Entropic Effects in the Very Low Temperature Regime of Diluted Ising Spin Glasses with Discrete Couplings

Jörg, Thomas; Ricci-Tersenghi, Federico

doi:10.1103/physrevlett.100.177203

Cited by 13 publications

(19 citation statements)

References 32 publications

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“…Hierarchical lattices provide exact renormalizationgroup solutions to network [21,22,26,27,28,29,30,31,32] and other diverse complex problems, as seen in recent works [33,34,35,36,37,38,39,40,41,42,43,44,45,46]. The percolation problem presented by the random network defined above is also readily solved by renormalization-group theory.…”

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confidence: 94%

Critical percolation phase and thermal Berezinskii-Kosterlitz-Thouless transition in a scale-free network with short-range and long-range random bonds

2009

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Percolation in a scale-free hierarchical network is solved exactly by renormalization-group theory in terms of the different probabilities of short-range and long-range bonds. A phase of critical percolation, with algebraic ͓Berezinskii-Kosterlitz-Thouless ͑BKT͔͒ geometric order, occurs in the phase diagram in addition to the ordinary ͑compact͒ percolating phase and the nonpercolating phase. It is found that no connection exists between, on the one hand, the onset of this geometric BKT behavior and, on the other hand, the onsets of the highly clustered small-world character of the network and of the thermal BKT transition of the Ising model on this network. Nevertheless, both geometric and thermal BKT behaviors have inverted characters, occurring where disorder is expected, namely, at low bond probability and high temperature, respectively. This may be a general property of long-range networks. Scale-free networks are of high current interest ͓1-5͔ due to their ubiquitous occurrence in physical, biological, social, and information systems and due to their distinctive geometric and thermal properties. The geometric properties reflect the connectivity of the points of the network. The thermal properties reflect the interactions, along the geometric lines of connectivity, between degrees of freedom located at the points of the network. These interacting degrees of freedom could be voters influencing each other, persons communicating a disease, etc. and can be represented by model systems. Among issues most recently addressed have been the occurrence of true or algebraic ͓6,7͔ order in the geometric or thermal long-range correlations, and the connection between these geometric and thermal characteristics. In Ising magnetic systems on a one-dimensional inhomogeneous lattice ͓8-11͔ and on an inhomogeneous growing network ͓12͔, a Berezinskii-Kosterlitz-Thouless ͑BKT͒ phase in which the thermal correlations between the spins decay algebraically with distance was found. In growing networks ͓13-20͔, geometric algebraic correlations were seen with the exponential ͑non-power-law͒ scaling of the size of the giant component above the percolation threshold. The connection between geometric and thermal properties was investigated with an Ising magnetic system on a hierarchical lattice that can be continuously tuned from non-small-world to highly clustered small-world via increase in the occurrence of quenchedrandom long-range bonds ͓21͔. Whereas in the non-smallworld regime a standard second-order phase transition was found, when the small-world regime is entered, an inverted BKT transition was found, with a high-temperature algebraically ordered phase and a low-temperature phase with true long-range order but delayed short-range order. Algebraic order in the thermal correlations has also been found in a community network ͓22͔. In the current work, the geometric percolation property of the quenched-random long-range bonds is studied, aiming to relate the geometric properties to the algebraic thermal properties. From an exact re...

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confidence: 94%

Critical percolation phase and thermal Berezinskii-Kosterlitz-Thouless transition in a scale-free network with short-range and long-range random bonds

2009

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“…Recently in Ref. [8] the same model has been solved exactly on the hierarchical lattice, showing that T = 0 computations can lead to misleading results.…”

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confidence: 99%

“…Given that the model is link diluted, we can eliminate recursively weakly connected spins, generalizing what was done in Refs. [6,8]. In the original model couplings are Tindependent, but, by decimating spins, effective couplings are created whose intensity will depend on temperature.…”

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Entropic long range order in a 3D spin glass model

Angelini¹,

Ricci-Tersenghi²

2011

J. Stat. Mech.

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We uncover a new kind of entropic long range order in finite dimensional spin glasses. We study the link-diluted version of the Edwards-Anderson spin glass model with bimodal couplings (J = ±1) on a 3D lattice. By using exact reduction algorithms, we prove that there exists a region of the phase diagram (at zero temperature and link density low enough), where spins are long range correlated, even if the ground states energy stiffness is null. In other words, in this region twisting the boundary conditions cost no energy, but spins are long range correlated by means of pure entropic effects.

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“…The reasons behind the instability of the discrete fixed point are, as first discussed by [10], entropic fluctuations that at any finite temperature make the zero-temperature and the vanishing-temperature limits different (a very similar phenomenon appears in diluted 3D spin glasses [14], where the critical dilution calculated by first setting the temperature to zero and then taking the system size to infinity is strictly larger than when first taking the system size to infinity at finite temperature and then taking the temperature to zero). This is in fact deeply connected with the notion of temperature chaos [15,16,17,18,24,29,30].…”

Section: Entropic Fluctuations and Temperature Chaosmentioning

confidence: 97%

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

Jörg¹,

Krząkała²

2012

J. Stat. Mech.

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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-...

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Entropic Effects in the Very Low Temperature Regime of Diluted Ising Spin Glasses with Discrete Couplings

Cited by 13 publications

References 32 publications

Critical percolation phase and thermal Berezinskii-Kosterlitz-Thouless transition in a scale-free network with short-range and long-range random bonds

Critical percolation phase and thermal Berezinskii-Kosterlitz-Thouless transition in a scale-free network with short-range and long-range random bonds

Entropic long range order in a 3D spin glass model

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

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