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Cardy J. Fluids in Random Media and Dimensional Augmentation. PHYSICAL REVIEW LETTERS 2023; 131:147102. [PMID: 37862635 DOI: 10.1103/physrevlett.131.147102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2023] [Revised: 08/11/2023] [Accepted: 09/11/2023] [Indexed: 10/22/2023]
Abstract
We propose a solution to the puzzle of dimensional reduction in the random field Ising model, asking the following: To what random problem in D=d+2 dimensions does a pure system in d dimensions correspond? For a continuum binary fluid and an Ising lattice gas, we prove that the mean density and other observables equal those of a similar model in D dimensions, but with infinite range interactions and correlated disorder in the extra two dimensions. There is no conflict with rigorous results that the finite range model orders in D=3. Our arguments avoid the use of replicas and perturbative field theory, being based on convergent cluster expansions, which, for the lattice gas, may be extended to the critical point by the Lee-Yang theorem. Although our results may be derived using supersymmetry, they follow more directly from the matrix-tree theorem.
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Affiliation(s)
- John Cardy
- All Souls College, University of Oxford, Oxford OX1 4AL, United Kingdom and Department of Physics, University of California, Berkeley, California 94720, USA
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2
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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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3
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d'Auriac JCA, Iglói F. Random-bond antiferromagnetic Ising model in a field. Phys Rev E 2022; 106:034117. [PMID: 36266800 DOI: 10.1103/physreve.106.034117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2022] [Accepted: 08/29/2022] [Indexed: 06/16/2023]
Abstract
Using combinatorial optimization techniques we study the critical properties of the two- and three-dimensional Ising models with uniformly distributed random antiferromagnetic couplings (1≤J_{i}≤2) in the presence of a homogeneous longitudinal field, h, at zero temperature. In finite systems of linear size, L, we measure the average correlation function, C_{L}(ℓ,h), when the sites are either on the same sublattice, or they belong to different sublattices. The phase transition, which is of first order in the pure system, turns to mixed order in two dimensions with critical exponents 1/ν≈0.5 and η≈0.7. In three dimensions we obtain 1/ν≈0.7, which is compatible with the value of the random-field Ising model, but we cannot discriminate between second-order and mixed-order transitions.
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Affiliation(s)
| | - Ferenc Iglói
- Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary
- Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary
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4
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Wiese KJ. Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 85:086502. [PMID: 35943081 DOI: 10.1088/1361-6633/ac4648] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2021] [Accepted: 12/23/2021] [Indexed: 06/15/2023]
Abstract
Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.
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Affiliation(s)
- Kay Jörg Wiese
- Laboratoire de physique, Département de physique de l'ENS, École normale supérieure, UPMC Univ. Paris 06, CNRS, PSL Research University, 75005 Paris, France
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5
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Kaviraj A, Rychkov S, Trevisani E. Parisi-Sourlas Supersymmetry in Random Field Models. PHYSICAL REVIEW LETTERS 2022; 129:045701. [PMID: 35939006 DOI: 10.1103/physrevlett.129.045701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2022] [Accepted: 06/27/2022] [Indexed: 06/15/2023]
Abstract
By the Parisi-Sourlas conjecture, the critical point of a theory with random field (RF) disorder is described by a supersymmeric (SUSY) conformal field theory (CFT), related to a d-2 dimensional CFT without SUSY. Numerical studies indicate that this is true for the RF ϕ^{3} model but not for the RF ϕ^{4} model in d<5 dimensions. Here we argue that the SUSY fixed point is not reached because of new relevant SUSY-breaking interactions. We use a perturbative renormalization group in a judiciously chosen field basis, allowing systematic exploration of the space of interactions. Our computations agree with the numerical results for both cubic and quartic potential.
