1
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Setty C, Baggioli M, Zaccone A. Anharmonic theory of superconductivity and its applications to emerging quantum materials. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:173002. [PMID: 38252997 DOI: 10.1088/1361-648x/ad2159] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2023] [Accepted: 01/22/2024] [Indexed: 01/24/2024]
Abstract
The role of anharmonicity on superconductivity has often been disregarded in the past. Recently, it has been recognized that anharmonic decoherence could play a fundamental role in determining the superconducting properties (electron-phonon coupling, critical temperature, etc) of a large class of materials, including systems close to structural soft-mode instabilities, amorphous solids and metals under extreme high-pressure conditions. Here, we review recent theoretical progress on the role of anharmonic effects, and in particular certain universal properties of anharmonic damping, on superconductivity. Our focus regards the combination of microscopic-agnostic effective theories for bosonic mediators with the well-established BCS theory and Migdal-Eliashberg theory for superconductivity. We discuss in detail the theoretical frameworks, their possible implementation within first-principles methods, and the experimental probes for anharmonic decoherence. Finally, we present several concrete applications to emerging quantum materials, including hydrides, ferroelectrics and systems with charge density wave instabilities.
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Affiliation(s)
- Chandan Setty
- Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice University, Houston, TX 77005, United States of America
| | - Matteo Baggioli
- Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China
- Shanghai Research Center for Quantum Sciences, Shanghai 201315, People's Republic of China
| | - Alessio Zaccone
- Department of Physics 'A. Pontremoli', University of Milan, via Celoria 16, 20133 Milan, Italy
- Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, United Kingdom
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2
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Newman CM, Read N, Stein DL. Proof of Single-Replica Equivalence in Short-Range Spin Glasses. PHYSICAL REVIEW LETTERS 2023; 130:077102. [PMID: 36867816 DOI: 10.1103/physrevlett.130.077102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2022] [Accepted: 01/19/2023] [Indexed: 06/18/2023]
Abstract
We consider short-range Ising spin glasses in equilibrium at infinite system size, and prove that, for fixed bond realization and a given Gibbs state drawn from a suitable metastate, each translation and locally invariant function (for example, self-overlaps) of a single pure state in the decomposition of the Gibbs state takes the same value for all the pure states in that Gibbs state. We describe several significant applications to spin glasses.
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Affiliation(s)
- C M Newman
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
- NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
| | - N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
- Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
| | - D L Stein
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
- NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- Department of Physics, New York University, New York, New York 10012, USA
- NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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3
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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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4
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Read N. Complexity as information in spin-glass Gibbs states and metastates: Upper bounds at nonzero temperature and long-range models. Phys Rev E 2022; 105:054134. [PMID: 35706314 DOI: 10.1103/physreve.105.054134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2022] [Accepted: 03/24/2022] [Indexed: 06/15/2023]
Abstract
In classical finite-range spin systems, especially those with disorder such as spin glasses, a low-temperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly as the logarithm of this number, assuming the question is meaningful in a finite system. As nontrivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] Höller and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all T≥0. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are found for each of these complexities for general finite-range (i.e., short- or long-range, in a sense we define) mixed p-spin interactions of discrete or continuous spins (such as m-vector models), but only for T>0. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.
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Affiliation(s)
- N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA and Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
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5
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Newman CM, Stein DL. Ground-state stability and the nature of the spin glass phase. Phys Rev E 2022; 105:044132. [PMID: 35590620 DOI: 10.1103/physreve.105.044132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Accepted: 02/23/2022] [Indexed: 06/15/2023]
Abstract
We propose an approach toward understanding the spin glass phase at zero and low temperature by studying the stability of a spin glass ground state against perturbations of a single coupling. After reviewing the concepts of flexibility, critical droplet, and related quantities for both finite- and infinite-volume ground states, we study some of their properties and review three models in which these quantities are partially or fully understood. We also review a recent result showing the connection between our approach and that of disorder chaos. We then view four proposed scenarios for the low-temperature spin glass phase-replica symmetry breaking, scaling-droplet, TNT, and chaotic pairs-through the lens of the predictions of each scenario for the lowest-energy large-lengthscale excitations above the ground state. Using a new concept called σ-criticality, which quantifies the sensitivity of ground states to single-bond coupling variations, we show that each of these four pictures can be identified with different critical droplet geometries and energies. We also investigate necessary and sufficient conditions for the existence of multiple incongruent ground states.
