Reply to "Comment on 'Effect of polydispersity on the ordering transition of adsorbed self- assembled rigid rods'"

We comment on the nature of the ordering transition of a model of equilibrium polydisperse rigid rods on the square lattice, which is reported by Lopez et al. to exhibit random percolation criticality in the canonical ensemble, in sharp contrast to (i) our results of Ising criticality for the same model in the grand canonical ensemble [Phys. Rev. E 82, 061117 (2010)] and (ii) the absence of exponent(s) renormalization for constrained systems with logarithmic specific-heat anomalies predicted on very general grounds by Fisher [Phys. Rev. 176, 257 (1968)].

Avalanches in a fluctuationless first-order phase transition in a random-bond Ising model

We study the behavior of the random-bond Ising model at zero temperature by numerical simulations for a variable amount of disorder. The model is an example of systems exhibiting a fluctuationless first-order phase transition similar to some field-induced phase transitions in ferromagnetic systems and the martensitic phase transition appearing in a number of metallic alloys. We focus on the study of the hysteresis cycles appearing when the external field is swept from positive to negative values. By using a finite-size scaling hypothesis, we analyze the disorder-induced phase transition between the phase exhibiting a discontinuity in the hysteresis cycle and the phase with the continuous hysteresis cycle. Critical exponents characterizing the transition are obtained. We also analyze the size and duration distributions of the magnetization jumps (avalanches).

Hysteresis and avalanches in the random anisotropy Ising model

ches. The critical point is characterized by a set of critical exponents, which are consistent with the universal values proposed from the study of other simpler models.

Diluted three-dimensional random field Ising model at zero temperature with metastable dynamics.

The influence of vacancy concentration on the behavior of the three-dimensional random field Ising model with metastable dynamics is studied. We have focused our analysis on the number of spanning avalanches which allows us a clear determination of the critical line where the hysteresis loops change from continuous to discontinuous. By a detailed finite-size scaling analysis we determine the phase diagram and numerically estimate the critical exponents along the whole critical line. Finally, we discuss the origin of the curvature of the critical line at high vacancy concentration.

Universality in models for disorder induced phase transitions

We have systematically analyzed six different reticular models with quenched disorder and no thermal fluctuations exhibiting a field-driven first-order phase transition. We have studied the nonequilibrium transition, appearing when varying the amount of disorder, characterized by the change from a discontinuous hysteresis cycle (with one or more large avalanches) to a smooth one (with only tiny avalanches). We have computed critical exponents using finite size scaling techniques and shown that they are consistent with universal values depending only on the space dimensionality d.

Nonequilibrium transitions induced by multiplicative noise

A simple model exhibiting a noise-induced ordering transition (NIOT) and a noise-induced disordering transition (NIDT), in which the noise is purely multiplicative, is presented. Both transitions are found in two dimensions as well as in one dimension. We show analytically and numerically that the critical behavior of these two transitions is described by the so called multiplicative noise (MN) universality class. A computation of the set of critical exponents is presented in both d=1 and d=2.

Potts fully frustrated model: Thermodynamics, percolation and dynamics in two dimensions

We consider a Potts model diluted by fully frustrated Ising spins. The model corresponds to a fully frustrated Potts model with variables having an integer absolute value and a sign. This model presents precursor phenomena of a glass transition in the high-temperature region. We show that the onset of these phenomena can be related to a thermodynamic transition. Furthermore, this transition can be mapped onto a percolation transition. We numerically study the phase diagram in two dimensions (2D) for this model with frustration and without disorder and we compare it to the phase diagram of (i) the model with frustration and disorder and (ii) the ferromagnetic model. Introducing a parameter that connects the three models, we generalize the exact expression of the ferromagnetic Potts transition temperature in 2D to the other cases. Finally, we estimate the dynamic critical exponents related to the Potts order parameter and to the energy.

