the Sherrington-Kirkpatrick model with centered Gaussian external field.Electron. Commun. Probab., 26:1–9, 2021

work page 2021

[14] [14]

W.-K. Chen, D. Panchenko, and E. Subag. The generalized TAP free energy.Communications on Pure and Applied Mathematics, 73(5):1011–1048, 2020

work page 2020

[15] [15]

W.-K. Chen, D. Panchenko, and E. Subag. Generalized TAP free energy II.Annals of Probability, 49(5):2277–2323, 2021

work page 2021

[16] [16]

Crawford

N. Crawford. Thermodynamics and universality for mean field quantum spin glasses.Communications in Mathematical Physics, 274(3):821–839, 2007

work page 2007

[17] [17]

Y. V. Fedorov and E. F. Shender. Quantum spin glasses in the Ising model with a transverse field. JETP Lett., 43:681 –684, 1986

work page 1986

[18] [18]

J. P. Garrahan, C. Manai, and S. Warzel. Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model.Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 381(2241):20210415, 12 2022

work page 2022

[19] [19]

Y. Y. Goldschmidt and P.-Y. Lai. Ising spin glass in a transverse field: Replica-symmetry-breaking solution.Phys. Rev. Lett., 64:2467–2470, 1990

work page 1990

[20] [20]

F. Guerra. Broken replica symmetry bounds in the mean field spin glass model.Communications in Mathematical Physics, 233(1):1–12, 2003

work page 2003

[21] [21]

Jagannath and I

A. Jagannath and I. Tobasco. A dynamic programming approach to the Parisi functional.Proceedings of the American Mathematical Society, 144(7):3135–3150, 2016

work page 2016

[22] [22]

Jagannath and I

A. Jagannath and I. Tobasco. Some properties of the phase diagram for mixedp-spin glasses. Probability Theory and Related Fields, 167(3–4):615–672, 2017

work page 2017

[23] [23]

Kouraich, C

A. Kouraich, C. Manai, and S. Warzel. The quantum random energy model is the limit of quantum 40 p-spin glasses, 2025

work page 2025

[24] [24]

Leschke, C

H. Leschke, C. Manai, R. Ruder, and S. Warzel. Existence of replica-symmetry breaking in quantum glasses.Physical Review Letters, 127(20):207204, 2021

work page 2021

[25] [25]

Leschke, S

H. Leschke, S. Rothlauf, R. Ruder, and W. Spitzer. The free energy of a quantum Sherrington–Kirkpatrick spin-glass model for weak disorder.Journal of Statistical Physics, 182(3):55, 2021

work page 2021

[26] [26]

P. Lopatto. Replica symmetry up to the de Almeida–Thouless line in the Sherrington–Kirkpatrick model. arXiv:2604.11921, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[27] [27]

C. Manai. Quantum mean-fields spin systems in a random external field. arXiv:2512.21502, 2025

work page arXiv 2025

[28] [28]

Manai and S

C. Manai and S. Warzel. Phase diagram of the quantum random energy model.Journal of Statistical Physics, 180(1):654–664, 2020

work page 2020

[29] [29]

Manai and S

C. Manai and S. Warzel. The de Almeida–Thouless line in hierarchical quantum spin glasses.Journal of Statistical Physics, 186(1):14, 2021

work page 2021

[30] [30]

Manai and S

C. Manai and S. Warzel. The quantum random energy model as a limit ofp-spin interactions.Reviews in Mathematical Physics, 33(01):2060013, 2021

work page 2021

[31] [31]

Manai and S

C. Manai and S. Warzel. Generalized random energy models in a transversal magnetic field: Free energy and phase diagrams.Probability and Mathematical Physics, 3:215—245, 2022

work page 2022

[32] [32]

Manai and S

C. Manai and S. Warzel. Spectral analysis of the quantum random energy model.Communications in Mathematical Physics, 402(2):1259–1306, 2023

work page 2023

[33] [33]

