Review of Recent Developments in the Random-Field Ising Model

A lot of progress has been made recently in our understanding of the random-field Ising model thanks to large-scale numerical simulations. In particular, it has been shown that, contrary to previous statements: the critical exponents for different probability distributions of the random fields and for diluted antiferromagnets in a field are the same. Therefore, critical universality, which is a perturbative renormalization-group prediction, holds beyond the validity regime of perturbation theory. Most notably, dimensional reduction is restored at five dimensions, i.e., the exponents of the random-field Ising model at five dimensions and those of the pure Ising ferromagnet at three dimensions are the same.     

A Quantum d-Dimensional Random-Field XY Ferromagnet

We propose a soluble quantum spherical XY ferromagnet with a random field in the boson space. We obtain a general expression of the critical temperat ure T_c below which the ordered ferromagnet phase appears. The Imry-Ma result concerning the lower critical dimension d_c = 4 is recovered, and the critical exponents near the critical temperature T_c are calculated. We show that the random-field fluctuations rather than the quantum fluctuations dominate the phase transition and critical behavior of the system. The entropy vanishes as T^d/2 at low temperatures, contrary to the classical spherical model.     

Universality in three dimensional random-field ground states

We investigate the critical behavior of three-dimensional random-field Ising systems with both Gauss and bimodal distribution of random fields and additional the three-dimensional diluted Ising antiferromagnet in an external field. These models are expected to be in the same universality class. We use exact ground-state calculations with an integer optimization algorithm and by a finite-size scaling analysis we calculate the critical exponents , , and . While the random-field model with Gauss distribution of random fields and the diluted antiferromagnet appear to be in same universality class, the critical exponents of the random-field model with bimodal distribution of random fields seem to be significantly different. Received: 9 July 1998 / Received in final form: 15 July 1998 / Accepted: 20 July 1998     

Order parameter criticality of the d = 3 random-field Ising antiferromagnet Fe0.85Zn0.15F2

The critical exponent beta=0.16+/-0.02 for the random-field Ising model order parameter is determined using extinction-free magnetic x-ray scattering for Fe0.85Zn0.15F2 in magnetic fields of 10 and 11 T. The observed value is consistent with other experimental random-field critical exponents, but disagrees sharply with Monte Carlo and exact ground state calculations on finite-sized systems.     

Modified scaling relation for the random-field Ising model

We investigate the low-temperature critical behavior of the three-dimensional random-field Ising ferromagnet. By a scaling analysis we find that in the limit of temperature T → 0 the usual scaling relations have to be modified as far as the exponent α of the specific heat is concerned. At zero temperature, the Rushbrooke equation is modified to α + 2β + γ = 1, an equation which we expect to be valid also for other systems with similar critical behavior. We test the scaling theory numerically for the three-dimensional random-field Ising system with Gaussian probability distribution of the random fields by a combination of calculations of exact ground states with an integer optimization algorithm and Monte Carlo methods. By a finite-size scaling analysis we calculate the critical exponents ν ≈ 1.0, β ≈ 0.05,

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≈ 2.9, γ ≈ 1.5 and α ≈ −0.55.     

Low-concentration series in general dimension

We discuss recent work on the development and analysis of low-concentration series. For many models, the recent breakthrough in the extremely efficient no- free-end method of series generation facilitates the derivation of 15th-order series for multiple moments in general dimension. The 15th-order series have been obtained for lattice animals, percolation, and the Edwards-Anderson Ising spin glass. In the latter cases multiple moments have been found. From complete graph tables through to 13th order, general dimension 13th-order series have been derived for the resistive susceptibility, the moments of the logarithms of the distribution of currents in resistor networks, and the average transmission coefficient in the quantum percolation problem, 11th-order series have been found for several other systems, including the crossover from animals to percolation, the full resistance distribution, nonlinear resistive susceptibility and current distribution in dilute resistor networks, diffusion on percolation clusters, the dilute Ising model, dilute antiferromagnet in a field, and random field Ising model and self-avoiding walks on percolation clusters. Series for the dilute spin-1/2 quantum Heisenberg ferromagnet are in the process of development. Analysis of these series gives estimates for critical thresholds, amplitude ratios, and critical exponents for all dimensions. Where comparisons are possible, our series results are in good agreement with both-expansion results near the upper critical dimension and with exact results (when available) in low dimensions, and are competitive with other numerical approaches in intermediate realistic dimensions.     