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Affiliation(s)
- C M Newman
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
| | - D L Stein
- Department of Physics and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA; NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China; and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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6
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Jensen S, Read N, Young AP. Nontrivial maturation metastate-average state in a one-dimensional long-range Ising spin glass: Above and below the upper critical range. Phys Rev E 2021; 104:034105. [PMID: 34654158 DOI: 10.1103/physreve.104.034105] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2021] [Accepted: 08/13/2021] [Indexed: 11/07/2022]
Abstract
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature T_{c} for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability P_{ij} that an edge {i,j} has nonvanishing interaction falls as a power law with chord distance, P_{ij}∝1/R_{ij}^{2σ}, and we study a range of values of σ with 1/2<σ<1. We consider a correlation function C_{4}(r,t). A dynamic correlation length that shows power-law growth with time ξ(t)∝t^{1/z} can be identified in the data and, for large time t, C_{4}(r,t) decays as a power law r^{-α_{d}} with distance r when r≪ξ(t). The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent α_{d} differentiates between a trivial MMAS (α_{d}=0), as expected in the droplet picture of spin glasses, and a nontrivial MMAS (α_{d}≠0), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero α_{d} even in the regime σ>2/3 which corresponds to short-range systems below six dimensions. For σ<2/3, the decay exponent α_{d} follows the RSB prediction for the decay exponent α_{s}=3-4σ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches α_{d}=1-σ in the regime 2/3<σ<1; however, it deviates from both lines in the vicinity of σ=2/3.
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Affiliation(s)
- S Jensen
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA.,Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
| | - A P Young
- Physics Department, University of California, Santa Cruz, Santa Cruz, California 95064, USA
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7
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Höller J, Read N. One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses. Phys Rev E 2020; 101:042114. [PMID: 32422847 DOI: 10.1103/physreve.101.042114] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2019] [Accepted: 03/16/2020] [Indexed: 06/11/2023]
Abstract
The de Almeida-Thouless (AT) line is the phase boundary in the temperature-magnetic field plane of an Ising spin glass at which a continuous (i.e., second-order) transition from a paramagnet to a replica-symmetry-breaking (RSB) phase occurs, according to mean-field theory. Here, using field-theoretic perturbative renormalization group methods on the Bray-Roberts reduced Landau-Ginzburg-type theory for a short-range Ising spin glass in space of dimension d, we show that at nonzero magnetic field the nature of the corresponding transition is modified as follows: (a) For d-6 small and positive, with increasing field on the AT line, first, the ordered phase just below the transition becomes the so-called one-step RSB, instead of the full RSB that occurs in mean-field theory; the transition on the AT line remains continuous with a diverging correlation length. Then at a higher field, a tricritical point separates the latter transition from a quasi-first-order one, that is one at which the correlation length does not diverge, and there is a jump in part of the order parameter, but no latent heat. The location of the tricritical point tends to zero as d→6^{+}. (b) For d≤6, we argue that the quasi-first-order transition could persist down to arbitrarily small nonzero fields, with a transition to full RSB still expected at lower temperature. Whenever the quasi-first-order transition occurs, it is at a higher temperature than the AT transition would be for the same field, preempting it as the temperature is lowered. These results may explain the reported absence of a diverging correlation length in the presence of a magnetic field in low-dimensional spin glasses in some simulations and in high-temperature series expansions. We also draw attention to the similarity of the "dynamically frozen" state, which occurs at temperatures just above the quasi-first-order transition, and the "metastate-average state" of the one-step RSB phase, and discuss the issue of the number of pure states in either.
