Monte Carlo Studies of Systems with Disorder

The main emphasis in this chapter is on two types of magnetic systems: on the percolation problem which corresponds to a zero-temperature dilute quenched ferro-magnet; and on the spin glass problem, where the interactions between nearest-neighbor Ising spins can be both ferromagnetic and antiferromagnetic. In both problems, a square or simple cubic lattice structure is assumed in most studies, and the distribution of ferromagnetic ions and exchange energies, respectively, is taken as random. Near the percolation threshold, the percolation cluster numbers seem well described by scaling assumptions analogous to usual critical phenomena. While for spin glasses the Monte Carlo work does not reliably answer the question whether or not a phase transition exists in true equilibrium, results for specific heat, susceptibility, remanent magnetization, etc., favorably compare with experimental data.

[1]  B. P. Watson,et al.  Conductivity in the two-dimensional-site percolation problem , 1974 .

[2]  M. Levinshtein.  Critical indexes of conductivity in two-dimensional percolation problems , 1977 .

[3]  U. Krey.  Excitations and critical phenomena of amorphous magnetic systems , 1978 .

[4]  R. Birgeneau,et al.  Spin Dynamics and Critical Fluctuations in a Two-Dimensional Random Antiferromagnet , 1975 .

[5]  J. Chalupa,et al.  Spin-glass behavior from Migdal's recursion relations , 1977 .

[6]  T. Lubensky,et al.  Renormalization-Group Approach to Percolation Problems , 1975 .

[7]  D. Thouless.  Percolation Theory and Electrical Conductivity , 1971 .

[8]  A. Bray,et al.  Monte Carlo evidence for the absence of a phase transition in the two-dimensional Ising spin glass , 1977 .

[9]  Ground State and Phase Transition in the Quenched Random Mixture of Ferro-Antiferromagnets , 1976 .

[10]  P. G. de Gennes,et al.  On a relation between percolation theory and the elasticity of gels , 1976 .

[11]  L. Wenger,et al.  Calorimetric investigation of a spin-glass alloy: CuMn , 1976 .

[12]  E. Stoll,et al.  Shape and size of clusters in the Ising model , 1977 .

[13]  J. Mydosh.  Spin glasses — recent experiments and systems☆ , 1978 .

[14]  B. Averbach,et al.  A Monte Carlo model of sputtered vitreous selenium thin films , 1973 .

[15]  D. Thouless,et al.  Electrons in disordered systems and the theory of localization , 1974 .

[16]  P. Leath.  Cluster Shape and Critical Exponents near Percolation Threshold , 1976 .

[17]  J. Essam,et al.  Thermodynamic scaling laws for a dilute ferromagnet in the percolation limit by series methods , 1976 .

[18]  A. Wolfendale,et al.  Spectral shape of cosmic rays over the Galaxy , 1978 .

[19]  Paul J. Flory,et al.  Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1 , 1941 .

[20]  K. Binder,et al.  Linear and nonlinear relaxation and cluster dynamics near critical points , 1976 .

[21]  H. Müller-krumbhaar.  Percolation in a lattice system with particle interaction , 1974 .

[22]  A. Bray,et al.  Critical behaviour of semi-infinite systems , 1977 .

[23]  J. Essam,et al.  The scaling laws for percolation processes , 1971 .

[24]  T. Williams,et al.  Bond percolation and cluster formation on random lattices , 1977 .

[25]  P. Steiner,et al.  Critical phenomena in Fe:Ni , 1971 .

[26]  Roy Kaplow,et al.  Atomic Arrangement in Vitreous Selenium , 1968 .

[27]  T. Odagaki,et al.  Percolation Approach to the Metal-Insulator Transition in Super-Critical Fluid Metals , 1975 .

[28]  Y. Shan,et al.  Electronic Conductivity and Percolation Theory in Aggregated Films , 1976 .

[29]  D. Fremlin,et al.  The clustering problem : some Monte Carlo results , 1976 .