Nonequilibrium critical behavior in unidirectionally coupled stochastic processes

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d(c)=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A-->A+A, A+A-->A, and A-->0. We study a hierarchy of such DP processes for particle species A,B,..., unidirectionally coupled via the reactions A-->B, ...(with rates mu(AB),...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta(i) which are markedly reduced at each hierarchy level i> or =2. This scenario can be understood on the basis of the mean-field rate equations, which yield beta(i)=1/2(i-1) at the multicritical point. Using field-theoretic renormalization-group techniques in d=4-epsilon dimensions, we identify a new crossover exponent phi, and compute phi=1+O(epsilon(2)) in the multicritical regime (for small mu(AB)) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, beta(2)=1/2-epsilon/8+O(epsilon(2)). Outside the multicritical region, we discuss the crossover to ordinary DP behavior, with the density exponent beta(1)=1-epsilon/6+O(epsilon(2)). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< or =3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.     

Critical properties of the half-filled Hubbard model in three dimensions

By means of the dynamical vertex approximation (DΓA) we include spatial correlations on all length scales beyond the dynamical mean-field theory (DMFT) for the half-filled Hubbard model in three dimensions. The most relevant changes due to nonlocal fluctuations are (i) a deviation from the mean-field critical behavior with the same critical exponents as for the three dimensional Heisenberg (anti)ferromagnet and (ii) a sizable reduction of the Néel temperature (T(N)) by ~30% for the onset of antiferromagnetic order. Finally, we give a quantitative estimate of the deviation of the spectra between DΓA and DMFT in different regions of the phase diagram.     

Random‐field ising systems: A survey of current theoretical views

Ising or Ising-like models in random fields are good representations of a large number of impure materials. The main attempts of theoretical treatments of these models--as far as they are relevant from an experimental point of view--are reviewed. A domain argument invented by Imry and Ma shows that the long-range order is not destroyed by weak random-fields in more than D = 2 dimensions. This result is supported by considerations of the roughening of an isolated domain wall in such systems: domain walls turn out to be well defined objects for D > 2, but arbitrarily convoluted for D < 2. Different approaches for calculating the roughness exponent ζ yield ζ= (5 - D)/3 in random-field systems. The application of ζ in incommensurate-commensurate critical behaviour is discussed.

Non-classical critical behaviour occurs in random-field systems below D = 6 dimensions which is determined in general by three independent exponents. The new exponent y_J = θ= D/2 - σ corresponds to random-field renormalization or, to say it differently, to the irrelevance of the temperature at the zero-temperature fixed point, which is believed to rule the critical behaviour. The inequalities satisfied by these exponents are investigated, as well as the relations between the eigenvalue and the critical exponents and their numerical values found in the literature.

Domain wail roughening due to random fields produces also metastability and hysteresis. It is shown that when cooling a 3D system into the low-temperature phase in an applied random field, the system runs into a metastable domain state, in agreement with the experimental observation. The further approach of the system to the ordered equilibrium state is hindered by pinning which leads to domain size increasing only logarithmically in time. Domain wall roughness and pinning in random bond systems is also considered.     

Renormalization group phase boundary of the three-dimensional bond-dilute Ising ferromagnet

The phase boundary (as well as the thermal-type critical exponents) associated to the quenched bond-dilute spin-1/2 Ising ferromagnet in the simple cubic lattice is approximately calculated within a real space renormalization group framework in two different versions. Both lead to qualitatively satisfactory critical frontiers, although one of them provides an unphysical fixed point (which seems to be related to the three-dimensionality of the system) besides the expected pure ones; its effects tend to disappear for increasingly large clusters. Through an extrapolation procedure the (unknown) critical phase boundary is approximately located.     

Field-theoretic approach to second-order phase transitions in two- and three-dimensional systems

We review the physical principles which are at the basis of recent field-theoretic computations of the critical exponents in two- and three-dimensional systems. We concentrate on those points that do not show up explicitly in the more standard-expansion: they must be discussed with care if one uses a perturbative approach at fixed space dimensions (the loop expansion). We present in detail simple computations of the critical exponents, while we summarize the results of longer and more accurate computations.     