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Affiliation(s)
- J Höller
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
| | - N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
- Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
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8
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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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9
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Charbonneau P, Hu Y, Raju A, Sethna JP, Yaida S. Morphology of renormalization-group flow for the de Almeida-Thouless-Gardner universality class. Phys Rev E 2019; 99:022132. [PMID: 30934357 DOI: 10.1103/physreve.99.022132] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2018] [Indexed: 11/07/2022]
Abstract
A replica-symmetry-breaking phase transition is predicted in a host of disordered media. The criticality of the transition has, however, long been questioned below its upper critical dimension, six, due to the absence of a critical fixed point in the renormalization-group flows at one-loop order. A recent two-loop analysis revealed a possible strong-coupling fixed point, but given the uncontrolled nature of perturbative analysis in the strong-coupling regime, debate persists. Here we examine the nature of the transition as a function of spatial dimension and show that the strong-coupling fixed point can go through a Hopf bifurcation, resulting in a critical limit cycle and a concomitant discrete scale invariance. We further investigate a different renormalization scheme and argue that the basin of attraction of the strong-coupling fixed point (or limit cycle) may stay finite for all dimensions.
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Affiliation(s)
- Patrick Charbonneau
- Department of Chemistry, Duke University, Durham, North Carolina 27708, USA.,Department of Physics, Duke University, Durham, North Carolina 27708, USA
| | - Yi Hu
- Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
| | - Archishman Raju
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
| | - James P Sethna
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
| | - Sho Yaida
- Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
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10
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Vaezi MS, Ortiz G, Weigel M, Nussinov Z. Binomial Spin Glass. PHYSICAL REVIEW LETTERS 2018; 121:080601. [PMID: 30192575 DOI: 10.1103/physrevlett.121.080601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2017] [Indexed: 06/08/2023]
Abstract
To establish a unified framework for studying both discrete and continuous coupling distributions, we introduce the binomial spin glass, a class of models where the couplings are sums of m identically distributed Bernoulli random variables. In the continuum limit m→∞, the class reduces to one with Gaussian couplings, while m=1 corresponds to the ±J spin glass. We demonstrate that for short-range Ising models on d-dimensional hypercubic lattices the ground-state entropy density for N spins is bounded from above by (sqrt[d/2m]+1/N)ln2, and further show that the actual entropies follow the scaling behavior implied by this bound. We thus uncover a fundamental noncommutativity of the thermodynamic and continuous coupling limits that leads to the presence or absence of degeneracies depending on the precise way the limits are taken. Exact calculations of defect energies reveal a crossover length scale L^{*}(m)∼L^{κ} below which the binomial spin glass is indistinguishable from the Gaussian system. Since κ=-1/(2θ), where θ is the spin-stiffness exponent, discrete couplings become irrelevant at large scales for systems with a finite-temperature spin-glass phase.
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Affiliation(s)
| | - Gerardo Ortiz
- Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
- Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801, USA
| | - Martin Weigel
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
| | - Zohar Nussinov
- Department of Physics, Washington University, St. Louis, Missouri 63160, USA
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11
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Amey C, Machta J. Analysis and optimization of population annealing. Phys Rev E 2018; 97:033301. [PMID: 29776070 DOI: 10.1103/physreve.97.033301] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2017] [Indexed: 11/07/2022]
Abstract
Population annealing is an easily parallelizable sequential Monte Carlo algorithm that is well suited for simulating the equilibrium properties of systems with rough free-energy landscapes. In this work we seek to understand and improve the performance of population annealing. We derive several useful relations between quantities that describe the performance of population annealing and use these relations to suggest methods to optimize the algorithm. These optimization methods were tested by performing large-scale simulations of the three-dimensional (3D) Edwards-Anderson (Ising) spin glass and measuring several observables. The optimization methods were found to substantially decrease the amount of computational work necessary as compared to previously used, unoptimized versions of population annealing. We also obtain more accurate values of several important observables for the 3D Edwards-Anderson model.