[30]  Scott Kirkpatrick,et al.  An introduction to percolation theory , 1971 .

[31]  K. Binder,et al.  Monte Carlo study of the surface area of liquid droplets , 1972 .

[32]  J. Gubernatis,et al.  Computer Experiments and Disordered Solids , 1980 .

[33]  Walter H. Stockmayer,et al.  Theory of Molecular Size Distribution and Gel Formation in Branched‐Chain Polymers , 1943 .

[34]  R. J. Bell.  The dynamics of disordered lattices , 1972 .

[35]  J. W. Essam,et al.  Some Cluster Size and Percolation Problems , 1961 .

[36]  H. Stanley,et al.  CORRIGENDUM: A test of scaling near the bond percolation threshold , 1979 .

[37]  G. Heber.  Spin glasses models and typical properties , 1976 .

[38]  J. Essam,et al.  Series expansion study of the pair connectedness in bond percolation models , 1975 .

[39]  Determination of the Transition Temperature for a Randomly Dilute Ising Ferromagnet by Computer Simulations , 1976 .

[40]  S. Kirkpatrick,et al.  Low-Frequency Response Functions of Random Magnetic Systems , 1977 .

[41]  Monte Carlo studies of the internal energy and specific heat of a classical Heisenberg spin glass , 1976 .

[42]  P. Dean.  A new Monte Carlo method for percolation problems on a lattice , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[43]  K. Binder,et al.  Monte Carlo study of a two-dimensional Ising “spin-glass” , 1976 .

[44]  M. W. Klein,et al.  Internal Field Distribution in Spin-Glasses , 1975 .

[45]  David P. Landau,et al.  Finite-size behavior of the Ising square lattice , 1976 .

[46]  D. Huber,et al.  Zero-temperature spin dynamics of model spin-glass Hamiltonians , 1977 .

[47]  Kurt Binder,et al.  Monte Carlo computer experiments on critical phenomena and metastable states , 1974 .

[48]  Michael E. Fisher,et al.  The theory of condensation and the critical point , 1967 .

[49]  A. Murani.  Nonuniqueness of Freezing Temperature of Spins in Binary Alloys , 1976 .

[50]  Spin Glasses and Mictomagnets — Revisited , 1977 .

[51]  R. Birgeneau,et al.  Spin correlations near the percolation concentration in two dimensions , 1976 .

[52]  S. Kirkpatrick,et al.  Infinite-ranged models of spin-glasses , 1978 .

[53]  H. Kunz,et al.  Essential singularity in percolation problems and asymptotic behavior of cluster size distribution , 1978 .

[54]  Douglas Poland,et al.  Theory of helix-coil transitions in biopolymers , 1970 .

[55]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[56]  B. Shklovskii,et al.  The Curie temperature of dilute ferromagnetic alloys near the percolation threshold , 1975 .

[57]  D. Stauffer.  Some Percolation Scaling Applications to Low Temperature Dilute Magnets , 1977 .

[58]  V. Cannella,et al.  Magnetic Ordering in Gold-Iron Alloys , 1972 .

[59]  E. Stoll,et al.  Cluster shapes in lattice gases and percolation , 1975 .

[60]  K. Onizuka.  Computer Experiment on a 3D Site Percolation Model of Porous Materials--Its Connectivity and Conductivity , 1975 .

[61]  L. Walker,et al.  Computer model of metallic spin-glasses , 1977 .

[62]  Spingläser — einige typische physikalische eigenschaften und modelle , 1975 .

[63]  E. Stoll,et al.  Cluster statistics of the lattice gas model in three and two dimensions , 1976 .

[64]  B. Shklovskii,et al.  TOPOLOGY OF AN INFINITE CLUSTER IN THE THEORY OF PERCOLATION AND ITS RELATIONSHIP TO THE THEORY OF HOPPING CONDUCTION. , 1975 .

