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arXiv:1401.1751 · Physical Review E

Ensemble renormalization group for the random-field hierarchical model

Aurélien Decelle, Giorgio Parisi A, Jacopo Rocchi

Published Mar 26, 2014• 10 citations
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Abstract

The renormalization group (RG) methods are still far from being completely understood in quenched disordered systems. In order to gain insight into the nature of the phase transition of these systems, it is common to investigate simple models. In this work we study a real-space RG transformation on the Dyson hierarchical lattice with a random field, which leads to a reconstruction of the RG flow and to an evaluation of the critical exponents of the model at T=0. We show that this method gives very accurate estimations of the critical exponents by comparing our results with those obtained by some of us using an independent method.

DOI
10.1103/physreve.89.032132

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The renormalization group (RG) methods are still far from being completely understood in quenched disordered systems. In order to gain insight into the nature of the phase transition of these systems, it is common to investigate simple models. In this work we study a real-space RG transformation on the Dyson hierarchical lattice with a random field, which leads to a reconstruction of the RG flow and to an evaluation of the critical exponents of the model at T=0. We show that this method gives very accurate estimations of the critical exponents by comparing our results with those obtained by some of us using an independent method.
Abstract We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: a sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method (Angelini M C, Parisi G and Ricci-Tersenghi F 2013 Ensemble renormalization group for disordered systems Phys. Rev . B 87 134201) in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical exponent related to the divergence of the correlation length, while it does not predict the correct qualitative behavior of below the upper critical dimension. On the other hand, the ERG procedure has been claimed to predict the correct behavior of both above and below the upper critical dimension. Here, we straighten out the reasons for the discrepancy between the two methods above, by demonstrating that the ERG method predicts a marginally stable critical fixed point, thus implying a prediction for the critical exponent given by . This prediction disagrees, on a qualitative and quantitative level, both with the mean-field value of above the critical dimension, and with numerical estimates of below the upper critical dimension. Therefore, our results show that finding a real-space RG method for spin glasses which yields the correct prediction for universal quantities below the upper critical dimension is still an open problem, for which our analysis may provide some general guidance for future studies.
Abstract In the paper Angelini M C et al (2013 Phys. Rev. B 87 134201) we introduced a real-space renormalization group called ensemble renormalization group (ERG) and we applied it to the Edwards–Anderson model, obtaining estimates for the critical exponents in good agreement with those from Monte Carlo simulations. Recently the paper Castellana M (2019 J. Phys. A: Math. Theor. 52 445002) re-examined the ERG method from a different perspective, concluding that the previous results were wrong, and claiming that the ERG method predicts trivially wrong critical exponents. In this comment we explain why the conclusions reached by Castellana are wrong, as they are based on a misinterpretation of finite-size effects. We conclude that the ERG method remains a good RG method to obtain critical exponents in strongly disordered models (if properly used).
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.
The Dyson hierarchical version of the quantum Ising chain with Long-Ranged power-law ferromagnetic couplings J ( r ) ∝ r −1− σ and pure or random transverse fields is studied via real-space renormalization. For the pure case, the critical exponents are explicitly obtained as a function of the parameter σ , and are compared with previous results of other approaches. For the random case, the RG rules are numerically applied and the critical behaviors are compared with previous strong disorder renormalization results.
Strong disorder renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as d > 1. In this paper, we propose to include strong disorder renormalization ideas within the more traditional fixed-cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed-cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent β (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed-cell-size procedure to two types of renormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimension D = ln3/ln2, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in d = 1; (ii) a hierarchical diamond lattice of fractal dimension D = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension d > 1. In both cases, we find that the transition is governed by an infinite disorder fixed point: besides the measure of the activated exponent ψ, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.
We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigourous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and functional RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large- N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review is to present a configuration space counterpart, suitable for lattice formulations, of functional RG equations formulated in momentum space (often called exact RG equations and abbreviated ERGE).