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Affiliation(s)
- Christopher Amey
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - Jonathan Machta
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA and Santa Fe Institute, Santa Fe, New Mexico 87501, USA
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12
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Read N. Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension. Phys Rev E 2018; 97:012134. [PMID: 29448455 DOI: 10.1103/physreve.97.012134] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2017] [Indexed: 11/07/2022]
Abstract
We consider the one-dimensional model of a spin glass with independent Gaussian-distributed random interactions, which have mean zero and variance 1/|i-j|^{2σ}, between the spins at sites i and j for all i≠j. It is known that, for σ>1, there is no phase transition at any nonzero temperature in this model. We prove rigorously that, for σ>3/2, any translation-covariant Newman-Stein metastate for the ground states (i.e., the frequencies with which distinct ground states are observed in finite-size samples in the limit of infinite size, for given disorder) is trivial and unique. In other words, for given disorder and asymptotically at large sizes, the same ground state, or its global spin flip, is obtained (almost) always. The proof consists of two parts: One is a theorem (based on one by Newman and Stein for short-range two-dimensional models), valid for all σ>1, that establishes triviality under a convergence hypothesis on something similar to the energies of domain walls and the other (based on older results for the one-dimensional model) establishes that the hypothesis is true for σ>3/2. In addition, we derive heuristic scaling arguments and rigorous exponent inequalities which tend to support the validity of the hypothesis under broader conditions. The constructions of various metastates are extended to all values σ>1/2. Triviality of the metastate in bond-diluted power-law models for σ>1 is proved directly.
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Affiliation(s)
- N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
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13
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Billoire A, Fernandez LA, Maiorano A, Marinari E, Martin-Mayor V, Moreno-Gordo J, Parisi G, Ricci-Tersenghi F, Ruiz-Lorenzo JJ. Numerical Construction of the Aizenman-Wehr Metastate. PHYSICAL REVIEW LETTERS 2017; 119:037203. [PMID: 28777596 DOI: 10.1103/physrevlett.119.037203] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2017] [Indexed: 06/07/2023]
Abstract
Chaotic size dependence makes it extremely difficult to take the thermodynamic limit in disordered systems. Instead, the metastate, which is a distribution over thermodynamic states, might have a smooth limit. So far, studies of the metastate have been mostly mathematical. We present a numerical construction of the metastate for the d=3 Ising spin glass. We work in equilibrium, below the critical temperature. Leveraging recent rigorous results, our numerical analysis gives evidence for a dispersed metastate, supported on many thermodynamic states.
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Affiliation(s)
- A Billoire
- Institute de Physique Théorique, CEA Saclay and CNRS, 91191 Gif-sur-Yvette, France
| | - L A Fernandez
- Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - A Maiorano
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
| | - E Marinari
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
- Nanotec, Consiglio Nazionale delle Ricerche, I-00185 Rome, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, I-00185 Rome, Italy
| | - V Martin-Mayor
- Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - J Moreno-Gordo
- Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - G Parisi
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
- Nanotec, Consiglio Nazionale delle Ricerche, I-00185 Rome, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, I-00185 Rome, Italy
| | - F Ricci-Tersenghi
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
- Nanotec, Consiglio Nazionale delle Ricerche, I-00185 Rome, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, I-00185 Rome, Italy
| | - J J Ruiz-Lorenzo
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, 06071 Badajoz, Spain
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14
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Fytas NG, Martín-Mayor V, Picco M, Sourlas N. Restoration of dimensional reduction in the random-field Ising model at five dimensions. Phys Rev E 2017; 95:042117. [PMID: 28505873 DOI: 10.1103/physreve.95.042117] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2016] [Indexed: 11/07/2022]
Abstract
The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.