[65]  H. Stanley,et al.  Renormalization-Group Approach to the Percolation Properties of the Triangular Ising Model , 1978 .

[66]  J. M. Hammersley,et al.  Critical Percolation Probabilities (Bond Problem) , 1961 .

[67]  R. J. Harrison,et al.  CORRIGENDUM: On the cluster size distribution for critical percolation , 1976 .

[68]  B. L. Averbach,et al.  Monte Carlo models of atomic arrangements in arsenic-selenium glasses , 1974 .

[69]  M. Gordon,et al.  The critically branched state in a covalent synthetic system and in the reversible gelation of gelatin , 1974 .

[70]  S. Kirkpatrick.  Percolation Phenomena in Higher Dimensions: Approach to the Mean-Field Limit , 1976 .

[71]  H. Böttger.  Vibrational Properties of Non‐Crystalline Solids , 1974 .

[72]  R. Levy,et al.  Amorphous Magnetism II , 1977 .

[73]  S. Edwards,et al.  Theory of spin glasses , 1975 .

[74]  Computer experiments for the glass-like phase of two-dimensional spin systems , 1977 .

[75]  E. Anderson,et al.  Critical exponent β for nickel and nickel-copper alloys , 1971 .

[76]  B. Averbach,et al.  Atomic arrangements in sputtered thin films of vitreous SeAs with 36 to 50 at% As , 1975 .

[77]  P. Dean.  The Vibrational Properties of Disordered Systems: Numerical Studies , 1972 .

[78]  Raoul Kopelman,et al.  Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm , 1976 .

[79]  S. Kobe.  Spontaneous Magnetization of an Amorphous Ferromagnet , 1970 .

[80]  P. Anderson.  Survey of Theories of Spin Glass , 1977 .

[81]  Morrell H. Cohen,et al.  Numerical simulation of electrical conductivity in microscopically inhomogeneous materials , 1975 .

[82]  M. Shur,et al.  The relation between the critical exponents of percolation theory , 1975 .

[83]  D. Mattis.  Solvable spin systems with random interactions , 1976 .

[84]  B. Averbach,et al.  Monte-Carlo models of amorphous materials , 1974 .

[85]  M. W. Klein.  Temperature-Dependent Internal Field Distribution and Magnetic Susceptibility of a Dilute Ising Spin System , 1968 .

[86]  D. Stauffer.  Violation of Dynamical Scaling for Randomly Dilute Ising Ferromagnets near Percolation Threshold , 1975 .

[87]  P. Dean.  Vibrations of glass-like disordered chains , 1964 .

[88]  J. Marro.  Universality test for critical amplitudes in two dimensional percolation , 1976 .

[89]  Ising ferromagnet with quenched impurities , 1976 .

[90]  K. Onizuka,et al.  Monte Carlo experiment for the two-dimensional site percolation network , 1978 .

[91]  P. Lightsey.  Percolation View of Transport Properties inNaxWO3 , 1973 .

[92]  Thermodynamic properties of a classical RKKY model of CuMn , 1978 .

[93]  A. Bray,et al.  Vanishing of the Edwards-Anderson order parameter in two- and three-dimensional Ising spin glasses , 1978 .

[94]  S. Kirkpatrick,et al.  Monte Carlo studies of percolation phenomena for a simple cubic lattice , 1976 .

[95]  S. Kirkpatrick,et al.  Measurement of the Spin-Glass Order Parameter in Amorphous Gd 0.37 Al 0.63 , 1977 .

[96]  T. Lubensky.  Critical properties of random-spin models from the ∊ expansion , 1975 .

[97]  I. Morgenstern,et al.  Statistical mechanics of Ising chains in random magnetic fields , 1978 .

[98]  D. Landau,et al.  Critical behavior of a 3‐D Ising model in a random field , 1978 .

[99]  K. Binder.  Clusters in the Ising Model, Metastable States and Essential Singularity , 1976 .

[100]  A. Young,et al.  Real space rescaling study of spin glass behaviour in three dimensions , 1977 .

