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Haga T. Machine learning analysis of dimensional reduction conjecture for nonequilibrium Berezinskii-Kosterlitz-Thouless transition in three dimensions. Phys Rev E 2024; 110:014137. [PMID: 39160920 DOI: 10.1103/physreve.110.014137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2023] [Accepted: 07/01/2024] [Indexed: 08/21/2024]
Abstract
We investigate the recently proposed dimensional reduction conjecture in driven disordered systems using a machine learning technique. The conjecture states that a static snapshot of a disordered system driven at a constant velocity is equal to a space-time trajectory of its lower-dimensional pure counterpart. This suggests that the three-dimensional (3D) random field XY model exhibits the Berezinskii-Kosterlitz-Thouless transition when driven out of equilibrium. To verify the conjecture directly by observing configurations of the system, we utilize the capacity of neural networks to detect subtle features of images. Specifically, we train neural networks to differentiate snapshots of the 3D driven random field XY model from space-time trajectories of the two-dimensional pure XY model. Our results demonstrate that the network cannot distinguish between the two, confirming the dimensional reduction conjecture.
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2
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Yabunaka S, Fleming C, Delamotte B. Incompleteness of the large-N analysis of the O(N) models: Nonperturbative cuspy fixed points and their nontrivial homotopy at finite N. Phys Rev E 2022; 106:054105. [PMID: 36559345 DOI: 10.1103/physreve.106.054105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Accepted: 09/21/2022] [Indexed: 11/06/2022]
Abstract
We summarize the usual implementations of the large-N limit of O(N) models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit N→∞. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as N increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of N. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all N>0 and show that they play an important role for the tricritical physics of O(N) models. Finally, we show that some of these fixed points are bivalued when they are considered as functions of d and N thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at N=∞ and d=3 is shown to play a crucial role for the internal consistency of all our results.
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Affiliation(s)
- S Yabunaka
- Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan
| | - C Fleming
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, F-75005, Paris, France
| | - B Delamotte
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, F-75005, Paris, France
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3
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Ruiz-Lorenzo JJ, Dudka M, Holovatch Y. Critical behavior of the three-dimensional random-anisotropy Heisenberg model. Phys Rev E 2022; 106:034123. [PMID: 36266892 DOI: 10.1103/physreve.106.034123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2021] [Accepted: 08/04/2022] [Indexed: 06/16/2023]
Abstract
We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.
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Affiliation(s)
- J J Ruiz-Lorenzo
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, 06071 Badajoz, Spain and Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50018 Zaragoza, Spain
| | - M Dudka
- Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, UA-79011 Lviv, Ukraine and L^{4} Collaboration and Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, D-04009 Leipzig, Germany
| | - Yu Holovatch
- Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, UA-79011 Lviv, Ukraine; L^{4} Collaboration and Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, D-04009 Leipzig, Germany; Centre for Fluid and Complex Systems, Coventry University, Coventry, CV1 5FB, United Kingdom; and Complexity Science Hub Vienna, 1080 Vienna, Austria
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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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Dupuis N, Daviet R. Bose-glass phase of a one-dimensional disordered Bose fluid: Metastable states, quantum tunneling, and droplets. Phys Rev E 2020; 101:042139. [PMID: 32422844 DOI: 10.1103/physreve.101.042139] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2019] [Accepted: 03/23/2020] [Indexed: 06/11/2023]
Abstract
We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. We find that the Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale k, quantum tunneling between the ground state and low-lying metastable states leads to a rounding of the cusp singularity into a quantum boundary layer (QBL). The width of the QBL depends on an effective Luttinger parameter K_{k}∼k^{θ} that vanishes with an exponent θ=z-1 related to the dynamical critical exponent z. The QBL encodes the existence of rare "superfluid" regions, controls the low-energy dynamics, and yields a (dissipative) conductivity vanishing as ω^{2} in the low-frequency limit. These results reveal the glassy properties (pinning, "shocks," or static avalanches) of the Bose-glass phase and can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.
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Affiliation(s)
- Nicolas Dupuis
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France
| | - Romain Daviet
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France
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6
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Dupuis N. Glassy properties of the Bose-glass phase of a one-dimensional disordered Bose fluid. Phys Rev E 2019; 100:030102. [PMID: 31640005 DOI: 10.1103/physreve.100.030102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2019] [Indexed: 06/10/2023]
Abstract
We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. The Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale, quantum tunneling between these metastable states leads to a rounding of the nonanalyticity in a quantum boundary layer that encodes the existence of rare superfluid regions responsible for the ω^{2} behavior of the (dissipative) conductivity in the low-frequency limit. These results can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.