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Affiliation(s)
- Nikolaos G Fytas
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
| | - Víctor Martín-Mayor
- Departamento de Física Téorica I, Universidad Complutense, 28040 Madrid, Spain.,Instituto de Biocomputacíon y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - Marco Picco
- Sorbonne Universités, Université Pierre et Marie Curie-Paris VI, Laboratoire de Physique Théorique et Hautes Energies, 4, Place Jussieu, 75252 Paris Cedex 05, France
| | - Nicolas Sourlas
- Laboratoire de Physique Théorique de l'Ecole Normale Supérieure (Unité Mixte de Recherche du CNRS et de l'Ecole Normale Supérieure, associée à l'Université Pierre et Marie Curie, PARIS VI) 24 rue Lhomond, 75231 Paris Cedex 05, France
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Fytas NG, Martín-Mayor V. Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions. Phys Rev E 2016; 93:063308. [PMID: 27415388 DOI: 10.1103/physreve.93.063308] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2015] [Indexed: 06/06/2023]
Abstract
It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.227201] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero- and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent α of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.
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Affiliation(s)
- Nikolaos G Fytas
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
| | - Víctor Martín-Mayor
- Departamento de Física Teórica I, Universidad Complutense, E-28040 Madrid, Spain and Instituto de Biocomputación and Física de Sistemas Complejos (BIFI), E-50009 Zaragoza, Spain
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Arguin LP, Newman CM, Stein DL. Thermodynamic Identities and Symmetry Breaking in Short-Range Spin Glasses. PHYSICAL REVIEW LETTERS 2015; 115:187202. [PMID: 26565493 DOI: 10.1103/physrevlett.115.187202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2015] [Indexed: 06/05/2023]
Abstract
We present a technique to generate relations connecting pure state weights, overlaps, and correlation functions in short-range spin glasses. These are obtained directly from the unperturbed Hamiltonian and hold for general coupling distributions. All are satisfied in phases with simple thermodynamic structure, such as the droplet-scaling and chaotic pairs pictures. If instead nontrivial mixed-state pictures hold, the relations suggest that replica symmetry is broken as described by a Derrida-Ruelle cascade, with pure state weights distributed as a Poisson-Dirichlet process.
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Affiliation(s)
- L-P Arguin
- Department of Mathematics, City University of New York, Baruch College and Graduate Center, New York, New York 10010, USA
| | - C M Newman
- Courant Institute of Mathematical Sciences, New York, New York 10012, USA and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
| | - D L Stein
- Department of Physics and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA and NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
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Mydosh JA. Spin glasses: redux: an updated experimental/materials survey. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2015; 78:052501. [PMID: 25872613 DOI: 10.1088/0034-4885/78/5/052501] [Citation(s) in RCA: 53] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
This article reviews the 40+ year old spin-glass field and one of its earliest model interpretations as a spin density wave. Our description is from an experimental phenomenological point of view with emphasis on new spin glass materials and their relation to topical problems and strongly correlated materials in condensed matter physics. We first simply define a spin glass (SG), give its basic ingredients and explain how the spin glasses enter into the statistical mechanics of classical phase transitions. We then consider the four basic experimental properties to solidly characterize canonical spin glass behavior and introduce the early theories and models. Here the spin density wave (SDW) concept is used to explain the difference between a short-range SDW, i.e. a SG and, in contrast, a long-range SDW, i.e. a conventional magnetic phase transition. We continue with the present state of SG, its massive computer simulations and recent proposals of chiral glasses and quantum SG. We then collect and mention the various SG 'spin-off's'. A major section uncovers the fashionable unconventional materials that display SG-like freezing and glassy ground states, such as (high temperature) superconductors, heavy fermions, intermetallics and Heuslers, pyrochlor and spinels, oxides and chalogenides and exotics, e.g. quasicrystals. Some conclusions and future directions complete the review.