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is constructed for two different models exhibiting self-organized criticality: the Bak-Tang-Wiesenfeld (BTW) sandpile model [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] and the Zhang model [Phys. Rev. Lett. 63, 470 (1989)]. The dynamic renormalization group (DRG) enables one to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

Dynamical properties of the Zhang model of self-organized criticality

Critical exponents of the infinitely slowly driven Zhang model of self-organized criticality are computed for d=2 and 3, with particular emphasis devoted to the various roughening exponents. Besides confirming recent estimates of some exponents, new quantities are monitored, and their critical exponents computed. Among other results, it is shown that the three-dimensional exponents do not coincide with the Bak-Tang-Wiesenfeld [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] (Abelian) model, and that the dynamical exponent as computed from the correlation length and from the roughness of the energy profile do not necessarily coincide, as is usually implicitly assumed. An explanation for this is provided. The possibility of comparing these results with those obtained from renormalization group arguments is also briefly addressed.

Nonequilibrium phase transition in a model for the propagation of innovations among economic agents

We characterize the different morphological phases that occur in a simple one-dimensional model of propagation of innovations among economic agents [X. Guardiola et al., Phys. Rev E 66, 026121 (2002)]. We show that the model can be regarded as a nonequilibrium surface growth model. This allows us to demonstrate the presence of a continuous roughening transition between a flat (system size independent fluctuations) and a rough phase (system size dependent fluctuations). Finite-size scaling studies at the transition strongly suggest that the dynamic critical transition does not belong to directed percolation and, in fact, critical exponents do not seem to fit in any of the known universality classes of nonequilibrium phase transitions. Finally, we present an explanation for the occurrence of the roughening transition and argue that avalanche driven dynamics is responsible for the novel critical behavior.

Phenomenological models of holographic superconductors and hall currents

We study general models of holographic superconductivity parametrized by four arbitrary functions of a neutral scalar field of the bulk theory. The models can accommodate several features of real superconductors, like arbitrary critical temperatures and critical exponents in a certain range, and perhaps impurities or boundary or thickness effects. We find analytical expressions for the critical exponents of the general model and show that they satisfy the Rushbrooke identity. An important subclass of models exhibit second order phase transitions. A study of the specific heat shows that general models can also describe holographic superconductors undergoing first, second and third (or higher) order phase transitions. We discuss how small deformations of the HHH model can lead to the appearance of resonance peaks in the conductivity, which increase in number and become narrower as the temperature is gradually decreased, without the need for tuning mass of the scalar to be close to the Breitenlohner-Freedman bound. Finally, we investigate the inclusion of a generalized ¿theta term¿ producing Hall effect without magnetic field.

Percolation of spatially constrained Erdos-R enyi networks with degree correlations

Motivated by experiments on activity in neuronal cultures [J. Soriano, M. Rodr ́ıguez Mart́ınez, T. Tlusty, and E. Moses, Proc. Natl. Acad. Sci. 105, 13758 (2008)], we investigate the percolation transition and critical exponents of spatially embedded Erd̋os-Ŕenyi networks with degree correlations. In our model networks, nodes are randomly distributed in a two-dimensional spatial domain, and the connection probability depends on Euclidian link length by a power law as well as on the degrees of linked nodes. Generally, spatial constraints lead to higher percolation thresholds in the sense that more links are needed to achieve global connectivity. However, degree correlations favor or do not favor percolation depending on the connectivity rules. We employ two construction methods to introduce degree correlations. In the first one, nodes stay homogeneously distributed and are connected via a distance- and degree-dependent probability. We observe that assortativity in the resulting network leads to a decrease of the percolation threshold. In the second construction methods, nodes are first spatially segregated depending on their degree and afterwards connected with a distance-dependent probability. In this segregated model, we find a threshold increase that accompanies the rising assortativity. Additionally, when the network is constructed in a disassortative way, we observe that this property has little effect on the percolation transition.

Monte Carlo study of the XY-model on quasi-periodic lattices

Monte Carlo Simulations were carried out using a nearest neighbour ferromagnetic XYmodel, on both 2-D and 3-D quasi-periodic lattices. In the case of 2-D, both the unfrustrated and frustrated XV-model were studied. For the unfrustrated 2-D XV-model, we have examined the magnetization, specific heat, linear susceptibility, helicity modulus and the derivative of the helicity modulus with respect to inverse temperature. The behaviour of all these quatities point to a Kosterlitz-Thouless transition occuring in temperature range Te == (1.0 -1.05) JlkB and with critical exponents that are consistent with previous results (obtained for crystalline lattices) . However, in the frustrated case, analysis of the spin glass susceptibility and EdwardsAnderson order parameter, in addition to the magnetization, specific heat and linear susceptibility, support a spin glass transition. In the case where the 'thin' rhombus is fully frustrated, a freezing transition occurs at Tf == 0.137 JlkB , which contradicts previous work suggesting the critical dimension of spin glasses to be de > 2 . In the 3-D systems, examination of the magnetization, specific heat and linear susceptibility reveal a conventional second order phase transition. Through a cumulant analysis and finite size scaling, a critical temperature of Te == (2.292 ± 0.003) JI kB and critical exponents of 0:' == 0.03 ± 0.03, f3 == 0.30 ± 0.01 and I == 1.31 ± 0.02 have been obtained.