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Affiliation(s)
- Nicolas Dupuis
- Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, Sorbonne Université, CNRS, F-75005 Paris, France
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7
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Andrade EC, Hoyos JA, Rachel S, Vojta M. Cluster-Glass Phase in Pyrochlore XY Antiferromagnets with Quenched Disorder. PHYSICAL REVIEW LETTERS 2018; 120:097204. [PMID: 29547304 DOI: 10.1103/physrevlett.120.097204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2017] [Indexed: 06/08/2023]
Abstract
We study the impact of quenched disorder (random exchange couplings or site dilution) on easy-plane pyrochlore antiferromagnets. In the clean system, order by disorder selects a magnetically ordered state from a classically degenerate manifold. In the presence of randomness, however, different orders can be chosen locally depending on details of the disorder configuration. Using a combination of analytical considerations and classical Monte Carlo simulations, we argue that any long-range-ordered magnetic state is destroyed beyond a critical level of randomness where the system breaks into magnetic domains due to random exchange anisotropies, becoming, therefore, a glass of spin clusters, in accordance with the available experimental data. These random anisotropies originate from off-diagonal exchange couplings in the microscopic Hamiltonian, establishing their relevance to other magnets with strong spin-orbit coupling.
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Affiliation(s)
- Eric C Andrade
- Instituto de Física de São Carlos, Universidade de São Paulo, C.P. 369, São Carlos, São Paulo 13560-970, Brazil
| | - José A Hoyos
- Instituto de Física de São Carlos, Universidade de São Paulo, C.P. 369, São Carlos, São Paulo 13560-970, Brazil
| | - Stephan Rachel
- Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
- School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
| | - Matthias Vojta
- Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
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8
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Haga T. Nonequilibrium quasi-long-range order of a driven random-field O(N) model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062113. [PMID: 26764638 DOI: 10.1103/physreve.92.062113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2015] [Indexed: 06/05/2023]
Abstract
We investigate three-dimensional O(N) spin models driven with a uniform velocity over a random field. Within a spin-wave approximation, it is shown that in the strong driving regime the model with N=2 exhibits a quasi-long-range order in which the spatial correlation function decays in a power-law form. Furthermore, for the cases that N=2 and 3, we numerically demonstrate a nonequilibrium phase transition between the quasi-long-range order phase and the disordered phase, which turns out to resemble the Kosterlitz-Thouless transition in the two-dimensional pure XY model in equilibrium.
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Affiliation(s)
- Taiki Haga
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
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9
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Tarjus G, Baczyk M, Tissier M. Avalanches and dimensional reduction breakdown in the critical behavior of disordered systems. PHYSICAL REVIEW LETTERS 2013; 110:135703. [PMID: 23581342 DOI: 10.1103/physrevlett.110.135703] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2012] [Indexed: 06/02/2023]
Abstract
We investigate the connection between a formal property of the critical behavior of several disordered systems, known as "dimensional reduction," and the presence in these systems at zero temperature of collective events known as "avalanches." Avalanches generically produce nonanalyticities in the functional dependence of the cumulants of the renormalized disorder. We show that this leads to a breakdown of the dimensional reduction predictions if and only if the fractal dimension characterizing the scaling properties of the avalanches is exactly equal to the difference between the dimension of space and the scaling dimension of the primary field. This is proven by combining scaling theory and the functional renormalization group. We therefore clarify the puzzle of why dimensional reduction remains valid in random field systems above a nontrivial dimension (but fails below), always applies to the statistics of branched polymer, and is always wrong in elastic models of interfaces in a random environment.
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Affiliation(s)
- Gilles Tarjus
- LPTMC, CNRS-UMR 7600, Université Pierre et Marie Curie, Boîte 121, 4 Place Jussieu, 75252 Paris Cédex 05, France.
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10
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Zhang Q, Radzihovsky L. Smectic order, pinning, and phase transition in a smectic-liquid-crystal cell with a random substrate. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022509. [PMID: 23496537 DOI: 10.1103/physreve.87.022509] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/2012] [Indexed: 06/01/2023]
Abstract
We study smectic-liquid-crystal order in a cell with a heterogeneous substrate imposing surface random positional and orientational pinnings. Proposing a minimal random elastic model, we demonstrate that, for a thick cell, the smectic state without a rubbed substrate is always unstable at long scales and, for weak random pinning, is replaced by a smectic glass state. We compute the statistics of the associated substrate-driven distortions and the characteristic smectic domain size on the heterogeneous substrate and in the bulk. We find that for weak disorder, the system exhibits a three-dimensional temperature-controlled phase transition between a weakly and strongly pinned smectic glass states akin to the Cardy-Ostlund phase transition. We explore experimental implications of the predicted phenomenology and suggest that it provides a plausible explanation for the experimental observations on polarized light microscopy and x-ray scattering.
