Ensemble renormalization group for the random-field hierarchical model

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 led 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 the ones obtained by some of us using an independent method.The Renormalization Group (RG) is a fundamental tool to study the changes of a system as observed at different scales, which has been successfully employed both in quantum field theory [1] and in the theory of second order phase transitions [2]. This process consists in integrating out small scale details of the physical systems: the original interactions between the fundamental degrees of freedom are replaced by renormalized interactions between effective degrees of freedom. Finally, the critical exponents may be computed repeating this transformation over and over [2,3]. The renormalization group has two main flavours. On the one hand, the process can be done in momentum space, by slowly integrating out high momenta [4]. Momentum space RG was first developed in quantum field theory and then it also became a highly developed tool in statistical mechanics, where it is usually performed on a perturbation expansion to compute critical exponents. On the other hand, a technically different approach is to integrate out small distance degrees of freedom in real space [2]. The advantage of the latter is to provide a more physical picture of the process despite the difficulty to obtain accurate results. In general, both methods need some approximations, but it is possible to find systems with a particular topology for which real space transformations can be performed exactly, such as the two dimensional triangular lattice [5] and the diamond hierarchical lattice [6]. These kinds of exactly soluble models could play a very important rôle to test new ideas for systems where the nature of the phase transition is difficult to understand.In this work, we focus on quenched disorder systems for which the renormalization group approach is still not deeply understood. Following the approach introduced with the diamond hierarchical lattice [7], we concentrate our effort on another hierarchical lattice which gave us the opportunity to define an approximate transformation for more realistic systems.Years ago, Dyson [8] introduced the so-called Hierarchical Model (HM) to study the problem of phase transitions in one dimensional long-range models. It was later understood that the topology of this model could be used to implement an exact real-space RG transformation [9]. Therefore, analytical and numerical studies were pursued in this direction [10], [11], [12][13][14][15]. Among the numerical works, an ...

Real-space renormalization-group methods for hierarchical spin glasses

“…In this regard, we observe that a variant of the ERG method has been recently applied to the ferromagnetic version of the HEA with a random magnetic field, where it has been claimed to yield accurate estimates of ν [19]. Specifically, this modification of the ERG method differs from the one of [18].…”

“…The sample-by-sample procedure predicts the correct mean-field value of the critical exponent ν related to the divergence of the correlation length above the upper critical dimension, while its predictions disagree with numerical simulations below the upper critical dimension [15,18]. On the other hand, the ERG procedure has been claimed to predict the correct behavior of ν , both above and below the upper critical dimension [18,19].…”

Statistical mechanics of the spherical hierarchical model with random fields

“…, P q k that interact through the off-diagonal elements of A k (s 1,...,q ), as can be noted from eq. (28). By assuming a > βλ…”

“…Dyson hierarchical models exhibit qualitatively a similar phenomenology, with the dimension D replaced by an exponent τ , the latter being responsible for controlling the power-law decay of the interactions as a function of the inter-site distance. Although this intuitive analogy has been recently exploited to study quenched disorder systems in the non-mean-field sector [17,18,19,20,21,22,23,24,25,26,27,28,29], it is not clear to which extent there is a clear mapping between the spatial dimension D and the exponent τ [25,26,27,29]. Indeed, a strict mapping holds in the mean-field region, whereas it does not give satisfying results for low enough dimensions [27,29].…”

Metastable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network

“…Far from artificial intelligence, but related to the task of bypassing mean-field limitations, there is currently renewed interest in hierarchical models, and specifically in models where closer spins result in stronger links (see figure 1). Recently, Dysonʼs pioneering work [30], where the hierarchical ferromagnet was introduced and its phase transition (splitting an ergodic region from a ferromagnetic one) was rigorously proven, has been extended to investigate extensions to spin-glasses [31,[33][34][35][36][37][38]. Although an analytical solution is still not available, researchers have made significant steps toward deep comprehension of hierarchical statistical mechanics [32,[39][40][41][42][43][44][45].…”