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Affiliation(s)
- Apratim Kaviraj
- Institut de Physique Théorique Philippe Meyer, ENS, Université PSL, CNRS Sorbonne Université, Université de Paris, F-75005 Paris, France
- Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS Sorbonne Université, Université de Paris, F-75005 Paris, France
- DESY Hamburg, Theory Group, Notkestraße 85, D-22607 Hamburg, Germany
| | - Slava Rychkov
- Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS Sorbonne Université, Université de Paris, F-75005 Paris, France
- Institut des Hautes Études Scientifiques, 91440 Bures-sur-Yvette, France
| | - Emilio Trevisani
- Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS Sorbonne Université, Université de Paris, F-75005 Paris, France
- CPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France
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6
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Vasilopoulos A, Fytas NG, Vatansever E, Malakis A, Weigel M. Universality in the two-dimensional dilute Baxter-Wu model. Phys Rev E 2022; 105:054143. [PMID: 35706204 DOI: 10.1103/physreve.105.054143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2022] [Accepted: 05/02/2022] [Indexed: 06/15/2023]
Abstract
We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field Δ. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of Δ and a parallelized variant of the multicanonical approach performed at constant temperature T. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram indicates that the transition belongs to the universality class of the four-state Potts model. Previous controversies with respect to the nature of the transition are discussed and attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.
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Affiliation(s)
| | - Nikolaos G Fytas
- Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, United Kingdom
- Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
| | - Erol Vatansever
- Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, United Kingdom
- Department of Physics, Dokuz Eylül University, TR-35160 Izmir, Turkey
| | - Anastasios Malakis
- Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, United Kingdom
- Department of Physics, University of Athens, Panepistimiopolis, GR-15784 Zografou, Greece
| | - Martin Weigel
- Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
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7
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Graovac S, Mijatović S, Spasojević D. Mechanism of subcritical avalanche propagation in three-dimensional disordered systems. Phys Rev E 2021; 103:062123. [PMID: 34271753 DOI: 10.1103/physreve.103.062123] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2021] [Accepted: 05/25/2021] [Indexed: 11/07/2022]
Abstract
We present a numerical study on necessary conditions for the appearance of infinite avalanche below the critical point in disordered systems that evolve throughout metastable states. The representative of those systems is the nonequilibrium athermal random-field Ising model. We investigate the impact on propagation of infinite avalanche of both the interface of flipped spins at the avalanche's starting point and the number of independent islands of flipped spins in the system at the moment when the avalanche starts. To deduce what effects are originated due to finite system's size, and to distinguish them from the real necessary conditions for the appearance of the infinite avalanche, we examined lattices of different sizes as well as other key parameters for the avalanche propagation.
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Affiliation(s)
- Stefan Graovac
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
| | - Svetislav Mijatović
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
| | - Djordje Spasojević
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
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8
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Jovković D, Janićević S, Mijatović S, Laurson L, Spasojević D. Effects of external noise on threshold-induced correlations in ferromagnetic systems. Phys Rev E 2021; 103:062114. [PMID: 34271613 DOI: 10.1103/physreve.103.062114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2020] [Accepted: 05/17/2021] [Indexed: 11/07/2022]
Abstract
In the present paper we investigate the impact of the external noise and detection threshold level on the simulation data for the systems that evolve through metastable states. As a representative model of such systems we chose the nonequilibrium athermal random-field Ising model with two types of the external noise, uniform white noise and Gaussian white noise with various different standard deviations, imposed on the original response signal obtained in model simulations. We applied a wide range of detection threshold levels in analysis of the signal and show how these quantities affect the values of exponent γ_{S/T} (describing the scaling of the average avalanche size with duration), the shift of waiting time between the avalanches, and finally the collapses of the waiting time distributions. The results are obtained via extensive numerical simulations on the equilateral three-dimensional cubic lattices of various sizes and disorders.
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Affiliation(s)
- Dragutin Jovković
- Faculty of Mining and Geology, University of Belgrade, P.O. Box 162, 11000 Belgrade, Serbia
| | - Sanja Janićević
- Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia
| | - Svetislav Mijatović
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
| | - Lasse Laurson
- Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland
| | - Djordje Spasojević
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
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9
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Mijatović S, Jovković D, Spasojević D. Nonequilibrium athermal random-field Ising model on hexagonal lattices. Phys Rev E 2021; 103:032147. [PMID: 33862757 DOI: 10.1103/physreve.103.032147] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2020] [Accepted: 03/08/2021] [Indexed: 11/07/2022]
Abstract
We present the results of a study providing numerical evidence for the absence of critical behavior of the nonequilibrium athermal random-field Ising model in adiabatic regime on the hexagonal two-dimensional lattice. The results are obtained on the systems containing up to 32768×32768 spins and are the averages of up to 1700 runs with different random-field configurations per each value of disorder. We analyzed regular systems as well as the systems with different preset conditions to capture behavior in thermodynamic limit. The superficial insight to the avalanche propagation in this type of lattice is given as a stimulus for further research on the topic of avalanche evolution. With obtained data we may conclude that there is no critical behavior of random-field Ising model on hexagonal lattice which is a result that differs from the ones found for the square and for the triangular lattices supporting the recent conjecture that the number of nearest neighbors affects the model criticality.