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Affiliation(s)
- Quan Zhang
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
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11
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Fedorenko AA. Random-field and random-anisotropy O(N) spin systems with a free surface. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:021131. [PMID: 23005746 DOI: 10.1103/physreve.86.021131] [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: 07/12/2012] [Indexed: 06/01/2023]
Abstract
We study the surface scaling behavior of a semi-infinite d-dimensional O(N) spin system in the presence of a quenched random field and random anisotropy disorders. It is known that above the lower critical dimension d(LC) = 4 the infinite models undergo a paramagnetic-ferromagnetic transition for N > N(c) (N(c) = 2.835 for the random field and N(c) =9.441 for random anisotropy). For N < N(c) and d < d(LC) there exists a quasi-long-range-order phase with a zero order parameter and a power-law decay of spin correlations. Using a functional renormalization group, we derive the surface scaling laws that describe the ordinary surface transition for d > d(LC) and the long-range behavior of spin correlations near the surface in the quasi-long-range-order phase for d < d(LC). The corresponding surface exponents are calculated to one-loop order. The obtained results can be applied to the surface scaling of periodic elastic systems in disordered media, amorphous magnets, and (3)He-A in aerogel.
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Affiliation(s)
- Andrei A Fedorenko
- Laboratoire de Physique de l'Ecole Normale Supérieure de Lyon, Unité Mixte de Recherche No 5672 Associée au Centre National de la Recherche Scientifique, 46 Allée d'Italie, 69007 Lyon, France
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12
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Mouhanna D, Tarjus G. Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:051101. [PMID: 20866179 DOI: 10.1103/physreve.81.051101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2009] [Indexed: 05/29/2023]
Abstract
We investigate the relation between spontaneous and explicit replica symmetry breaking in the theory of disordered systems. On general ground, we prove the equivalence between the replicon operator associated with the stability of the replica-symmetric solution in the standard replica scheme and the operator signaling a breakdown of the solution with analytic field dependence in a scheme in which replica symmetry is explicitly broken by applied sources. This opens the possibility to study, via the recently developed functional renormalization group, unresolved questions related to spontaneous replica symmetry breaking and spin-glass behavior in finite-dimensional disordered systems.
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Affiliation(s)
- D Mouhanna
- LPTMC, CNRS UMR 7600, UPMC, 4 Place Jussieu, 75252 Paris Cedex 05, France.
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13
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Zhang Q, Radzihovsky L. Stability and distortions of liquid crystal order in a cell with a heterogeneous substrate. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:051701. [PMID: 20866241 DOI: 10.1103/physreve.81.051701] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2009] [Indexed: 05/29/2023]
Abstract
We study stability and distortions of liquid crystal nematic order in a cell with a random heterogeneous substrate. Modeling this system as a bulk xy model with quenched disorder confined to a surface, we find that nematic order is marginally unstable to such surface pinning. We compute the length scale beyond which nematic distortions become large and calculate orientational correlation functions using the functional renormalization-group and matching methods, finding universal logarithmic and double-logarithmic distortions in two and three dimensions, respectively. We extend these results to a finite-thickness liquid crystal cell with a second homogeneous substrate, detailing crossovers as a function of random pinning strength and cell thickness. We conclude with analysis of experimental signatures of these distortions in a conventional crossed-polarizer-analyzer light microscopy.
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Affiliation(s)
- Quan Zhang
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
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14
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Radzihovsky L, Zhang Q. Liquid crystal cells with "dirty" substrates. PHYSICAL REVIEW LETTERS 2009; 103:167802. [PMID: 19905725 DOI: 10.1103/physrevlett.103.167802] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2009] [Indexed: 05/28/2023]
Abstract
We explore liquid crystal order in a cell with a "dirty" substrate imposing a random surface pinning. Modeling such systems by a random-field xy model with surface heterogeneity, we find that orientational order in the three-dimensional system is marginally unstable to such surface pinning. We compute the Larkin length scale, and the corresponding surface and bulk distortions. On longer scales we calculate correlation functions using the functional renormalization group and matching methods, finding a universal logarithmic and double-logarithmic roughness in two and three dimensions, respectively. For a finite thickness cell, we explore the interplay of homogeneous-heterogeneous substrate pair and detail crossovers as a function of disorder strength and cell thickness.