Heisenberg ferromagnet with Dzyaloshinsky-Moriya interactions: A real space renormalization group approach

We study the anisotropic Heisenberg ferromagnet with anti-symmetric Dzyaloshinsky-Moriya (DM) interactions for arbitrary dimensions. We use a real space renormalization group approach of the Migdal-Kadanoff type in order to obtain the phase diagrams and critical exponents. The effect of the Dzyaloshinsky-Moriya term in the global phase diagram is worked out in detail. Our results suggest that in all dimensions the effect of the DM interaction is to renormalize the parameters of the anisotropic exchange Hamiltonian. Finally we discuss the modification of hyperscaling associated with the zero temperature Heisenberg-like fixed point.     

Phase transitions in Sr 0.61 Ba 0.39 Nb 2 O 6 :Ce 3+ : II. Linear birefringence studies of spontaneous and precursor polarization

The linear birefringence (LB) of Sr _0.61-x Ba _0.39 Nb ₂ O ₆ :Ce ³⁺ x (SBN61:Ce) has been measured as a function of temperature within the range of . Large tails have been observed above the ferroelectric phase transition temperatures T _c = 350, 328, 320 and 291 K for the concentrations x = 0, 0.0066, 0.0113 and 0.0207, respectively. Within an Ornstein-Zernike analysis the critical exponents , and are determined. It suggests that pure SBN61 belongs to the 3D Ising universality class. Doping with Ce ³⁺ ions, which seem to act as random fields, enhances the relaxor properties. The critical exponents and of SBN61:Ce shift against those of the three-dimensional random-field Ising model. Received 1 October 1999     

Critical dimensions of the diffusion equation

We study the evolution of a random initial field under pure diffusion in various space dimensions. From numerical calculations we find that the persistence properties of the system show sharp transitions at critical dimensions d(1) approximately 26 and d(2) approximately 46. We also give refined measurements of the persistence exponents for low dimensions.     

Unified picture of ferromagnetism, quasi-long-range order, and criticality in random-field models

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.     

Diagrammatic expansion and metastability in the random-field Ising model

Within the perturbation diagrammatic expansion we discuss the origin of differences in determinations of the lower critical dimension of the random-field Ising model and show that below four dimensions metastability and hysteresis occur. We also explain the occurrence of a quasicritical d=2 behavior at weak random fields, which is responsible for local stability of the ordered state above two dimensions.     

Dynamic renormalization-group analysis of the d+1 dimensional Kuramoto-Sivashinsky equation with both conservative and nonconservative noises

The long-wavelength properties of the (d + 1)-dimensional Kuramoto-Sivashinsky (KS) equation with both conservative and nonconservative noises are investigated by use of the dynamic renormalization-group (DRG) theory. The dynamic exponent z and roughness exponent α are calculated for substrate dimensions d = 1 and d = 2, respectively. In the case of d = 1, we arrive at the critical exponents z = 1.5 and α = 0.5 , which are consistent with the results obtained by Ueno et al. in the discussion of the same noisy KS equation in 1+1 dimensions [Phys. Rev. E 71, 046138 (2005)] and are believed to be identical with the dynamic scaling of the Kardar-Parisi-Zhang (KPZ) in 1+1 dimensions. In the case of d = 2, we find a fixed point with the dynamic exponents z = 2.866 and α = -0.866 , which show that, as in the 1 + 1 dimensions situation, the existence of the conservative noise in 2 + 1 or higher dimensional KS equation can also lead to new fixed points with different dynamic scaling exponents. In addition, since a higher order approximation is adopted, our calculations in this paper have improved the results obtained previously by Cuerno and Lauritsen [Phys. Rev. E 52, 4853 (1995)] in the DRG analysis of the noisy KS equation, where the conservative noise is not taken into account.     

Criticality in inhomogeneous magnetic systems: application to quantum ferromagnets

We consider a phi4 theory with a position-dependent distance from the critical point. One realization of this model is a classical ferromagnet subject to nonuniform mechanical stress. We find a sharp phase transition where the envelope of the local magnetization vanishes uniformly. The first-order transition in a quantum ferromagnet also remains sharp. The universal mechanism leading to a tricritical point in an itinerant quantum ferromagnet is suppressed, and in principle, one can recover a quantum critical point with mean-field exponents. Observable consequences of these results are discussed.