Un modèle à criticalité auto-régulée de la magnétosphère terrestre

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Exposants géométriques des modèles de boucles dilués et idempotents des TL-modules de la chaîne de spins XXZ

Cette thèse porte sur les phénomènes critiques survenant dans les modèles bidimensionnels sur réseau. Les résultats sont l'objet de deux articles : le premier porte sur la mesure d'exposants critiques décrivant des objets géométriques du réseau et, le second, sur la construction d'idempotents projetant sur des modules indécomposables de l'algèbre de Temperley-Lieb pour la chaîne de spins XXZ. Le premier article présente des expériences numériques Monte Carlo effectuées pour une famille de modèles de boucles en phase diluée. Baptisés "dilute loop models (DLM)", ceux-ci sont inspirés du modèle O(n) introduit par Nienhuis (1990). La famille est étiquetée par les entiers relativement premiers p et p' ainsi que par un paramètre d'anisotropie. Dans la limite thermodynamique, il est pressenti que le modèle DLM(p,p') soit décrit par une théorie logarithmique des champs conformes de charge centrale c(\kappa)=13-6(\kappa+1/\kappa), où \kappa=p/p' est lié à la fugacité du gaz de boucles \beta=-2\cos\pi/\kappa, pour toute valeur du paramètre d'anisotropie. Les mesures portent sur les exposants critiques représentant la loi d'échelle des objets géométriques suivants : l'interface, le périmètre externe et les liens rouges. L'algorithme Metropolis-Hastings employé, pour lequel nous avons introduit de nombreuses améliorations spécifiques aux modèles dilués, est détaillé. Un traitement statistique rigoureux des données permet des extrapolations coïncidant avec les prédictions théoriques à trois ou quatre chiffres significatifs, malgré des courbes d'extrapolation aux pentes abruptes. Le deuxième article porte sur la décomposition de l'espace de Hilbert \otimes^nC^2 sur lequel la chaîne XXZ de n spins 1/2 agit. La version étudiée ici (Pasquier et Saleur (1990)) est décrite par un hamiltonien H_{XXZ}(q) dépendant d'un paramètre q\in C^\times et s'exprimant comme une somme d'éléments de l'algèbre de Temperley-Lieb TL_n(q). Comme pour les modèles dilués, le spectre de la limite continue de H_{XXZ}(q) semble relié aux théories des champs conformes, le paramètre q déterminant la charge centrale. Les idempotents primitifs de End_{TL_n}\otimes^nC^2 sont obtenus, pour tout q, en termes d'éléments de l'algèbre quantique U_qsl_2 (ou d'une extension) par la dualité de Schur-Weyl quantique. Ces idempotents permettent de construire explicitement les TL_n-modules indécomposables de \otimes^nC^2. Ceux-ci sont tous irréductibles, sauf si q est une racine de l'unité. Cette exception est traitée séparément du cas où q est générique. Les problèmes résolus par ces articles nécessitent une grande variété de résultats et d'outils. Pour cette raison, la thèse comporte plusieurs chapitres préparatoires. Sa structure est la suivante. Le premier chapitre introduit certains concepts communs aux deux articles, notamment une description des phénomènes critiques et de la théorie des champs conformes. Le deuxième chapitre aborde brièvement la question des champs logarithmiques, l'évolution de Schramm-Loewner ainsi que l'algorithme de Metropolis-Hastings. Ces sujets sont nécessaires à la lecture de l'article "Geometric Exponents of Dilute Loop Models" au chapitre 3. Le quatrième chapitre présente les outils algébriques utilisés dans le deuxième article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2", constituant le chapitre 5. La thèse conclut par un résumé des résultats importants et la proposition d'avenues de recherche qui en découlent.