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Affiliation(s)
- Svetislav Mijatović
- Faculty of Physics, University of Belgrade, P.O.B. 44, 11001 Belgrade, Serbia
| | - Dragutin Jovković
- Faculty of Mining and Geology, University of Belgrade, P.O.B. 162, 11000 Belgrade, Serbia
| | - Djordje Spasojević
- Faculty of Physics, University of Belgrade, P.O.B. 44, 11001 Belgrade, Serbia
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10
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Vatansever E, Vatansever ZD, Theodorakis PE, Fytas NG. Ising universality in the two-dimensional Blume-Capel model with quenched random crystal field. Phys Rev E 2020; 102:062138. [PMID: 33466068 DOI: 10.1103/physreve.102.062138] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2020] [Accepted: 12/04/2020] [Indexed: 06/12/2023]
Abstract
Using high-precision Monte Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques, we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on a square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of a pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the scaling of the specific heat. Our results are in agreement with an early real-space renormalization-group study of the model as well as a very recent numerical work where quenched randomness was introduced in the energy exchange coupling. Finally, by properly fine tuning the control parameters of the randomness distribution we also qualitatively investigate the part of the phase diagram where the pure model undergoes a first-order phase transition. For this region, preliminary evidence indicate a smoothing of the transition to second-order with the presence of strong scaling corrections.
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Affiliation(s)
- Erol Vatansever
- Department of Physics, Dokuz Eylül University, TR-35160 Izmir, Turkey
| | | | | | - Nikolaos G Fytas
- Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, United Kingdom
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11
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Balog I, Tarjus G, Tissier M. Dimensional reduction breakdown and correction to scaling in the random-field Ising model. Phys Rev E 2020; 102:062154. [PMID: 33466013 DOI: 10.1103/physreve.102.062154] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2020] [Accepted: 11/30/2020] [Indexed: 11/07/2022]
Abstract
We provide a theoretical analysis by means of the nonperturbative functional renormalization group (NP-FRG) of the corrections to scaling in the critical behavior of the random-field Ising model (RFIM) near the dimension d_{DR}≈5.1 that separates a region where the renormalized theory at the fixed point is supersymmetric and critical scaling satisfies the d→d-2 dimensional reduction property (d>d_{DR}) from a region where both supersymmetry and dimensional reduction break down at criticality (d<d_{DR}). We show that the NP-FRG results are in very good agreement with recent large-scale lattice simulations of the RFIM in d=5 and we detail the consequences for the leading correction-to-scaling exponent of the peculiar boundary-layer mechanism by which the dimensional-reduction fixed point disappears and the dimensional-reduction-broken fixed point emerges in d_{DR}.
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Affiliation(s)
- Ivan Balog
- Institute of Physics, P.O. Box 304, Bijenička cesta 46, HR-10001 Zagreb, Croatia
| | - Gilles Tarjus
- LPTMC, CNRS-UMR 7600, Sorbonne Université, Boîte 121, 4 Pl. Jussieu, 75252 Paris cedex 05, France
| | - Matthieu Tissier
- LPTMC, CNRS-UMR 7600, Sorbonne Université, Boîte 121, 4 Pl. Jussieu, 75252 Paris cedex 05, France
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12
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Mijatović S, Branković M, Graovac S, Spasojević D. Avalanche properties in striplike ferromagnetic systems. Phys Rev E 2020; 102:022124. [PMID: 32942372 DOI: 10.1103/physreve.102.022124] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2020] [Accepted: 07/24/2020] [Indexed: 11/07/2022]
Abstract
We present numerical findings on the behavior of the athermal nonequilibrium random-field Ising model of spins at the thin striplike L_{1}×L_{2}×L_{3} cubic lattices with L_{1}<L_{2}<L_{3}. Changing of system sizes highly influences the evolution and shape of avalanches. The smallest avalanches [classified as three-dimension- (3D) like] are unaffected by the system boundaries, the larger are sandwiched between the top and bottom system faces so are 2D-like, while the largest are extended over the system lateral cross section and propagate along the length L_{3} like in 1D systems. Such a structure of avalanches causes double power-law distributions of their size, duration, and energy with larger effective critical exponent corresponding to 3D-like and smaller to 2D-like avalanches. The distributions scale with thickness L_{1} and are collapsible following the proposed scaling predictions which, together with the distributions' shape, might be important for analysis of the Barkhausen noise experimental data for striplike samples. Finally, the impact of system size on external field that triggers the largest avalanche for a given disorder is presented and discussed.