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Affiliation(s)
- J A Mydosh
- Kamerlingh Onnes Laboratory and Institute Lorentz, Leiden University, PO Box 9504, 2300RA Leiden, The Netherlands
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Read N. Short-range Ising spin glasses: the metastate interpretation of replica symmetry breaking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032142. [PMID: 25314430 DOI: 10.1103/physreve.90.032142] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2014] [Indexed: 06/04/2023]
Abstract
Parisi's formal replica-symmetry-breaking (RSB) scheme for mean-field spin glasses has long been interpreted in terms of many pure states organized ultrametrically. However, the early version of this interpretation, as applied to the short-range Edwards-Anderson model, runs into problems because as shown by Newman and Stein (NS) it does not allow for chaotic size dependence, and predicts non-self-averaging that cannot occur. NS proposed the concept of the metastate (a probability distribution over infinite-size Gibbs states in a given sample that captures the effects of chaotic size dependence) and a nonstandard interpretation of the RSB results in which the metastate is nontrivial and is responsible for what was called non-self-averaging. In this picture, each state drawn from the metastate has the ultrametric properties of the old theory, but when the state is averaged using the metastate, the resulting mixed state has little structure. This picture was constructed so as to agree both with the earlier RSB results and with rigorous results. Here we use the effective field theory of RSB, in conjunction with the rigorous definitions of pure states and the metastate in infinite-size systems, to show that the nonstandard picture follows directly from the RSB mean-field theory. In addition, the metastate-averaged state possesses power-law correlations throughout the low-temperature phase; the corresponding exponent ζ takes the value 4 according to the field theory in high dimensions d, and describes the effective fractal dimension of clusters of spins. Further, the logarithm of the number of pure states in the decomposition of the metastate-averaged state that can be distinguished if only correlations in a window of size W can be observed is of order W(d-ζ). These results extend the nonstandard picture quantitatively; we show that arguments against this scenario are inconclusive. More generally, in terms of Parisi's function q(x), if q(0)≠∫(0)(1)dxq(x), then the metastate is nontrivial. In an Appendix, we also prove rigorously that the metastate-averaged state of the Sherrington-Kirkpatrick model is a uniform distribution on all spin configurations at all temperatures.
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Affiliation(s)
- N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
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Sasaki M, Hukushima K, Yoshino H, Takayama H. Scaling analysis of domain-wall free energy in the Edwards-Anderson Ising spin glass in a magnetic field. PHYSICAL REVIEW LETTERS 2007; 99:137202. [PMID: 17930628 DOI: 10.1103/physrevlett.99.137202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2007] [Indexed: 05/25/2023]
Abstract
The stability of the spin-glass phase against a magnetic field is studied in the three- and four-dimensional Edwards-Anderson Ising spin glasses. Effective couplings J(eff) and effective fields H(eff) associated with length scale L are measured by a numerical domain-wall renormalization-group method. The results obtained by scaling analysis of the data strongly indicate the existence of a crossover length beyond which the spin-glass order is destroyed by field H. The crossover length well obeys a power law of H which diverges as H --> 0 but remains finite for any nonzero H, implying that the spin-glass phase is absent even in an infinitesimal field. These results are well consistent with the droplet theory for short-range spin glasses.
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Affiliation(s)
- M Sasaki
- Department of Applied Physics, Tohoku University, Sendai, 980-8579, Japan
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Read N. Minimum spanning trees and random resistor networks in d dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:036114. [PMID: 16241522 DOI: 10.1103/physreve.72.036114] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2005] [Indexed: 05/05/2023]
Abstract
We consider minimum-cost spanning trees, both in lattice and Euclidean models, in d dimensions. For the cost of the optimum tree in a box of size L , we show that there is a correction of order L(theta) , where theta< or =0 is a universal d -dependent exponent. There is a similar form for the change in optimum cost under a change in boundary condition. At nonzero temperature T , there is a crossover length xi approximately T(-nu) , such that on length scales larger than xi, the behavior becomes that of uniform spanning trees. There is a scaling relation theta=-1/nu, and we provide several arguments that show that nu and -1/theta both equal nu(perc) , the correlation length exponent for ordinary percolation in the same dimension d , in all dimensions d> or =1 . The arguments all rely on the close relation of Kruskal's greedy algorithm for the minimum spanning tree, percolation, and (for some arguments) random resistor networks. The scaling of the entropy and free energy at small nonzero T , and hence of the number of near-optimal solutions, is also discussed. We suggest that the Steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions, as is the traveling salesman problem in two dimensions. Hence all will have the same value of theta=-3/4 in two dimensions.
