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Fernandez LA, Gonzalez-Adalid Pemartin I, Martin-Mayor V, Parisi G, Ricci-Tersenghi F, Rizzo T, Ruiz-Lorenzo JJ, Veca M. Numerical test of the replica-symmetric Hamiltonian for correlations of the critical state of spin glasses in a field. Phys Rev E 2022; 105:054106. [PMID: 35706223 DOI: 10.1103/physreve.105.054106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/23/2021] [Accepted: 03/28/2022] [Indexed: 06/15/2023]
Abstract
A growing body of evidence indicates that the sluggish low-temperature dynamics of glass formers (e.g., supercooled liquids, colloids, or spin glasses) is due to a growing correlation length. Which is the effective field theory that describes these correlations? The natural field theory was drastically simplified by Bray and Roberts in 1980. More than 40 years later, we confirm the tenets of Bray and Roberts's theory by studying the Ising spin glass in an externally applied magnetic field, both in four spatial dimensions (data obtained from the Janus collaboration) and on the Bethe lattice.
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Affiliation(s)
- L A Fernandez
- Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50018 Zaragoza, Spain
| | | | - V Martin-Mayor
- Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50018 Zaragoza, Spain
| | - G Parisi
- Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 5, 00185 Rome, Italy
- INFN, Sezione di Roma 1, P.le A. Moro 5, 00185 Rome, Italy
- CNR-Nanotec, Unità di Roma, P.le A. Moro 5, 00185 Rome, Italy
| | - F Ricci-Tersenghi
- Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 5, 00185 Rome, Italy
- INFN, Sezione di Roma 1, P.le A. Moro 5, 00185 Rome, Italy
- CNR-Nanotec, Unità di Roma, P.le A. Moro 5, 00185 Rome, Italy
| | - T Rizzo
- Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 5, 00185 Rome, Italy
- Institute of Complex Systems (ISC) - CNR, Rome Unit, P.le A. Moro 5, 00185 Rome, Italy
| | - J J Ruiz-Lorenzo
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50018 Zaragoza, Spain
- Departamento de Física, Universidad de Extremadura, 06006 Badajoz, Spain
- Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, 06006 Badajoz, Spain
| | - M Veca
- Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 5, 00185 Rome, Italy
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Dilucca M, Leuzzi L, Parisi G, Ricci-Tersenghi F, Ruiz-Lorenzo JJ. Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field). ENTROPY 2020; 22:e22020250. [PMID: 33286024 PMCID: PMC7516687 DOI: 10.3390/e22020250] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/23/2020] [Revised: 02/18/2020] [Accepted: 02/20/2020] [Indexed: 11/16/2022]
Abstract
We discuss a phase transition in spin glass models that have been rarely considered in the past, namely, the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e., at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that, even in a paramagnetic phase, the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite dimensional spin glasses. This is a notoriously difficult computational problem, because of considerable finite size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, h > 0 and T c ( h ) < T < T c ( h = 0 ) , of the d = 1 spin glass model with long range interactions decaying fast enough to be outside the regime of validity of the mean field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.
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Affiliation(s)
- Maddalena Dilucca
- Dipartimento di Fisica, Sapienza Universitá di Roma, P.le A. Moro 2, I-00185 Roma, Italy;
| | - Luca Leuzzi
- Dipartimento di Fisica, Sapienza Universitá di Roma, P.le A. Moro 2, I-00185 Roma, Italy;
- CNR, Nanotec, Rome unit, P.le A. Moro 2, I-00185 Roma, Italy
- Correspondence: (L.L.); (G.P.); (F.R.-T.); (J.J.R.-L.)
| | - Giorgio Parisi
- Dipartimento di Fisica, Sapienza Universitá di Roma, P.le A. Moro 2, I-00185 Roma, Italy;
- CNR, Nanotec, Rome unit, P.le A. Moro 2, I-00185 Roma, Italy
- INFN, Sezione di Roma I, P.le A. Moro 2, I-00185 Roma, Italy
- Correspondence: (L.L.); (G.P.); (F.R.-T.); (J.J.R.-L.)