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Affiliation(s)
- Svetislav Mijatović
- Faculty of Physics, University of Belgrade, P. O. Box 44, 11001 Belgrade, Serbia
| | - Milica Branković
- Faculty of Physics, University of Belgrade, P. O. Box 44, 11001 Belgrade, Serbia
| | - Stefan Graovac
- Faculty of Physics, University of Belgrade, P. O. Box 44, 11001 Belgrade, Serbia
| | - Djordje Spasojević
- Faculty of Physics, University of Belgrade, P. O. Box 44, 11001 Belgrade, Serbia
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13
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Fisch R. Behavior of the random-field XY model on simple cubic lattices at h_{r}=1.5. Phys Rev E 2020; 101:062134. [PMID: 32688498 DOI: 10.1103/physreve.101.062134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2019] [Accepted: 06/05/2020] [Indexed: 06/11/2023]
Abstract
We have performed studies of the three-dimensional random-field XY model on 32 samples of L×L×L simple cubic lattices with periodic boundary conditions, with a random field strength of h_{r} = 1.5, for L= 128, using a parallelized Monte Carlo algorithm. We present results for the sample-averaged magnetic structure factor S(k[over ⃗]) over a range of temperature, using both random hot start and ferromagnetic cold start initial states, and k[over ⃗] along the [1,0,0] and [1,1,1] directions. At T= 1.875, S(k[over ⃗]) shows a broad peak near |k[over ⃗]|=0, with a correlation length which is limited by thermal fluctuations, rather than the lattice size. As T is lowered, this peak grows and sharpens. By T= 1.5, it is clear that the correlation length is larger than L= 128. The lowest temperature for which S(k[over ⃗]) was calculated is T= 1.421875, where the hot start and cold start initial conditions usually do not find the same local minimum in the phase space. Our results are consistent with the idea that there is a finite value of T below which S(k[over ⃗]) diverges slowly as |k[over ⃗]| goes to zero. This divergence would imply that the relaxation time of the spins is also diverging. That is the signature of an ergodicity-breaking phase transition.
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Affiliation(s)
- Ronald Fisch
- 382 Willowbrook Drive, North Brunswick, New Jersey 08902, USA
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14
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Acosta Diaz RJ, Rodríguez-Camargo CD, Svaiter NF. Directed Polymers and Interfaces in Disordered Media. Polymers (Basel) 2020; 12:polym12051066. [PMID: 32384732 PMCID: PMC7284553 DOI: 10.3390/polym12051066] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2020] [Revised: 04/08/2020] [Accepted: 04/09/2020] [Indexed: 11/16/2022] Open
Abstract
We consider field theory formulation for directed polymers and interfaces in the presence of quenched disorder. We write a series representation for the averaged free energy, where all the integer moments of the partition function of the model contribute. The structure of field space is analysed for polymers and interfaces at finite temperature using the saddle-point equations derived from each integer moments of the partition function. For the case of an interface we obtain the wandering exponent ξ = ( 4 - d ) / 2 , also obtained by the conventional replica method for the replica symmetric scenario.