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Affiliation(s)
- Leo Radzihovsky
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
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Le Doussal P, Middleton AA, Wiese KJ. Statistics of static avalanches in a random pinning landscape. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:050101. [PMID: 19518396 DOI: 10.1103/physreve.79.050101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2008] [Indexed: 05/27/2023]
Abstract
We study the minimum-energy configuration of a d -dimensional elastic interface in a random potential tied to a harmonic spring. As a function of the spring position, the center of mass of the interface changes in discrete jumps, also called shocks or "static avalanches." We obtain analytically the distribution of avalanche sizes and its cumulants within an =4-d expansion from a tree and one-loop resummation using functional renormalization. This is compared with exact numerical minimizations of interface energies for random-field disorder in d=2,3 . Connections to dynamic avalanches are mentioned.
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Affiliation(s)
- Pierre Le Doussal
- Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, CNRS, 24 rue Lhomond, 75005 Paris, France
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16
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Le Doussal P, Wiese KJ. Size distributions of shocks and static avalanches from the functional renormalization group. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051106. [PMID: 19518415 DOI: 10.1103/physreve.79.051106] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2009] [Indexed: 05/27/2023]
Abstract
Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution P(S) of static avalanches in an expansion in the internal dimension d of the interface. Near and above d=4 this yields the mean-field distribution P(S) approximately S;{-3/2}e;{-S4S_{m}} , where S_{m} is a large-scale cutoff, in some cases calculable. Resumming all one-loop contributions, we find P(S) approximately S;{-tau}exp(C(SS_{m});{1/2}-B/4(S/S_{m});{delta}) , where B , C , delta , and tau are obtained to first order in =4-d . Our result is consistent to O() with the relation tau=tau_{zeta}:=2-2/d+zeta , where zeta is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field, and random-periodic disorders. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with tau=2-1/d+zeta to O(=2-d) . We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the relation tau=tau_{zeta} be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyperplane of codimension one is in mean field (valid close to and above d=4 ) given by P(S) approximately K_{13}(S)S , where K is the Bessel- K function, thus tau_{hyperplane}=4/3 .
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Affiliation(s)
- Pierre Le Doussal
- Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, CNRS, 24 rue Lhomond, 75231 Paris Cedex, France
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17
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Le Doussal P, Wiese KJ. Driven particle in a random landscape: disorder correlator, avalanche distribution, and extreme value statistics of records. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051105. [PMID: 19518414 DOI: 10.1103/physreve.79.051105] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2008] [Indexed: 05/27/2023]
Abstract
We review how the renormalized force correlator Delta(micro) , the function computed in the functional renormalization-group (RG) field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in the presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a one-dimensional random-force landscape. The limit of small velocity allows one to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Fréchet. For each class we obtain analytically the universal function Delta(micro) , the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w . We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model, avalanche distributions and Delta(micro) can be computed for any velocity. For two-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(micro[over ]) , and to extract the anisotropic scaling exponents zeta_{x}>zeta_{y} . We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered, e.g., in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w .
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Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
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18
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Tiampo KF, Rundle JB, Klein W, Holliday J, Sá Martins JS, Ferguson CD. Ergodicity in natural earthquake fault networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:066107. [PMID: 17677325 DOI: 10.1103/physreve.75.066107] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2006] [Revised: 04/08/2007] [Indexed: 05/16/2023]
Abstract
Numerical simulations have shown that certain driven nonlinear systems can be characterized by mean-field statistical properties often associated with ergodic dynamics [C. D. Ferguson, W. Klein, and J. B. Rundle, Phys. Rev. E 60, 1359 (1999); D. Egolf, Science 287, 101 (2000)]. These driven mean-field threshold systems feature long-range interactions and can be treated as equilibriumlike systems with statistically stationary dynamics over long time intervals. Recently the equilibrium property of ergodicity was identified in an earthquake fault system, a natural driven threshold system, by means of the Thirumalai-Mountain (TM) fluctuation metric developed in the study of diffusive systems [K. F. Tiampo, J. B. Rundle, W. Klein, J. S. Sá Martins, and C. D. Ferguson, Phys. Rev. Lett. 91, 238501 (2003)]. We analyze the seismicity of three naturally occurring earthquake fault networks from a variety of tectonic settings in an attempt to investigate the range of applicability of effective ergodicity, using the TM metric and other related statistics. Results suggest that, once variations in the catalog data resulting from technical and network issues are accounted for, all of these natural earthquake systems display stationary periods of metastable equilibrium and effective ergodicity that are disrupted by large events. We conclude that a constant rate of events is an important prerequisite for these periods of punctuated ergodicity and that, while the level of temporal variability in the spatial statistics is the controlling factor in the ergodic behavior of seismic networks, no single statistic is sufficient to ensure quantification of ergodicity. Ergodicity in this application not only requires that the system be stationary for these networks at the applicable spatial and temporal scales, but also implies that they are in a state of metastable equilibrium, one in which the ensemble averages can be substituted for temporal averages in studying their spatiotemporal evolution.