| | - Federico Ricci-Tersenghi
- Dipartimento di Fisica, Sapienza Universitá di Roma, P.le A. Moro 2, I-00185 Roma, Italy;
- CNR, Nanotec, Rome unit, P.le A. Moro 2, I-00185 Roma, Italy
- INFN, Sezione di Roma I, P.le A. Moro 2, I-00185 Roma, Italy
- Correspondence: (L.L.); (G.P.); (F.R.-T.); (J.J.R.-L.)
| | - Juan J. Ruiz-Lorenzo
- Departamendo de Física and ICCAEx, Universidad de Extremadura, 06006 Badajoz, Spain
- Instituto de Biocomputación y Física de los Sistemas Complejos (BIFI), 50018 Zaragoza, Spain
- Correspondence: (L.L.); (G.P.); (F.R.-T.); (J.J.R.-L.)
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Andresen JC, Thomas CK, Katzgraber HG, Schechter M. Novel disordering mechanism in ferromagnetic systems with competing interactions. PHYSICAL REVIEW LETTERS 2013; 111:177202. [PMID: 24206516 DOI: 10.1103/physrevlett.111.177202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2012] [Revised: 07/11/2013] [Indexed: 06/02/2023]
Abstract
Ferromagnetic Ising systems with competing interactions are considered in the presence of a random field. We find that in three space dimensions the ferromagnetic phase is disordered by a random field which is considerably smaller than the typical interaction strength between the spins. This is the result of a novel disordering mechanism triggered by an underlying spin-glass phase. Calculations for the specific case of the long-range dipolar LiHo(x)Y(1-x)F(4) compound suggest that the above mechanism is responsible for the peculiar dependence of the critical temperature on the strength of the random field and the broadening of the susceptibility peaks as temperature is decreased, as found in recent experiments by Silevitch et al.. [Nature (London) 448, 567 (2007)]. Our results thus emphasize the need to go beyond the standard Imry-Ma argument when studying general random-field systems.
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Leuzzi L, Parisi G, Ricci-Tersenghi F, Ruiz-Lorenzo JJ. Ising spin-glass transition in a magnetic field outside the limit of validity of mean-field theory. PHYSICAL REVIEW LETTERS 2009; 103:267201. [PMID: 20366338 DOI: 10.1103/physrevlett.103.267201] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2008] [Revised: 09/15/2009] [Indexed: 05/29/2023]
Abstract
The spin-glass transition in a magnetic field is studied both in and out of the limit of validity of mean-field theory on a diluted one dimensional chain of Ising spins where exchange bonds occur with a probability decaying as the inverse power of the distance. Varying the power in this long-range model corresponds, in a one-to-one relationship, to changing the dimension in spin-glass short-range models. Evidence for a spin-glass transition in a magnetic field is found also for systems whose equivalent dimension is below the upper critical dimension in a zero magnetic field.
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Affiliation(s)
- L Leuzzi
- Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 2, I-00185 Roma, Italy
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Jörg T, Katzgraber HG, Krzakała F. Behavior of Ising spin glasses in a magnetic field. PHYSICAL REVIEW LETTERS 2008; 100:197202. [PMID: 18518482 DOI: 10.1103/physrevlett.100.197202] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2007] [Revised: 02/25/2008] [Indexed: 05/26/2023]
Abstract
We study the existence of a spin-glass phase in a field using Monte Carlo simulations performed along a nontrivial path in the field-temperature plane that must cross any putative de Almeida-Thouless instability line. The method is first tested on the Ising spin glass on a Bethe lattice where the instability line separating the spin glass from the paramagnetic state is also computed analytically. While the instability line is reproduced by our simulations on the mean-field Bethe lattice, no such instability line can be found numerically for the short-range three-dimensional model.
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Affiliation(s)
- Thomas Jörg
- LPTMS, UMR 8626 CNRS et Université Paris-Sud, Orsay Cedex, France
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Sasaki M, Hukushima K, Yoshino H, Takayama H. Scaling analysis of domain-wall free energy in the Edwards-Anderson Ising spin glass in a magnetic field. PHYSICAL REVIEW LETTERS 2007; 99:137202. [PMID: 17930628 DOI: 10.1103/physrevlett.99.137202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2007] [Indexed: 05/25/2023]
Abstract
The stability of the spin-glass phase against a magnetic field is studied in the three- and four-dimensional Edwards-Anderson Ising spin glasses. Effective couplings J(eff) and effective fields H(eff) associated with length scale L are measured by a numerical domain-wall renormalization-group method. The results obtained by scaling analysis of the data strongly indicate the existence of a crossover length beyond which the spin-glass order is destroyed by field H. The crossover length well obeys a power law of H which diverges as H --> 0 but remains finite for any nonzero H, implying that the spin-glass phase is absent even in an infinitesimal field. These results are well consistent with the droplet theory for short-range spin glasses.