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Affiliation(s)
- Róbinson J. Acosta Diaz
- Instituto de Física, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Rio de Janeiro 21941-972, Brazil;
| | - Christian D. Rodríguez-Camargo
- Departamento de Ingeniería Industrial, Corporación Universitaria Minuto de Dios, Carrera 73A No. 81B-70, Bogotá 111021, Colombia;
- Facultad de Ingeniería, Universidad Nacional de Colombia—Sede Bogotá, Calle 44 No. 45-67, Bogotá 111321, Colombia
| | - Nami F. Svaiter
- Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
- Correspondence:
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15
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Mijatović S, Jovković D, Janićević S, Spasojević D. Critical disorder and critical magnetic field of the nonequilibrium athermal random-field Ising model in thin systems. Phys Rev E 2019; 100:032113. [PMID: 31639960 DOI: 10.1103/physreve.100.032113] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2019] [Indexed: 11/07/2022]
Abstract
In the present study of the nonequilibrium athermal random-field Ising model we focus on the behavior of the critical disorder R_{c}(l) and the critical magnetic field H_{c}(l) under different boundary conditions when the system thickness l varies. We propose expressions for R_{c}(l) and H_{c}(l) as well as for the effective critical disorder R_{c}^{eff}(l,L) and effective critical magnetic field H_{c}^{eff}(l,L) playing the role of the effective critical parameters for the L×L×l lattices of finite lateral size L. We support these expressions by the scaling collapses of the magnetization and susceptibility curves obtained in extensive simulations. The collapses are achieved with the two-dimensional (2D) exponents for l below some characteristic value, providing thus a numerical evidence that the thin systems exhibit a 2D-like criticality which should be relevant for the experimental analyses of thin ferromagnetic samples.
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Affiliation(s)
| | - Dragutin Jovković
- Faculty of Mining and Geology, University of Belgrade, POB 162, 11000 Belgrade, Serbia
| | - Sanja Janićević
- Faculty of Science, University of Kragujevac, POB 60, 34000 Kragujevac, Serbia
| | - Djordje Spasojević
- Faculty of Physics, University of Belgrade, POB 44, 11001 Belgrade, Serbia
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16
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Fytas NG, Martín-Mayor V, Parisi G, Picco M, Sourlas N. Evidence for Supersymmetry in the Random-Field Ising Model at D=5. PHYSICAL REVIEW LETTERS 2019; 122:240603. [PMID: 31322399 DOI: 10.1103/physrevlett.122.240603] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2019] [Indexed: 06/10/2023]
Abstract
We provide a nontrivial test of supersymmetry in the random-field Ising model at five spatial dimensions, by means of extensive zero-temperature numerical simulations. Indeed, supersymmetry relates correlation functions in a D-dimensional disordered system with some other correlation functions in a D-2 clean system. We first show how to check these relationships in a finite-size scaling calculation and then perform a high-accuracy test. While the supersymmetric predictions are satisfied even to our high accuracy at D=5, they fail to describe our results at D=4.
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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
| | - Giorgio Parisi
- Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Rome, Italy and INFN, Sezione di Roma I, IPCF-CNR, Piazzale 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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17
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Fytas NG, Zierenberg J, Theodorakis PE, Weigel M, Janke W, Malakis A. Universality from disorder in the random-bond Blume-Capel model. Phys Rev E 2018; 97:040102. [PMID: 29758610 DOI: 10.1103/physreve.97.040102] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2017] [Indexed: 01/23/2023]
Abstract
Using high-precision Monte Carlo simulations and finite-size scaling we study the effect of quenched disorder in the exchange couplings on the Blume-Capel model on the square lattice. The first-order transition for large crystal-field coupling is softened to become continuous, with a divergent correlation length. An analysis of the scaling of the correlation length as well as the susceptibility and specific heat reveals that it belongs to the universality class of the Ising model with additional logarithmic corrections which is also observed for the Ising model itself if coupled to weak disorder. While the leading scaling behavior of the disordered system is therefore identical between the second-order and first-order segments of the phase diagram of the pure model, the finite-size scaling in the ex-first-order regime is affected by strong transient effects with a crossover length scale L^{*}≈32 for the chosen parameters.
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Affiliation(s)
- N G Fytas
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
| | - J Zierenberg
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig, Germany.,Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany.,Bernstein Center for Computational Neuroscience, 37077 Göttingen, Germany
| | - P E Theodorakis
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
| | - M Weigel
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
| | - W Janke
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig, Germany
| | - A Malakis
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom.,Department of Physics, Section of Solid State Physics, University of Athens, Panepistimiopolis, GR 15784 Zografou, Greece
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