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Affiliation(s)
- K F Tiampo
- Department of Earth Sciences, University of Western Ontario, London, Ontario, N6A 5B7 Canada.
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Middleton AA, Le Doussal P, Wiese KJ. Measuring functional renormalization group fixed-point functions for pinned manifolds. PHYSICAL REVIEW LETTERS 2007; 98:155701. [PMID: 17501361 DOI: 10.1103/physrevlett.98.155701] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2006] [Indexed: 05/15/2023]
Abstract
Exact numerical minimization of interface energies is used to test the functional renormalization group analysis for interfaces pinned by quenched disorder. The fixed-point function R(u) (the correlator of the coarse-grained disorder) is computed. In dimensions D=d+1, a linear cusp in R''(u) is confirmed for random bond (d=1, 2, 3), random field (d=0, 2, 3), and periodic (d=2, 3) disorders. The functional shocks that lead to this cusp are seen. Small, but significant, deviations from the 1-loop calculation are compared to 2-loop corrections and chaos is measured.
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Affiliation(s)
- A Alan Middleton
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA
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Le Doussal P, Wiese KJ. Random-field spin models beyond 1 loop: a mechanism for decreasing the lower critical dimension. PHYSICAL REVIEW LETTERS 2006; 96:197202. [PMID: 16803135 DOI: 10.1103/physrevlett.96.197202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2005] [Indexed: 05/10/2023]
Abstract
The functional renormalization group for the random-field and random-anisotropy O(N) sigma models is studied to 2 loop. The ferromagnetic-disordered (F-D) transition fixed point is found to next order in d = 4 + epsilon for N > N(c) (N(c) = 2.834 740 8 for random field, N(c) = 9.441 21 for random anisotropy). For N < N(c) the lower critical dimension d = d(lc) plunges below d(lc) = 4: we find two fixed points, one describing the quasiordered phase, the other is novel and describes the F-D transition. d(lc) can be obtained in an (N(c)-N) expansion. The theory is also analyzed at large N and a glassy regime is found.
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Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris, France
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21
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Tissier M, Tarjus G. Unified picture of ferromagnetism, quasi-long-range order, and criticality in random-field models. PHYSICAL REVIEW LETTERS 2006; 96:087202. [PMID: 16606218 DOI: 10.1103/physrevlett.96.087202] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2005] [Indexed: 05/08/2023]
Abstract
By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the d-dimensional random-field O(N) model in the whole (N, d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N) model, with, however, no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely, a topologically ordered "Bragg glass" phase, is absent in the 3-dimensional random-field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around d = 4.
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Affiliation(s)
- Matthieu Tissier
- LPTMC, Université Pierre et Marie Curie, boîte 121, 4 Place Jussieu, 75252 Paris cédex 05, France
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22
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Tarjus G, Tissier M. Nonperturbative functional renormalization group for random-field models: the way out of dimensional reduction. PHYSICAL REVIEW LETTERS 2004; 93:267008. [PMID: 15698014 DOI: 10.1103/physrevlett.93.267008] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2004] [Indexed: 05/24/2023]
Abstract
We develop a nonperturbative functional renormalization group approach for the random-field O(N) model that allows us to investigate the ordering transition in any dimension and for any value of N including the Ising case. We show that the failure of dimensional reduction and standard perturbation theory is due to the nonanalytic nature of the zero-temperature fixed point controlling the critical behavior, nonanalyticity, which is associated with the existence of many metastable states. We find that this nonanalyticity leads to critical exponents differing from the dimensional reduction prediction only below a critical dimension dc(N)<6, with dc(N=1)>3.
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Affiliation(s)
- Gilles Tarjus
- LPTL, Université Pierre et Marie Curie, boîte 121, 4 Place Jussieu, 75252 Paris cédex 05, France.
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23
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Calabrese P, Pelissetto A, Vicari E. Spin models with random anisotropy and reflection symmetry. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:036104. [PMID: 15524584 DOI: 10.1103/physreve.70.036104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2003] [Indexed: 05/24/2023]
Abstract
We study the critical behavior of a general class of cubic-symmetric spin systems in which disorder preserves the reflection symmetry s(a) --> -s(a) , s(b) --> s(b) for b not equala . This includes spin models in the presence of random cubic-symmetric anisotropy with probability distribution vanishing outside the lattice axes. Using nonperturbative arguments we show the existence of a stable fixed point corresponding to the random-exchange Ising universality class. The field-theoretical renormalization-group flow is investigated in the framework of a fixed-dimension expansion in powers of appropriate quartic couplings, computing the corresponding beta functions to five loops. This analysis shows that the random Ising fixed point is the only stable fixed point that is accessible from the relevant parameter region. Therefore, if the system undergoes a continuous transition, it belongs to the random-exchange Ising universality class. The approach to the asymptotic critical behavior is controlled by scaling corrections with exponent Delta= -alpha(r) , where alpha(r) approximately -0.05 is the specific-heat exponent of the random-exchange Ising model.