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Affiliation(s)
- N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
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Aspelmeier T, Bray AJ, Moore MA. Why temperature chaos in spin glasses is hard to observe. PHYSICAL REVIEW LETTERS 2002; 89:197202. [PMID: 12443143 DOI: 10.1103/physrevlett.89.197202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2002] [Indexed: 05/24/2023]
Abstract
The overlap length of a three-dimensional Ising spin glass on a cubic lattice with Gaussian interactions has been estimated numerically by transfer matrix methods and within a Migdal-Kadanoff renormalization group scheme. We find that the overlap length is large, explaining why it has been difficult to observe spin glass chaos in numerical simulations and experiment.
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Affiliation(s)
- T Aspelmeier
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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Houdayer J, Martin OC. Hierarchical approach for computing spin glass ground states. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:056704. [PMID: 11736138 DOI: 10.1103/physreve.64.056704] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2001] [Indexed: 05/23/2023]
Abstract
We describe a numerical algorithm for computing spin glass ground states with a high level of reliability. The proposed method uses a population based search and applies optimization on multiple scales. Benchmarks are given leading to estimates of the performance on large lattices.
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Affiliation(s)
- J Houdayer
- Institut für Physik/Max Planck Institut für Polymerforschung, Mainz, Germany
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Morris BW, Colborne SG, Moore MA, Bray AJ, Canisius J. Zero-temperature critical behaviour of vector spin glasses. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/19/8/014] [Citation(s) in RCA: 130] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Caflisch RG, Banavar JR, Cieplak M. Critical behaviour of the scaling stiffness coefficient of spin glasses. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/18/31/003] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Curado EM, Nobre FD, Coutinho S. Ground-state degeneracies of Ising spin glasses on diamond hierarchical lattices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3761-70. [PMID: 11970209 DOI: 10.1103/physreve.60.3761] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/1998] [Revised: 05/10/1999] [Indexed: 11/07/2022]
Abstract
The total number of ground states for short-range Ising spin glasses, defined on diamond hierarchical lattices of fractal dimensions d=2, 3, 4, 5, and 2.58, is estimated by means of analytic calculations (three last hierarchy levels of the d=2 lattice) and numerical simulations (lower hierarchies for d=2 and all remaining cases). It is shown that in the case of continuous probability distributions for the couplings, the number of ground states is finite in the thermodynamic limit. However, for a bimodal probability distribution (+/-J with probabilities p and 1-p, respectively), the average number of ground states is maximum for a wide range of values of p around p=1 / 2 and depends on the total number of sites at hierarchy level n, Nn. In this case, for all lattices investigated, it is shown that the ground-state degeneracy behaves like exp[h(d)Nn], in the limit Nn large, where h(d) is a positive number which depends on the lattice fractal dimension. The probability of finding frustrated cells at a given hierarchy level n, Fn(p), is calculated analytically (three last hierarchy levels for d=2 and the last hierarchy of the d=3 lattice, with 0<or=p<or=1), as well as numerically (all other cases, with p=1 / 2). Except for d=2, in which case Fn(1 / 2) increases by decreasing the hierarchy level, all other dimensions investigated present an exponential decrease in Fn(1 / 2) for decreasing values of n. For d=2 our results refer to the paramagnetic phase, whereas for all other dimensions considered [which are greater than the lower critical dimension dl (dl approximately 2.5)], our results refer to the spin-glass phase at zero temperature; in the latter cases h(d) increases with the fractal dimension. For n>>1, only the last hierarchies contribute significantly to the ground-state degeneracy; such a dominant behavior becomes stronger for high fractal dimensions. The exponential increase of the number of ground states with the total number of sites is in agreement with the mean-field picture of spin glasses.
