Stability of the Sherrington-Kirkpatrick solution of a spin glass model
Abstract
The stationary point used by Sherrington and Kirkpatrick (1975) in their evaluation of the free energy of a spin glass by the method of steepest descent is examined carefully. It is found that, although this point is a maximum of the integrand at high temperatures, it is not a maximum in the spin glass phase nor in the ferromagnetic phase at low temperatures. The instability persists in the presence of a magnetic field. Results are given for the limit of stability both for a partly ferromagnetic interaction in the absence of an external field and for a purely random interaction in the presence of a field.
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Although the replica-symmetric ansatz (5.9) gives the correct form of saddles for computing Z_m when m is a positive integer, it becomes unstable for sufficiently small m<1: an eigenvalue of the Hessian evaluated on the ansatz becomes positive in the limit m→0.
“Indeed, though (5.9) does give the correct form of the saddles for computing Z m when m is a positive integer, it becomes unstable for sufficiently small m < 1: an eigenvalue of the Hessian ∂ 2 H eff =∂q ðα;γÞ ∂q ðβ;δÞ evaluated on the ansatz q ðα;γÞ ¼ q becomes positive in the limit m → 0 [68].”
From Free energy from replica wormholes (2021) · cites this paper as [68]
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The sphericalp-spin interaction spin glass model: the statics
Crisanti, A., & Sommers, H. J. (1992) · Zeitschrift f�r Physik B Condensed Matter · doi:10.1007/bf01309287 · Cited by 481 · see citing works
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Read, N., Sachdev, S., & Ye, J. (1995) · Physical Review B · doi:10.1103/physrevb.52.384 · Cited by 165 · see citing works
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Engelhardt, N., Fischetti, S., & Maloney, A. (2021) · Physical Review D · doi:10.1103/physrevd.103.046021 · Cited by 90 · see citing works
- Authors
- Almeida, J. R. L. D., & Thouless, D. J.
- Affiliations
- University of Birmingham (GB)
- ORCID
- Almeida, J R L de
- Venue
- Journal of Physics A: Mathematical and General
- Publisher
- IOP Publishing
- Published
- 1978-05
- Volume
- 11
- Issue
- 5
- Pages
- 983-990
- ISSN
- 0305-4470 (ISSN-L)
- 1361-6447
- OpenAlex
- W1991904924
- Cited by
- 2,061
- References
- 0
- Citations by year
- 2026: 22 · 2025: 52 · 2024: 37 · 2023: 49 · 2022: 46 · 2021: 46 · 2020: 51 · 2019: 76 · 2018: 48 · 2017: 42 · 2016: 55 · 2015: 42 · 2014: 62 · 2013: 47 · 2012: 57
- FWCI
- 18.72
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Physics A Mathematical and General (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Physical Sciences › Materials Science › Materials Chemistry › Material Dynamics and Properties
- Physical Sciences › Mathematics › Statistics and Probability › Random Matrices and Applications
- Keywords
- Sherrington-Kirkpatrick model, spin glass, free energy, gradient descent, ferromagnetic phase, random interactions, magnetic field, phase coexistence, replica symmetry breaking
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