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Affiliation(s)
- Pasquale Calabrese
- Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy.
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24
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Le Doussal P, Wiese KJ, Chauve P. Functional renormalization group and the field theory of disordered elastic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:026112. [PMID: 14995525 DOI: 10.1103/physreve.69.026112] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2003] [Indexed: 05/24/2023]
Abstract
We study elastic systems, such as interfaces or lattices, pinned by quenched disorder. To escape triviality as a result of "dimensional reduction," we use the functional renormalization group. Difficulties arise in the calculation of the renormalization group functions beyond one-loop order. Even worse, observables such as the two-point correlation function exhibit the same problem already at one-loop order. These difficulties are due to the nonanalyticity of the renormalized disorder correlator at zero temperature, which is inherent to the physics beyond the Larkin length, characterized by many metastable states. As a result, two-loop diagrams, which involve derivatives of the disorder correlator at the nonanalytic point, are naively "ambiguous." We examine several routes out of this dilemma, which lead to a unique renormalizable field theory at two-loop order. It is also the only theory consistent with the potentiality of the problem. The beta function differs from previous work and the one at depinning by novel "anomalous terms." For interfaces and random-bond disorder we find a roughness exponent zeta=0.208 298 04epsilon+0.006 858epsilon(2), epsilon=4-d. For random-field disorder we find zeta=epsilon/3 and compute universal amplitudes to order O(epsilon(2)). For periodic systems we evaluate the universal amplitude of the two-point function. We also clarify the dependence of universal amplitudes on the boundary conditions at large scale. All predictions are in good agreement with numerical and exact results and are an improvement over one loop. Finally we calculate higher correlation functions, which turn out to be equivalent to those at depinning to leading order in epsilon.
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Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
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Tiampo KF, Rundle JB, Klein W, Martins JSS, Ferguson CD. Ergodic dynamics in a natural threshold system. PHYSICAL REVIEW LETTERS 2003; 91:238501. [PMID: 14683219 DOI: 10.1103/physrevlett.91.238501] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2003] [Indexed: 05/24/2023]
Abstract
Numerical simulations suggest that certain driven, dissipative mean-field threshold systems, including earthquake models, can be characterized by statistical properties often associated with ergodic dynamics, in the same sense as stochastic Brownian motion. We applied a fluctuation metric proposed by Thirumalai and Mountain [Phys. Rev. E 47, 479 (1993)]] for statistically stationary systems and find that the natural earthquake fault system in California demonstrates similar ergodic dynamics.
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Affiliation(s)
- K F Tiampo
- CIRES, University of Colorado, Boulder, Colorado 80309, USA
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26
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Schehr G, Le Doussal P. Exact multilocal renormalization of the effective action: application to the random sine Gordon model statics and nonequilibrium dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046101. [PMID: 14682996 DOI: 10.1103/physreve.68.046101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2003] [Indexed: 05/24/2023]
Abstract
We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the eta exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent z are recovered and several results are obtained, such as the equilibrium two-point scaling functions. The nonequilibrium, finite momentum, two-time t,t' response and correlations are computed. They are shown to exhibit scaling forms, characterized by exponents lambda(R) not equal lambda(C), as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be nontrivial and of the form X[q(z)(t-t'),t/t']. Analogies and differences with pure critical models are discussed.
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Affiliation(s)
- Gregory Schehr
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris, France
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27
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Le Doussal P, Wiese KJ. Higher correlations, universal distributions, and finite size scaling in the field theory of depinning. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046118. [PMID: 14683013 DOI: 10.1103/physreve.68.046118] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2003] [Indexed: 05/24/2023]
Abstract
Recently we constructed a renormalizable field theory up to two loops for the quasistatic depinning of elastic manifolds in a disordered environment. Here we explore further properties of the theory. We show how higher correlation functions of the displacement field can be computed. Drastic simplifications occur, unveiling much simpler diagrammatic rules than anticipated. This is applied to the universal scaled width distribution. The expansion in d=4-epsilon predicts that the scaled distribution coincides to the lowest orders with the one for a Gaussian theory with propagator G(q)=1/q(d+2 zeta), zeta being the roughness exponent. The deviations from this Gaussian result are small and involve higher correlation functions, which are computed here for different boundary conditions. Other universal quantities are defined and evaluated: We perform a general analysis of the stability of the fixed point. We find that the correction-to-scaling exponent is omega=-epsilon and not -epsilon/3 as used in the analysis of some simulations. A more detailed study of the upper critical dimension is given, where the roughness of interfaces grows as a power of a logarithm instead of a pure power.
