The nature of the different zero-temperature phases in discrete two-dimensional spin glasses: entropy, universality, chaos and cascades in the renormalization group flow

Abstract

The properties of discrete two-dimensional spin glasses depend strongly on the way the zero-temperature limit is taken. We discuss this phenomenon in the context of the Migdal-Kadanoff renormalization group. We see, in particular, how these properties are connected with the presence of a cascade of fixed points in the renormalization group flow. Of particular interest are two unstable fixed points that correspond to two different spin-glass phases at zero temperature. We discuss how these phenomena are related with the presence of entropy fluctuations and temperature chaos, and universality in this model. PACS numbers: 75.50. Lk,05.70.Fh,64.60.Fr Since the beginning of the study of disordered systems with renormalization and scaling methods the question of universality and of the relevance of the realization of the disorder have been a key issue [1]. Why indeed should we accept that some abstract models are the archetype of a very broad class of systems if their behavior depends drastically on tiny details? The Edwards-Anderson [2] model is one of the models that are widely regarded as a prototype of disordered systems in statistical physics. It is an Ising model with disordered and competitive interactions and has been the source of many surprises and developments in the last thirty years [3,4].Its two-dimensional (2D) version, one of its simplest settings, has very special properties with respect to universality. It is now agreed that the spin-glass phase exists only at zero temperature [5,6,7], where the spin-glass susceptibility diverges. However, the behavior of the model seems to depend drastically on microscopic details and in particular discrete and continuous couplings leads to different properties [8,9] at zero temperature. On the other hand, for nonzero temperatures, strong evidence for universal critical behavior has been observed [10,11,12]. This rises questions on the very nature of universality, if any, in strongly disordered bidimensional systems. Why such a difference? What are the mechanisms behind this behavior? The key to understand these features lies in the difference between strictly zero and vanishing temperature [13,14]. When temperature is finite, entropy fluctuations play a major role and, for large enough sizes, bring back discrete models to the continuous class [10]. These mechanisms were further exploited in [15], with a special emphasis on the so-called temperature chaos effect [16,17,18].In this Letter we study the behavior of continuous and discrete 2D spin glasses in the context of the Migdal-Kadanoff renormalization group (MKRG) [19]. We observe that indeed continuous and discrete 2D spin glasses are eventually associated with the very same physically relevant fixed point. However, there are also key differences that are the consequence of a cascade of two repulsive fixed points in the MKRG flow in the case of the discrete model. In a real-space picture, these phenomena are associated with two different temperature-dependent crossover length scales, so that two different zero-...

Universal critical behavior of the two-dimensional Ising spin glass

“…Several renormalization group (RG) fixed points appear at T = 0, depending on the nature of the couplings distribution [26]. However, most of these fixed points are unstable even for the tiniest positive temperature: the only remaining universality class is the one of the continuous coupling constants [28,[32][33][34][35] (the very same effect is found in the random field Ising model [40]). …”

Domain-wall excitations in the two-dimensional Ising spin glass

“…The couplings J ij are quenched random variables. At zero temperature, two distinct types of behavior are expected, one for discrete and commensurate allowed coupling values and a second class for distributions with incommensurate or continuous support 15,[17][18][19]57 . We consider one representative of each class, namely the symmetric bimodal (±J) distribution,…”

Chaos properties of the one-dimensional long-range Ising spin-glass

“…In the field of spin-glasses, the notion of 'chaos' has been introduced as the sensitivity of the renormalization flow seen as a 'dynamical system', with respect to the initial conditions (the random couplings) or with respect to external parameters like the temperature T or the magnetic field H. On hierarchical lattices where explicit renormalization rules exist for the renormalized couplings J R , the chaos properties have been thus much studied [1][2][3][4][5][6][7][8][9][10]. For other lattices without explicit renormalization rules, the droplet scaling theory [11][12][13] allows to define the chaos properties as follows :…”