Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition

Finite-size scaling of the random-field Ising model above the upper critical dimension

Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of statistical physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension D=7, i.e., above its upper critical dimension D_{u}=6, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions D>D_{u}, linear length scale L should be replaced in finite-size scaling expressions by the effective scale L_{eff}=L^{D/D_{u}}. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.

Noise-to-noise ratios in correlation length calculations near criticality

For finite quenched random systems, on regular lattices, it is possible to define two types of variances (noises). It is demonstrated that their ratio is useful in calculating the correlation length of an infinite and rather general random system, as a function of temperature. The numerical method of obtaining those variables is not relevant. It can be real-space numerical renormalization, simulation, or any other method. It does not matter. The correlation length obtained by this technique may then be used to obtain directly the critical correlation exponent, ν, rather than indirectly, using scaling relations, as is often done. The method is demonstrated by applying it to the random field Ising model.

Effect of Confinement on Capillary Phase Transition in Granular Aggregates

Using a 3D mean-field lattice-gas model, we analyze the effect of confinement on the nature of capillary phase transition in granular aggregates with varying disorder and their inverse porous structures obtained by interchanging particles and pores. Surprisingly, the confinement effects are found to be much less pronounced in granular aggregates as opposed to porous structures. We show that this discrepancy can be understood in terms of the surface-surface correlation length with a connected path through the fluid domain, suggesting that this length captures the true degree of confinement. We also find that the liquid-gas phase transition in these porous materials is of second order nature near capillary critical temperature, which is shown to represent a true critical temperature, i.e., independent of the degree of disorder and the nature of the solid matrix, discrete or continuous. The critical exponents estimated here from finite-size scaling analysis suggest that this transition belongs to the 3D random field Ising model universality class as hypothesized by F. Brochard and P.G. de Gennes, with the underlying random fields induced by local disorder in fluid-solid interactions.

Monte Carlo Renormalization Group for Classical Lattice Models with Quenched Disorder

We extend to quenched-disordered systems the variational scheme for real-space renormalization group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. Key to the variational approach is the bias potential found by minimizing a convex functional in statistical mechanics. This potential reduces dramatically the Monte Carlo relaxation time in large disordered systems. We demonstrate the method with applications to the two-dimensional dilute Ising model, the random transverse field quantum Ising chain, and the random field Ising in two- and three-dimensional lattices.

News on fluctuations and correlations from the NA61/SHINE experiment

In this contribution the latest NA61/SHINE results on fluctuations and correlations from the p+p, Be+Be, and Ar+Sc energy scans will be presented. The NA61 experimental results will be compared with existing NA49 data and with model predictions.

Aging in the three-dimensional random-field Ising model

We studied the nonequilibrium aging behavior of the random-field Ising model in three dimensions for various values of the disorder strength. This allowed us to investigate how the aging behavior changes across the ferromagnetic-paramagnetic phase transition. We investigated a large system size of N=256^{3} spins and up to 10^{8} Monte Carlo sweeps. To reach these necessary long simulation times, we employed an implementation running on Intel Xeon Phi coprocessors, reaching single-spin-flip times as short as 6 ps. We measured typical correlation functions in space and time to extract a growing length scale and corresponding exponents.

3D Imaging of Ferroelectric Kinetics during Electrically Driven Switching

Čerenkov-type second-harmonic generation microscopy is used for in situ 3D imaging of the switching process of ferroelectric domains in a relaxor-type ferroelectric crystal. The lateral and longitudinal domain wall motion is optically measured and compared with the hysteresis loop to reveal the connection between the domain kinetics and the ferroelectric aging effect in strontium barium niobate.

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

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.

Random-field Ising model on isometric lattices: Ground states and non-Porod scattering

We use a computationally efficient graph cut method to obtain ground state morphologies of the random-field Ising model (RFIM) on (i) simple cubic (SC), (ii) body-centered cubic (BCC), and (iii) face-centered cubic (FCC) lattices. We determine the critical disorder strength Δ_{c} at zero temperature with high accuracy. For the SC lattice, our estimate (Δ_{c}=2.278±0.002) is consistent with earlier reports. For the BCC and FCC lattices, Δ_{c}=3.316±0.002 and 5.160±0.002, respectively, which are the most accurate estimates in the literature to date. The small-r behavior of the correlation function exhibits a cusp regime characterized by a cusp exponent α signifying fractal interfaces. In the paramagnetic phase, α=0.5±0.01 for all three lattices. In the ferromagnetic phase, the cusp exponent shows small variations due to the lattice structure. Consequently, the interfacial energy E_{i}(L) for an interface of size L is significantly different for the three lattices. This has important implications for nonequilibrium properties.

Fluids with quenched disorder: scaling of the free energy barrier near critical points

In the context of Monte Carlo simulations, the analysis of the probability distribution P(L)(m) of the order parameter m, as obtained in simulation boxes of finite linear extension L, allows for an easy estimation of the location of the critical point and the critical exponents. For Ising-like systems without quenched disorder, P(L)(m) becomes scale-invariant at the critical point, where it assumes a characteristic bimodal shape featuring two overlapping peaks. In particular, the ratio between the value of P(L)(m) at the peaks (P(L, max)) and the value at the minimum in between (P(L, min)) becomes L-independent at criticality. However, for Ising-like systems with quenched random fields, we argue that instead ΔF(L) := ln(P(L, max)/P(L, min)) proportional to L(θ) should be observed, where θ > 0 is the 'violation of hyperscaling' exponent. Since θ is substantially non-zero, the scaling of ΔF(L) with system size should be easily detectable in simulations. For two fluid models with quenched disorder, ΔF(L) versus L was measured and the expected scaling was confirmed. This provides further evidence that fluids with quenched disorder belong to the universality class of the random field Ising model.