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Affiliation(s)
- M Sasaki
- Department of Applied Physics, Tohoku University, Sendai, 980-8579, Japan
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Young AP, Katzgraber HG. Absence of an Almeida-Thouless line in three-dimensional spin glasses. PHYSICAL REVIEW LETTERS 2004; 93:207203. [PMID: 15600963 DOI: 10.1103/physrevlett.93.207203] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2004] [Indexed: 05/24/2023]
Abstract
We present results of Monte Carlo simulations of the three-dimensional Edwards-Anderson Ising spin glass in the presence of a (random) field. A finite-size scaling analysis of the correlation length shows no indication of a transition, in contrast with the zero-field case. This suggests that there is no Almeida-Thouless line for short-range Ising spin glasses.
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Affiliation(s)
- A P Young
- Department of Physics, University of California, Santa Cruz, California 95064, USA.
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Billoire A, Coluzzi B. Numerical study of the Sherrington-Kirkpatrick model in a magnetic field. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:026131. [PMID: 14525073 DOI: 10.1103/physreve.68.026131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2003] [Indexed: 05/24/2023]
Abstract
We study numerically the Sherrington-Kirkpatrick model as a function of the magnetic field h, with fixed temperature T=0.6T(c). We investigate the finite size scaling behavior of several quantities, such as the spin-glass susceptibility, searching for numerical evidences of the transition on the de Almeida-Thouless line. We find strong corrections to scaling which make difficult to locate the transition point. This shows, in a simple case, the extreme difficulties of spin-glass simulations in a nonzero magnetic field. Next, we study various sum rules (consequences of stochastic stability) involving overlaps between three and four replicas, which appear to be numerically well satisfied, and in a nontrivial way. Finally, we present data on P(q) for a large lattice size (N=3200) at low temperature T=0.4T(c), where the shape predicted by the replica symmetry breaking solution of the model for a nonzero magnetic field is visible.
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Affiliation(s)
- Alain Billoire
- Service de Physique Théorique, CEA-Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France
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Drossel B, Bokil H, Moore MA. Spin glasses without time-reversal symmetry and the absence of a genuine structural glass transition. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:7690-9. [PMID: 11138040 DOI: 10.1103/physreve.62.7690] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2000] [Indexed: 11/07/2022]
Abstract
We study the three-spin model and the Ising spin glass in a field using the Migdal-Kadanoff approximation. The flows of the couplings and fields indicate no phase transition, but they show even for the three-spin model a slow crossover to the asymptotic high-temperature behavior for large values of the coupling. We have also evaluated a quantity that is a measure of the degree of non-self-averaging, and we found that it can become large for certain ranges of the parameters and the system sizes. For a spin glass in a field the maximum of non-self-averaging follows a line for given system size that resembles the de Almeida-Thouless line. We conclude that non-self-averaging found in Monte Carlo simulations cannot be taken as evidence for the existence of a low-temperature phase with replica symmetry breaking. Models similar to the three-spin model have been extensively discussed in order to provide a description of structural glasses. Their theory at mean-field level resembles the mode-coupling theory of real glasses. At that level the approach via one-step replica symmetry breaking predicts two transitions, the first transition being dynamic and the second thermodynamic. Our results suggest that in real finite-dimensional glasses there will be no genuine transitions at all, but that some features of mean-field theory could still provide some useful insights.
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Affiliation(s)
- B Drossel
- School of Physics and Astronomy, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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Marinari E, Parisi G, Zuliani F. Comment on "ising spin glasses in a magnetic field". PHYSICAL REVIEW LETTERS 2000; 84:1056-1057. [PMID: 11017440 DOI: 10.1103/physrevlett.84.1056] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/1999] [Indexed: 05/23/2023]
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