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Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
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28
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Feldman DE. Critical exponents of the random-field O(N) model. PHYSICAL REVIEW LETTERS 2002; 88:177202. [PMID: 12005781 DOI: 10.1103/physrevlett.88.177202] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2000] [Indexed: 05/23/2023]
Abstract
The critical behavior of the random-field Ising model has long been a puzzle. Different methods predict that its critical exponents in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components contrary to the experiments and simulations. We calculate the exponents of the random-field O(N) model with the (4+epsilon)-expansion and obtain values different from the exponents of the pure ferromagnet in 2+epsilon dimensions. An infinite set of relevant operators missed in previous studies leads to a breakdown of the (6-epsilon)-expansion.
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Affiliation(s)
- D E Feldman
- Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA
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29
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Rundle JB, Tiampo KF, Klein W, Sa Martins JS. Self-organization in leaky threshold systems: the influence of near-mean field dynamics and its implications for earthquakes, neurobiology, and forecasting. Proc Natl Acad Sci U S A 2002; 99 Suppl 1:2514-21. [PMID: 11875204 PMCID: PMC128570 DOI: 10.1073/pnas.012581899] [Citation(s) in RCA: 122] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Threshold systems are known to be some of the most important nonlinear self-organizing systems in nature, including networks of earthquake faults, neural networks, superconductors and semiconductors, and the World Wide Web, as well as political, social, and ecological systems. All of these systems have dynamics that are strongly correlated in space and time, and all typically display a multiplicity of spatial and temporal scales. Here we discuss the physics of self-organization in earthquake threshold systems at two distinct scales: (i) The "microscopic" laboratory scale, in which consideration of results from simulations leads to dynamical equations that can be used to derive the results obtained from sliding friction experiments, and (ii) the "macroscopic" earthquake fault-system scale, in which the physics of strongly correlated earthquake fault systems can be understood by using time-dependent state vectors defined in a Hilbert space of eigenstates, similar in many respects to the mathematics of quantum mechanics. In all of these systems, long-range interactions induce the existence of locally ergodic dynamics. The existence of dissipative effects leads to the appearance of a "leaky threshold" dynamics, equivalent to a new scaling field that controls the size of nucleation events relative to the size of background fluctuations. At the macroscopic earthquake fault-system scale, these ideas show considerable promise as a means of forecasting future earthquake activity.
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Affiliation(s)
- J B Rundle
- Colorado Center for Chaos and Complexity and Cooperative Institute for Research in Environmental Sciences (CIRES) and Department of Physics, 216 UCB, University of Colorado, Boulder, CO 80309-0216, USA.
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30
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Bellini T, Radzihovsky L, Toner J, Clark NA. Universality and scaling in the disordering of a smectic liquid crystal. Science 2001; 294:1074-9. [PMID: 11691985 DOI: 10.1126/science.1057480] [Citation(s) in RCA: 175] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
We present experimental and theoretical studies of the effects of quenched disorder on one-dimensional crystal ordering in three dimensions. This fragile smectic liquid crystal layering, the material with the simplest positional order, is also the most easily deformed periodic structure and is, therefore, profoundly affected by disorder, introduced here by confinement in silica aerogel. Theory and experiment combine to characterize this system to an extraordinary degree, their close accord producing a coherent picture: crystal ordering is lost, giving way to extended short-range correlations that exhibit universal structure and scaling, anomalous layer elasticity, and glassy dynamics.
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Affiliation(s)
- T Bellini
- Department of Physics, University of Colorado, Boulder, CO 80309, USA
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31
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Chauve P, Le Doussal P. Exact multilocal renormalization group and applications to disordered problems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:051102. [PMID: 11735895 DOI: 10.1103/physreve.64.051102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2000] [Revised: 01/02/2001] [Indexed: 05/23/2023]
Abstract
We develop a method, the exact multilocal renormalization group (EMRG) which applies to a broad set of theories. It is based on the systematic multilocal expansion of the Polchinski-Wilson exact renormalization group (ERG) equation together with a scheme to compute correlation functions. Integrating out explicitly the nonlocal interactions, we reduce the ERG equation obeyed by the full interaction functional to a flow equation for a function, its local part. This is done perturbatively around fixed points, but exactly to any given order in the local part. It is thus controlled, at variance with projection methods, e.g., derivative expansions or local potential approximations. Our EMRG method is well-suited to problems such as the pinning of disordered elastic systems, previously described via functional renormalization group (FRG) approach based on a hard cutoff scheme. Since it involves arbitrary cutoff functions, we explicitly verify universality to O(epsilon=4-D), both of the T=0 FRG equation and of correlations. Extension to finite temperature T yields the finite size (L) susceptibility fluctuations characterizing mesoscopic behavior (Deltachi)2 approximately L(straight theta)/T, where straight theta is the energy exponent. Finally, we obtain the universal scaling function to O(epsilon(1/3)) which describes the ground state of a domain wall in a random field confined by a field gradient, compare with exact results and variational method. Explicit two loop exact RG equations are derived and the application to the FRG problem is sketched.
