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arXiv:1605.05072 · Physical Review Letters

Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions

Nikolaos G. Fytas, Víctor Martín-Mayor, Marco Picco, Nicolas Sourlas

Published Jun 3, 2016• 62 citations
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Abstract

By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.

DOI
10.1103/physrevlett.116.227201

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We solve a long-standing puzzle in statistical mechanics of disordered systems. By performing a high-statistics simulation of the D=3 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute the complete set of critical exponents for this class, including the correction-to-scaling exponent, and we show, to high numerical accuracy, that scaling is described by two independent exponents. Discrepancies with previous works are explained in terms of strong scaling corrections.
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.
We present real-space renormalization-group (RG) calculations of the critical properties of the random-field Ising model on a cubic lattice in three dimensions. We calculate the RG flows in a two-parameter truncation of the Hamiltonian space. As predicted, the transition at finite randomness is controlled by a zero-temperature, disordered critical fixed point, and we exhibit the universal crossover trajectory from the pure Ising critical point. We extract scaling fields and critical exponents, and study the distribution of barrier heights between states as a function of length scale.
We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ≃ 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.
Abstract We present a dynamical numerical study on a three-dimensional (3D) Gaussian distribution random-field Ising model in a wide range of disorder strengths by means of the finite-time dynamics method. By validating the Rushbrooke scaling law and analyzing the variation of exponents with disorder amplitudes, the asymptotic critical regime controlled by a hysteresis fixed point is identified with exponents , , and . The phase transition is found to be continuous and there is not a tricritical point in this version of Gaussian distribution. Moreover, our results support that the hysteresis and zero-temperature fixed points belong to a different universality class.