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Affiliation(s)
- E M Curado
- Centro Brasiliro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
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Andersson JO, Jonsson T, Mattsson J. Monte Carlo studies of the dynamics of reentrant Ising spin glasses. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:9912-9919. [PMID: 9984726 DOI: 10.1103/physrevb.54.9912] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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34
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Khurana A. Ising spin glass: Replica-symmetric cluster expansion in finite dimensions. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:7295-7300. [PMID: 9984351 DOI: 10.1103/physrevb.54.7295] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Kisker J, Santen L, Schreckenberg M, Rieger H. Off-equilibrium dynamics in finite-dimensional spin-glass models. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:6418-6428. [PMID: 9982040 DOI: 10.1103/physrevb.53.6418] [Citation(s) in RCA: 88] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Kawashima N, Young AP. Phase transition in the three-dimensional +/-J Ising spin glass. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:R484-R487. [PMID: 9983061 DOI: 10.1103/physrevb.53.r484] [Citation(s) in RCA: 87] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Andersson J, Mattsson J, Svedlindh P. Monte Carlo studies of Ising spin-glass systems: Investigations of the overlap length in spin glasses. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:1120-1126. [PMID: 10010417 DOI: 10.1103/physrevb.49.1120] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Schins AG, Dons EM, Arts AF, Vincent E, Leylekian L, Hammann J. Aging in two-dimensional Ising spin glasses. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:16524-16532. [PMID: 10008235 DOI: 10.1103/physrevb.48.16524] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Reger JD, Tokuyasu TA, Young AP, Fisher MP. Vortex-glass transition in three dimensions. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:7147-7150. [PMID: 9998620 DOI: 10.1103/physrevb.44.7147] [Citation(s) in RCA: 62] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Blackman JA, Poulter J. Gauge-invariant method for the +/-J spin-glass model. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:4374-4386. [PMID: 10000087 DOI: 10.1103/physrevb.44.4374] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Golubovic L, Feng S. Dynamics of droplets in random Ising magnetic systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:972-992. [PMID: 9996292 DOI: 10.1103/physrevb.43.972] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Bhanot GV, Moriarty KJ, Salvador R. Evidence against a spin-glass transition in three dimensions. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 39:356-360. [PMID: 9947161 DOI: 10.1103/physrevb.39.356] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Stinchcombe RB, Pimentel IR. Transfer-matrix scaling for anomalous dynamics of a vector spin-glass chain. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 38:4980-4983. [PMID: 9946892 DOI: 10.1103/physrevb.38.4980] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Banavar JR, Bray AJ. Heisenberg and Potts spin glasses: A renormalization-group study. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 38:2564-2569. [PMID: 9946566 DOI: 10.1103/physrevb.38.2564] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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46
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Bhatt RN, Young AP. Numerical studies of Ising spin glasses in two, three, and four dimensions. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:5606-5614. [PMID: 9943754 DOI: 10.1103/physrevb.37.5606] [Citation(s) in RCA: 173] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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47
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Kawamura H, Tanemura M. Chiral order in a two-dimensional XY spin glass. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 36:7177-7180. [PMID: 9942455 DOI: 10.1103/physrevb.36.7177] [Citation(s) in RCA: 31] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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48
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Jacobs AE, Coram CM. Ferromagnetic random-bond Ising model: Metastable states and complexity of the energy surface. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 36:3844-3850. [PMID: 9943322 DOI: 10.1103/physrevb.36.3844] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Harris AB, Meir Y. Recursive enumeration of clusters in general dimension on hypercubic lattices. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 36:1840-1848. [PMID: 9899066 DOI: 10.1103/physreva.36.1840] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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50
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Sannemo U, Chao KA. Real-space renormalization study of a quantum spin glass with Ruderman-Kittel-Kasuya-Yosida interactions. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 35:7090-7096. [PMID: 9940969 DOI: 10.1103/physrevb.35.7090] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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