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Fytas NG, Martín-Mayor V, Parisi G, Picco M, Sourlas N. Finite-size scaling of the random-field Ising model above the upper critical dimension. Phys Rev E 2023; 108:044146. [PMID: 37978671 DOI: 10.1103/physreve.108.044146] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2023] [Accepted: 10/05/2023] [Indexed: 11/19/2023]
Abstract
Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of statistical physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension D=7, i.e., above its upper critical dimension D_{u}=6, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions D>D_{u}, linear length scale L should be replaced in finite-size scaling expressions by the effective scale L_{eff}=L^{D/D_{u}}. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.
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Affiliation(s)
- Nikolaos G Fytas
- Department of Mathematical Sciences, University of Essex, Colchester CO4 3SQ, United Kingdom
| | - Víctor Martín-Mayor
- Departamento de Física Téorica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputacíon y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - Giorgio Parisi
- Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy and INFN, Sezione di Roma I, IPCF-CNR, P.le A. Moro 2, 00185 Rome, Italy
| | - Marco Picco
- Laboratoire de Physique Théorique et Hautes Energies, UMR7589, Sorbonne Université et CNRS, 4 Place Jussieu, 75252 Paris Cedex 05, France
| | - Nicolas Sourlas
- Laboratoire de Physique Théorique de l'Ecole Normale Supérieure (Unité Mixte de Recherche du CNRS et de l'Ecole Normale Supérieure, associée à l'Université Pierre et Marie Curie, PARIS VI) 24 rue Lhomond, 75231 Paris Cedex 05, France
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Zhang K, Li X, Jin Y, Jiang Y. Machine learning glass caging order parameters with an artificial nested neural network. SOFT MATTER 2022; 18:6270-6277. [PMID: 35959881 DOI: 10.1039/d2sm00310d] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Around a glass transition, the dynamics of a supercooled liquid dramatically slow down, exhibited by caging of particles, while the structural changes remain subtle. Alternative to recent machine learning studies searching for structural predictors of glassy dynamics, here we propose to learn directly particle caging features defined purely according to dynamics. We focus on three transitions in a simulated hard sphere glass model, the melting of ultra-stable glasses, the Gardner transition and the liquid to ordinary glass transition. Implementing the machine learning algorithm based on a two-level nested neural network, we attain not only appropriate caging order parameters for all three transitions, but also a phase classification for input samples. A finite-size scaling analysis of the phase classification results identifies the order of melting (first) and Gardner (second) transitions. A false positive is avoided, as the liquid to glass transition is indicated as a crossover, rather than a phase transition with a well-defined transition point. This study paves the way to a generic approach for learning dynamical features in glassy systems, with a minimum requirement of system-specific knowledge.
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Affiliation(s)
- Kaihua Zhang
- School of Chemistry, Beihang University, Beijing 100191, China.
| | - Xinyang Li
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China.
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Yuliang Jin
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China.
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325000, China
| | - Ying Jiang
- School of Chemistry, Beihang University, Beijing 100191, China.
- Center of Soft Matter Physics and Its Applications, Beihang University, Beijing 100191, China
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Fytas NG, Martín-Mayor V, Picco M, Sourlas N. Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions. PHYSICAL REVIEW LETTERS 2016; 116:227201. [PMID: 27314735 DOI: 10.1103/physrevlett.116.227201] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2016] [Indexed: 06/06/2023]
Abstract
By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.
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Affiliation(s)
- Nikolaos G Fytas
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
| | - Víctor Martín-Mayor
- Departamento de Física Téorica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputacíon y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - Marco Picco
- LPTHE (Unité mixte de recherche du CNRS UMR 7589), Université Pierre et Marie Curie-Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France
| | - Nicolas Sourlas
- Laboratoire de Physique Théorique de l'Ecole Normale Supérieure (Unité Mixte de Recherche du CNRS et de l'Ecole Normale Supérieure, associée à l'Université Pierre et Marie Curie, PARIS VI) 24 rue Lhomond, 75231 Paris Cedex 05, France
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Fytas NG, Martín-Mayor V. Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions. Phys Rev E 2016; 93:063308. [PMID: 27415388 DOI: 10.1103/physreve.93.063308] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2015] [Indexed: 06/06/2023]
Abstract
It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.227201] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero- and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent α of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.
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Affiliation(s)
- Nikolaos G Fytas
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
| | - Víctor Martín-Mayor
- Departamento de Física Teórica I, Universidad Complutense, E-28040 Madrid, Spain and Instituto de Biocomputación and Física de Sistemas Complejos (BIFI), E-50009 Zaragoza, Spain
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Fytas NG, Martín-Mayor V. Universality in the three-dimensional random-field Ising model. PHYSICAL REVIEW LETTERS 2013; 110:227201. [PMID: 23767743 DOI: 10.1103/physrevlett.110.227201] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2013] [Indexed: 06/02/2023]
Abstract
We solve a long-standing puzzle in statistical mechanics of disordered systems. By performing a high-statistics simulation of the D=3 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute the complete set of critical exponents for this class, including the correction-to-scaling exponent, and we show, to high numerical accuracy, that scaling is described by two independent exponents. Discrepancies with previous works are explained in terms of strong scaling corrections.