Finite-size scaling in Ising-like systems with quenched random fields: evidence of hyperscaling violation

In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost ΔF of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, ΔF∝L(θ), with θ as the violation of hyperscaling critical exponent and L as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.

Critical behavior of colloid-polymer mixtures in random porous media

We show that the critical behavior of a colloid-polymer mixture inside a random porous matrix of quenched hard spheres belongs to the universality class of the random-field Ising model. We also demonstrate that random-field effects in colloid-polymer mixtures are surprisingly strong. This makes these systems attractive candidates to study random-field behavior experimentally.

Random-energy model in random fields

The random-energy model is studied in the presence of random fields. The problem is solved exactly both in the microcanonical ensemble, without recourse to the replica method, and in the canonical ensemble using the replica formalism. The phase diagrams for bimodal and Gaussian random fields are investigated in detail. In contrast to the Gaussian case, the bimodal random field may lead to a tricritical point and a first-order transition. An interesting feature of the phase diagram is the possibility of a first-order transition from paramagnetic to mixed phase.

Lack of self-averaging of the specific heat in the three-dimensional random-field Ising model

We apply the recently developed critical minimum-energy subspace scheme for the investigation of the random-field Ising model. We point out that this method is well suited for the study of this model. The density of states is obtained via the Wang-Landau and broad histogram methods in a unified implementation by employing the N-fold version of the Wang-Landau scheme. The random fields are obtained from a bimodal distribution (hi = +/-2), and the scaling of the specific heat maxima is studied on cubic lattices with sizes ranging from L=4 to L=32. Observing the finite-size scaling behavior of the maxima of the specific heats we examine the question of saturation of the specific heat. The lack of self-averaging of this quantity is fully illustrated, and it is shown that this property may be related to the question mentioned above.

Ground states and thermal states of the random field Ising model

The random field Ising model is studied numerically at both zero and positive temperature. Ground states are mapped out in a region of random field and external field strength. Thermal states and thermodynamic properties are obtained for all temperatures using the Wang-Landau algorithm. The specific heat and susceptibility typically display sharp peaks in the critical region for large systems and strong disorder. These sharp peaks result from large domains flipping. For a given realization of disorder, ground states and thermal states near the critical line are found to be strongly correlated--a concrete manifestation of the zero temperature fixed point scenario.

Change from 3D-Ising to random field-Ising-model criticality in a uniaxial relaxor ferroelectric

Pyroelectric measurements of polarization have been used to determine the temperature dependence of the polarization in strontium barium niobate, SBN:Ce, close to its phase transition temperature T(c) approximately 317 K. A gradual increase of the critical exponent from beta approximately 0.13 to beta approximately 0.30 is observed when decreasing the initial polarization from 100% to 0.8% of the saturation value. A change from three-dimensional random-field Ising to pure Ising model behavior is conjectured and explained by a gradual compensation of quenched random electric fields by those emerging from charged fractal nanodomain walls.

Full reduction of large finite random Ising systems by real space renormalization group

We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain Gamma(r)=<sigma(0)sigma(r)>-<sigma(0)><sigma(r)>, the "connected" correlation function, and S(r)=<sigma(0)sigma(r)>, the "disconnected" correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function Gamma(r) along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such nu or eta). The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.

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.

Numerical study of the disorder-driven roughening transition in an elastic manifold in a periodic potential

We study the roughening transition of a (3+1)-dimensional elastic manifold, which is driven by the competition between a periodic pinning potential and a random impurity potential. The elastic manifold is modeled by a solid-on-solid-type interface model, and the universal properties of the transition from a flat phase (for strong periodic potential) to a rough phase (for strong random potential) are investigated at zero temperature using a combinatorial optimization algorithm technique. We find that the transition is a continuous one. Critical exponents are estimated numerically, and compared with analytic results and those for a periodic elastic medium.

Disorder-driven critical behavior of periodic elastic media in a crystal potential

A lattice model of a three-dimensional periodic elastic medium at zero temperature is studied with exact combinatorial optimization methods. A competition between pinning of the elastic medium, representing magnetic flux lines in a superconductor or charge density waves in a crystal, by randomly distributed impurities and a periodic lattice potential gives rise to a continuous roughening transition from a flat to a rough phase. A finite size scaling analysis yields the critical exponents nu approximately 1.3, beta approximately 0.05, gamma/nu approximately 2.9 that are universal with respect to the periodicity of the lattice potential. The small order parameter exponent is reminiscent of the random field Ising critical behavior in 3D.

Replica-exchange algorithm and results for the three-dimensional random field ising model

The random field Ising model with Gaussian disorder is studied using a different Monte Carlo algorithm. The algorithm combines the advantages of the replica-exchange method and the two-replica cluster method and is much more efficient than the Metropolis algorithm for some disorder realizations. Three-dimensional systems of size 24(3) are studied. Each realization of disorder is simulated at a value of temperature and uniform field that is adjusted to the phase-transition region for that disorder realization. Energy and magnetization distributions show large variations from one realization of disorder to another. For some realizations of disorder there are three well separated peaks in the magnetization distribution and two well separated peaks in the energy distribution suggesting a first-order transition.