Manai and S

C. Manai and S. Warzel. A Parisi formula for quantum spin glasses.Electronic Journal of Probability, 30:1 – 39, 2025

work page 2025

[34] [34]

Manai and S

C. Manai and S. Warzel. Dynamical phase diagram of the REM under independent spin-flips.Annales Henri Poincar´ e, 2025. Published online December 2025

work page 2025

[35] [35]

Mezard, G

M. Mezard, G. Parisi, and M. Virasoro.Spin Glass Theory and Beyond. World Scientific Lecture Notes in Physics. World Scientific, 1986

work page 1986

[36] [36]

Panchenko

D. Panchenko. On differentiability of the Parisi formula.Electronic Communications in Probability, 13:241–247, 2008

work page 2008

[37] [37]

Panchenko

D. Panchenko. The Parisi ultrametricity conjecture.Annals of Mathematics, 177(1):383–393, 2013

work page 2013

[38] [38]

Panchenko.The Sherrington-Kirkpatrick model

D. Panchenko.The Sherrington-Kirkpatrick model. Springer New York, NY, 2013

work page 2013

[39] [39]

Panchenko

D. Panchenko. Free energy in the mixed p-spin models with vector spins.Ann. Probab., 46:865–896, 2018

work page 2018

[40] [40]

G. Parisi. A sequence of approximated solutions to the S-K model for spin glasses.Journal of Physics A: Mathematical and General, 13(4):L115, 1980

work page 1980

[41] [41]

I. Pinelis. Optimum bounds for the distributions of martingales in Banach spaces.The Annals of Probability, 22(4):1679–1706, 1994

work page 1994

[42] [42]

P. Ray, B. K. Chakrabarti, and A. Chakrabarti. Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations.Phys. Rev. B, 39:11828–11832, Jun 1989

work page 1989

[43] [43]

Shcherbina, B

M. Shcherbina, B. Tirozzi, and C. Tassi. Quantum Hopfield model.Physics, 2(2):184–196, 2020

work page 2020

[44] [44]

Talagrand

M. Talagrand. The Parisi formula.Annals of Mathematics, 163:221–263, 2006

work page 2006

[45] [45]

Talagrand.Mean Field Models for Spin Glasses I.Springer, 2011

M. Talagrand.Mean Field Models for Spin Glasses I.Springer, 2011

work page 2011

[46] [46]

Talagrand.Mean Field Models for Spin Glasses II.Springer, 2011

M. Talagrand.Mean Field Models for Spin Glasses II.Springer, 2011

work page 2011

[47] [47]

Tanaka, R

S. Tanaka, R. Tamura, and B. K. Chakrabarti.Quantum spin glasses, annealing and computation. Cambridge UP, 2017

work page 2017

[48] [48]

F. L. Toninelli. About the Almeida-Thouless transition line in the Sherrington-Kirkpatrick mean-field spin glass model.Europhysics Letters, 60(5):764, 2002

work page 2002

[49] [49]

K. D. Usadel and B. Schmitz. Quantum fluctuations in an Ising spin glass with transverse field.Solid State Communications, 64(6):975–977, 1987. 41

work page 1987

[50] [50]

Yamamoto and H

T. Yamamoto and H. Ishii. A perturbation expansion for the Sherrington-Kirkpatrick model with a transverse field.Journal of Physics C: Solid State Physics, 20(35):6053, 1987

work page 1987

[51] [51]

A. P. Young. Stability of the quantum Sherrington-Kirkpatrick spin glass model.Physical Review E, 96(3):032112–, 2017

work page 2017

[52] [52]

Y. Zhou. On the Gardner transition in the Ising purep-spin glass. arXiv:2408.14630, 2024

work page arXiv 2024

[53] [53]

Y. Zhou. Existence of full replica symmetry breaking for the Sherrington–Kirkpatrick model at low temperature. arXiv:2504.00269, 2025. 42

work page arXiv 2025