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Affiliation(s)
- Nikolaos G Fytas
- Departamento de Física Teórica I, Universidad Complutense, E-28040 Madrid, Spain
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Fischer T, Vink RLC. Fluids with quenched disorder: scaling of the free energy barrier near critical points. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2011; 23:234117. [PMID: 21613708 DOI: 10.1088/0953-8984/23/23/234117] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
In the context of Monte Carlo simulations, the analysis of the probability distribution P(L)(m) of the order parameter m, as obtained in simulation boxes of finite linear extension L, allows for an easy estimation of the location of the critical point and the critical exponents. For Ising-like systems without quenched disorder, P(L)(m) becomes scale-invariant at the critical point, where it assumes a characteristic bimodal shape featuring two overlapping peaks. In particular, the ratio between the value of P(L)(m) at the peaks (P(L, max)) and the value at the minimum in between (P(L, min)) becomes L-independent at criticality. However, for Ising-like systems with quenched random fields, we argue that instead ΔF(L) := ln(P(L, max)/P(L, min)) proportional to L(θ) should be observed, where θ > 0 is the 'violation of hyperscaling' exponent. Since θ is substantially non-zero, the scaling of ΔF(L) with system size should be easily detectable in simulations. For two fluid models with quenched disorder, ΔF(L) versus L was measured and the expected scaling was confirmed. This provides further evidence that fluids with quenched disorder belong to the universality class of the random field Ising model.
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Affiliation(s)
- T Fischer
- Institute of Theoretical Physics, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
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Vink RLC, Fischer T, Binder K. Finite-size scaling in Ising-like systems with quenched random fields: evidence of hyperscaling violation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:051134. [PMID: 21230464 DOI: 10.1103/physreve.82.051134] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2010] [Indexed: 05/30/2023]
Abstract
In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost ΔF of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, ΔF∝L(θ), with θ as the violation of hyperscaling critical exponent and L as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.
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Affiliation(s)
- R L C Vink
- Institute of Theoretical Physics, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
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Vink RLC, Binder K, Löwen H. Critical behavior of colloid-polymer mixtures in random porous media. PHYSICAL REVIEW LETTERS 2006; 97:230603. [PMID: 17280188 DOI: 10.1103/physrevlett.97.230603] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2006] [Indexed: 05/13/2023]
Abstract
We show that the critical behavior of a colloid-polymer mixture inside a random porous matrix of quenched hard spheres belongs to the universality class of the random-field Ising model. We also demonstrate that random-field effects in colloid-polymer mixtures are surprisingly strong. This makes these systems attractive candidates to study random-field behavior experimentally.
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Affiliation(s)
- R L C Vink
- Institut für Theoretische Physik II, Heinrich-Heine-Universität Düsseldorf, Universitätsstrabe 1, 40225 Düsseldorf, Germany
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Efrat A, Schwartz M. Full reduction of large finite random Ising systems by real space renormalization group. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:026114. [PMID: 14525056 DOI: 10.1103/physreve.68.026114] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2003] [Indexed: 11/07/2022]
Abstract
We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain Gamma(r)=<sigma(0)sigma(r)>-<sigma(0)><sigma(r)>, the "connected" correlation function, and S(r)=<sigma(0)sigma(r)>, the "disconnected" correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function Gamma(r) along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such nu or eta). The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.
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Affiliation(s)
- Avishay Efrat
- Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Israel
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Gofman M, Adler J, Aharony A, Harris AB, Schwartz M. Critical behavior of the random-field Ising model. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:6362-6384. [PMID: 9982034 DOI: 10.1103/physrevb.53.6362] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Newman ME, Barkema GT. Monte Carlo study of the random-field Ising model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:393-404. [PMID: 9964270 DOI: 10.1103/physreve.53.393] [Citation(s) in RCA: 48] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Harris QJ, Feng Q, Birgeneau RJ, Hirota K, Shirane G, Hase M, Uchinokura K. Large length-scale fluctuations at the spin-Peierls transition in CuGeO3. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:15420-15425. [PMID: 9980900 DOI: 10.1103/physrevb.52.15420] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Belitz D, Kirkpatrick TR. Anderson-Mott transition as a quantum-glass problem. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:13922-13935. [PMID: 9980608 DOI: 10.1103/physrevb.52.13922] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Rieger H. Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:6659-6667. [PMID: 9981896 DOI: 10.1103/physrevb.52.6659] [Citation(s) in RCA: 64] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Read N, Sachdev S, Ye J. Landau theory of quantum spin glasses of rotors and Ising spins. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:384-410. [PMID: 9979617 DOI: 10.1103/physrevb.52.384] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Pereyra V, Nielaba P, Binder K. Spin-one-Ising model for (CO)1?x (N2) x mixtures: A finite size scaling study of random-field-type critical phenomena. ACTA ACUST UNITED AC 1995. [DOI: 10.1007/bf01307468] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Pitard E, Rosinberg ML, Stell G, Tarjus G. Critical behavior of a fluid in a disordered porous matrix: An Ornstein-Zernike approach. PHYSICAL REVIEW LETTERS 1995; 74:4361-4364. [PMID: 10058487 DOI: 10.1103/physrevlett.74.4361] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Chang I, Lev Z, Harris AB, Adler J, Aharony A. Localization length exponent in quantum percolation. PHYSICAL REVIEW LETTERS 1995; 74:2094-2097. [PMID: 10057840 DOI: 10.1103/physrevlett.74.2094] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Korzhenevskii AL, Luzhkov AA, Schirmacher W. Critical behavior of crystals with long-range correlations caused by point defects with degenerate internal degrees of freedom. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:3661-3666. [PMID: 9976645 DOI: 10.1103/physrevb.50.3661] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Vojta T, Schreiber M. Critical correlations and susceptibilities in the random-field spherical model. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:1272-1274. [PMID: 9975801 DOI: 10.1103/physrevb.50.1272] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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