We introduce a novel approach to study the critical behavior of equilibrium and non-equilibrium systems which is based on the concept of an instantaneous correlation length. We analyze in detail two classical statistical mechanical systems: the XY model and the Ising model, and one of the prototype models of Self-Organized Criticality: the forest fire model (FFM). The proposed method can both capture the critical behavior of the XY model and the Ising model and discriminate between the nature of the phase transition in the two scenarios. When applied to the FFM, it gives surprising results, suggesting that the model could be critical despite displaying broken scaling in the distribution of cluster sizes.
We introduce a novel approach to study the critical behavior of equilibrium and non-equilibrium systems which is based on the concept of an instantaneous correlation length. We analyze in detail two classical statistical mechanical systems: the XY model and the Ising model, and one of the prototype models of Self-Organized Criticality: the forest fire model (FFM). The proposed method can both capture the critical behavior of the XY model and the Ising model and discriminate between the nature of the phase transition in the two scenarios. When applied to the FFM, it gives surprising results, suggesting that the model could be critical despite displaying broken scaling in the distribution of cluster sizes.
In 1989 Janssen, Schaub and Schmittmann have shown that universality and scaling hold already in the early stage of the dynamical evolution of statistical systems, if the system, initially at a very high temperature, is suddenly quenched to the critical temperature and then released to the dynamic evolution according to model A. This allows for a measurement of all the static and dynamic critical exponents and even for the critical point already in the short-time regime, i.e., far from equilibrium. Since the correlation length is still small here, the simulations do not suffer from critical slowing down, a problem encountered in the usual measurements in equilibrium. The concept has been successfully applied to a variety of statistical systems. We will report about recent results for the fully frustrated XY model, where the short-time approach is particularly efficient, since the standard cluster algorithm does not apply because of the frustration.
We propose an interpretation of previous experiments and numerical experiments showing that, for a large class of systems, distributions of global quantities are similar to a distribution originally obtained for the magnetization in the 2D-XY model (Bramwell S. T. et al. , Nature , 396 (1998) 512). This approach, developed for the Ising model, is based on previous numerical observations (Clusel M. et al. , Phys. Rev. E , 70 (2004) 046112). We obtain an effective action using a perturbative method, which successfully describes the order parameter fluctuations near the phase transition. This leads to a direct link between the D-dimensional Ising model and the XY model in the same dimension, which appears to be a generic feature of many equilibrium critical systems and which is at the heart of the above observations.
In statistical physics, if we divide successively an equilibrium system into two parts, we will face a situation that, to a certain length ξ, the physics of a subsystem is no longer the same as the original one. The extensive property of the thermal entropy S (A ∪ B) = S (A) + S (B) is then violated. This observation motivates us to introduce a concept of correlation entropy between two points, as measured by the mutual information in information theory, to study the critical phenomena. A rigorous relation is established to display some drastic features of the non-vanishing correlation entropy of a subsystem formed by any two distant particles with long-range correlation. This relation actually indicates a universal role played by the correlation entropy for understanding the critical phenomena. We also verify these analytical studies in terms of two well-studied models for both the thermal and quantum phase transitions: the two-dimensional Ising model and the one-dimensional transverse-field Ising model. Therefore, the correlation entropy provides us with a new physical intuition of the critical phenomena from the point of view of information theory.
Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.
Clearly, in nature, but also in technological applications, complex systems built in an entirely ordered and regular fashion are the exception rather than the rule. In this thesis we explore how critical phenomena are influenced by quenched spatial randomness. Specifically, we consider physical systems undergoing a continuous phase transition in the presence of topological disorder, where the underlying structure, on which the system evolves, is given by a non-regular, discrete lattice. We therefore endeavour to achieve a thorough understanding of the interplay between collective dynamics and quenched randomness. According to the intriguing concept of universality, certain laws emerge from collectively behaving many-body systems at criticality, almost regardless of the precise microscopic realization of interactions in those systems. As a consequence, vastly different phenomena show striking similarities at their respective phase transitions. In this dissertation we pursue the question of whether the universal properties of critical phenomena are preserved when the system is subjected to topological perturbations. For this purpose, we perform numerical simulations of several prototypical systems of statistical physics which show a continuous phase transition. In particular, the equilibrium spin-1/2 Ising model and its generalizations represent -- among other applications -- fairly natural approaches to model magnetism in solids, whereas the non-equilibrium contact process serves as a toy model for percolation in porous media and epidemic spreading. Finally, the Manna sandpile model is strongly related to the concept of self-organized criticality, where a complex dynamic system reaches a critical state without fine-tuning of external variables. Our results reveal that the prevailing understanding of the influence of topological randomness on critical phenomena is insufficient. In particular, by considering very specific and newly developed lattice structures, we are able to show that -- contrary to the popular opinion -- spatial correlations in the number of interacting neighbours are not a key measure for predicting whether disorder ultimately alters the behaviour of a given critical system.
We study numerically the non-equilibrium critical properties of the Ising model defined on direct products of graphs, obtained from factor graphs without phase transition (Tc = 0). On this class of product graphs, the Ising model features a finite temperature phase transition, and we find a pattern of scaling behaviors analogous to the one known on regular lattices: Observables take a scaling form in terms of a function L(t) of time, with the meaning of a growing length inside which a coherent fractal structure, the critical state, is progressively formed. Computing universal quantities, such as the critical exponents and the limiting fluctuation-dissipation ratio X_\infty, allows us to comment on the possibility to extend universality concepts to the critical behavior on inhomogeneous substrates.