[101]  H. C. Benski,et al.  New Mössbauer-Effect Measurements onFe57in a Ni Host: The Critical Exponentβfor Ni , 1972 .

[102]  Monte Carlo simulation of a three-dimensional spin glass , 1976 .

[103]  E. Shender.  Impurities in singlet magnets , 1975 .

[104]  R. Palmer,et al.  Solution of 'Solvable model of a spin glass' , 1977 .

[105]  Heather J. Ruskin,et al.  Percolation processes in d-dimensions , 1976 .

[106]  Boris I Shklovskii,et al.  Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non-Metal Transition Threshold , 1976 .

[107]  K. Bennemann,et al.  Local critical behaviour at impurities in Ising-type systems , 1977 .

[108]  Yoseph Imry,et al.  Random-Field Instability of the Ordered State of Continuous Symmetry , 1975 .

[109]  J. W. Essam,et al.  Scaling theory for the pair-connectedness in percolation models , 1975 .

[110]  V. Cannella,et al.  Calculations of Superparamagnetic Clustering in Random Alloys of Magnetic Atoms in Nonmagnetic Matrices , 1973 .

[111]  M. Sykes,et al.  Percolation processes in two dimensions. I. Low-density series expansions , 1976 .

[112]  G. Toulouse,et al.  Perspectives from the theory of phase transitions , 1974 .

[113]  D. Weaire.  The structure of amorphous solids , 1976 .

[114]  P. Peretto,et al.  Hyperfine critical exponents and d of tantalum in nickel , 1973 .

[115]  D. Huber,et al.  Monte Carlo studies of the critical behavior of site-dilute two-dimensional Ising models , 1976 .

[116]  Joshua Jortner,et al.  Critical exponents for percolation conductivity in resistor networks , 1977 .

[117]  Peter Reynolds,et al.  Percolation by position-space renormalisation group with large cells , 1978 .

[118]  H. Muller-Krumbhaar.  Critical properties of Heisenberg ferromagnets with lattice inhomogeneity , 1976 .

[119]  E. Kneller.  Theorie der Magnetisierungskurve kleiner Kristalle , 1966 .

[120]  A. Bray,et al.  Defect energies of two-, three- and four-dimensional Ising spin glasses , 1978 .

[121]  Shang‐keng Ma.  Modern Theory of Critical Phenomena , 1976 .

[122]  R. Tournier,et al.  The Remanent Magnetization of Spin Glasses and the Dipolar Coupling , 1977 .

[123]  B. Halperin,et al.  Defects and the central peak near structural phase transitions , 1976 .

[124]  P. Leath,et al.  Scaling form for percolation cluster sizes and perimeters , 1978 .

[125]  D. Stauffer,et al.  Essential singularities in dilute magnets , 1976 .

[126]  Benjamin Abeles,et al.  Percolation Conductivity in W-Al 2 O 3 Granular Metal Films , 1975 .

[127]  Y. Yuge.  Three-dimensional site percolation problem and effective-medium theory: A computer study , 1977 .

[128]  S. Kirkpatrick,et al.  Theory of the Spin Excitations of Rb 2 Mn x Ni 1−x F 4 , 1975 .

[129]  Susceptibilities of spin glass models: a Monte Carlo study , 1978 .

[130]  H. Scheraga,et al.  Theory of helix-coil transitions in biopolymers : statistical mechanical theory of order-disorder transitions in biological macromolecules , 1970 .

[131]  K. Binder,et al.  Classical Heisenberg ferromagnets with defects , 1973 .

[132]  Kurt Binder,et al.  Monte Carlo study of thin magnetic Ising films , 1974 .

[133]  E. Stoll,et al.  Monte Carlo studies of two-dimensional percolation , 1978 .

[134]  S. Kirkpatrick.  Frustration and ground-state degeneracy in spin glasses , 1977 .

[135]  S. Kirkpatrick.  Percolation and Conduction , 1973 .