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Affiliation(s)
- P Chauve
- Laboratoire de Physique des Solides, Université Paris-Sud, Bâtiment 510, 91405 Orsay, France
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Belitz D, Kirkpatrick TR. Order parameter description of the Anderson-Mott transition. ACTA ACUST UNITED AC 1995. [DOI: 10.1007/bf01320853] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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35
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Balents L, Nelson DR. Quantum smectic and supersolid order in helium films and vortex arrays. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:12951-12968. [PMID: 9980467 DOI: 10.1103/physrevb.52.12951] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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36
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Fisch R. Critical behavior of randomly pinned spin-density waves. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:11507-11514. [PMID: 9977882 DOI: 10.1103/physrevb.51.11507] [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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37
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Kirkpatrick TR, Belitz D. Anderson-Mott transition as a random-field problem. PHYSICAL REVIEW LETTERS 1995; 74:1178-1181. [PMID: 10058954 DOI: 10.1103/physrevlett.74.1178] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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38
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Lee KM, O'Shea MJ. Phase transitions and critical phenomena in alloys with random anisotropy. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:13614-13624. [PMID: 10007762 DOI: 10.1103/physrevb.48.13614] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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39
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Cao MS, Machta J. Migdal-Kadanoff study of the random-field Ising model. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:3177-3182. [PMID: 10008741 DOI: 10.1103/physrevb.48.3177] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Narayan O, Fisher DS. Critical behavior of sliding charge-density waves in 4- epsilon dimensions. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:11520-11549. [PMID: 10003040 DOI: 10.1103/physrevb.46.11520] [Citation(s) in RCA: 165] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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41
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Fisch R. Ferromagnetism in the XY model with random threefold anisotropy. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:242-246. [PMID: 10002205 DOI: 10.1103/physrevb.46.242] [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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42
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Fisch R. Infinite susceptibility phase in planar random-anisotropy magnets. PHYSICAL REVIEW LETTERS 1991; 66:2041-2044. [PMID: 10043375 DOI: 10.1103/physrevlett.66.2041] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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43
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Halpin-Healy T. Disorder-induced roughening of diverse manifolds. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:711-722. [PMID: 9904079 DOI: 10.1103/physreva.42.711] [Citation(s) in RCA: 65] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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44
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Fisch R. Low-temperature behavior of random-anisotropy magnets. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 42:540-544. [PMID: 9994571 DOI: 10.1103/physrevb.42.540] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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45
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Gehring PM, Salamon MB, Arnaudas JI. Magnetic static and scaling properties of the weak random-axis magnet (DyxY1-x)Al2. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 41:9134-9147. [PMID: 9993257 DOI: 10.1103/physrevb.41.9134] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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46
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Thirumalai D, Kirkpatrick TR. Mean-field Potts glass model: Initial-condition effects on dynamics and properties of metastable states. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 38:4881-4892. [PMID: 9946882 DOI: 10.1103/physrevb.38.4881] [Citation(s) in RCA: 52] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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47
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Golubovic L, Kulic M. Random fields and orientational order in models with a continuous-energy minimum set in q space and in magnetic systems favoring an incommensurate order. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:7582-7596. [PMID: 9944054 DOI: 10.1103/physrevb.37.7582] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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48
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Busiello G, Rabuffo I, D'Auria AC. Large-n -limit model for quantum structural phase transitions with correlated random fields. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 35:3536-3544. [PMID: 9941857 DOI: 10.1103/physrevb.35.3536] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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49
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Sellmyer DJ, Nafis S. Phase-transition behavior in a random-anisotropy system. PHYSICAL REVIEW LETTERS 1986; 57:1173-1176. [PMID: 10034266 DOI: 10.1103/physrevlett.57.1173] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Serota RA, Lee PA. Continuous-symmetry ferromagnet with weak random magnetic anisotropy: A numerical study. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 34:1806-1810. [PMID: 9939836 DOI: 10.1103/physrevb.34.1806] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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