Phase transitions are typically accompanied by non-analytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real-axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method. Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size. We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models, and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality. We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.
In the framework of the Tsallis nonextensive statistical mechanics we study an assembly of N spins, first in a background magnetic field, and then assuming them to interact via a long-range homogeneous mean field. To take into account the spin fluctuations the dynamical field coefficient is considered to be linearly dependent on the temperature. The physical quantities are evaluated using a perturbative expansion in the nonextensivity parameter (1-q). The extended Curie-Weiss law in the mean field case has been generalized. The critical temperature and the Curie- Weiss constant are found to be…
The Curie-Weiss model is an exactly soluble model of ferromagnetism that allows one to study in detail the thermodynamic functions, in particular their properties in the neighbourhood of the critical temperature. In this model every magnetic moment interacts with every other magnetic moment. Because of its simplicity and because of the correctness of at least of some of its predictions, the Curie-Weiss model occupies an important place in the statistical mechanics literature and its application to information theory. It is frequently presented as an introduction to the Ising model or to spin g…
Using the theory of large deviations, we analyze the phase transition structure of the Curie-Weiss-Potts spin model, which is a mean-field approximation to the Potts model. This analysis is carried out both for the canonical ensemble and the microcanonical ensemble. Besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles, we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates, relating these to concavity and support properties of the microcanonical entropy. The Curie-Weiss-Potts mo…
The object of this study is a cell model with Curie-Weiss interaction potential. We have already proved the possibility of a mathematically rigorous transition from a continuous system of interacting particles to such a model and made an accurate calculation of its grand partition function. In the present research, we derive an explicit accurate equation of state of the cell model with Curie-Weiss potential. It turned out that this model has a sequence of first-order phase transitions at temperatures below the critical. We analyzed the mechanism of these transitions based on the behavior of th…
We explain the Curie Weiss model in Statistical Mechanics within the Ergodic viewpoint. More precisely, we simultaneously define in $\{-1,+1\}^{\mathbb{N}}$, on the one hand a generalized Curie Weiss model within the Statistical Mechanics viewpoint and on the other hand, quadratic free energy and quadratic pressure within the Ergodic Theory viewpoint. We show that there are finitely many invariant measures which maximize the quadratic free energy. They are all Dynamical Gibbs Measures. Moreover, the Probabilistic Gibbs measures for generalized Curie Weiss model converge to a determined combina…
The microcanonical entropy s(e,m) as a function of the energy e and the magnetization m is computed analytically for the anisotropic quantum Heisenberg model with Curie-Weiss-type interactions. The result shows a number of interesting properties which are peculiar to long-range interacting systems, including nonequivalence of ensembles and partial equivalence. Furthermore, from the shape of the entropy it follows that the Curie-Weiss Heisenberg model is indistinguishable from the Curie-Weiss Ising model in canonical thermodynamics, although their microcanonical thermodynamics in general differ…
The Curie-Weiss Potts model is a mean field version of the well-known Potts model. In this model, the critical line $β= β_c (h)$ is explicitly known and corresponds to a first order transition when $q > 2$. In the present paper we describe the fluctuations of the density vector in the whole domain $β\geqslant 0$ and $h \geqslant 0$, including the conditional fluctuations on the critical line and the non-Gaussian fluctuations at the extremity of the critical line. The probabilities of each of the two thermodynamically stable states on the critical line are also computed. Similar results are inf…
The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy s(e,m) is obtained as a function of the energy e and the magnetization vector m in the thermodynamic limit. Since, for this model, e is uniquely determined by m, the same information can be encoded either in s(m) or s(e,m1,m2). Although these two entropies correspond to the same physical settin…
We study the large deviations of the magnetization at some finite time in the Curie-Weiss Random Field Ising Model with parallel updating. While relaxation dynamics in an infinite time horizon gives rise to unique dynamical trajectories (specified by initial conditions and governed by first-order dynamics of the form $m_{t+1}=f(m_t)$), we observe that the introduction of a finite time horizon and the specification of terminal conditions can generate a host of metastable solutions obeying \textit{second-order} dynamics. We show that these solutions are governed by a Newtonian-like dynamics in d…
The coherence length of the medium-range order (MRO) in metallic liquids is known to display a Curie-Weiss temperature dependence; its inverse is linearly related to temperature, and when extrapolated from temperatures above the glass transition, the coherence length diverges at a negative temperature with a critical exponent of unity. We propose a mean-field pseudospin model that explains this behavior. Specifically, we model the atoms and their local environment as Ising spins with antiferromagnetic exchange interactions. We further superimpose an exchange interaction between dynamical heter…
The mean field theory, in its different hues, form one of the most useful tools for calculating the single-body physical properties of a many-body system. It provides important information, like critical exponents, of the systems that do not yield to an exact analytical treatment. Here we propose an entanglement mean field theory (EMFT) to obtain the behavior of the two-body physical properties of such systems. We apply this theory to predict the phases in paradigmatic strongly correlated systems, viz. the transverse anisotropic XY, the transverse XX, and the Heisenberg models. We find the cri…
Among the Markov chains breaking detailed-balance that have been proposed in the field of Monte-Carlo sampling in order to accelerate the convergence towards the steady state with respect to the detailed-balance dynamics, the idea of 'Lifting' consists in duplicating the configuration space into two copies $σ=\pm$ and in imposing directed flows in each copy in order to explore the configuration space more efficiently. The skew-detailed-balance Lifted-Markov-chain introduced by K. S. Turitsyn, M. Chertkov and M. Vucelja [Physica D Nonlinear Phenomena 240 , 410 (2011)] is revisited for the Curie…
Motivated by the anisotropic long-range nature of the interactions between cold dipolar atoms or molecules in an optical lattice, we study the anisotropic quantum Heisenberg model with Curie-Weiss-type long-range interactions. Absence of a heat bath in optical lattice experiments suggests a study of this model within the microcanonical ensemble. The microcanonical entropy is calculated analytically, and nonequivalence of microcanonical and canonical ensembles is found for a range of anisotropy parameters. From the shape of the entropy it follows that the Curie-Weiss Heisenberg model is indisti…
A hamiltonian model is solved, which satisfies all requirements for a realistic ideal quantum measurement. The system S is a spin-$\half$, whose $z$-component is measured through coupling with an apparatus A=M+B, consisting of a magnet $\RM$ formed by a set of $N\gg 1$ spins with quartic infinite-range Ising interactions, and a phonon bath $\RB$ at temperature $T$. Initially A is in a metastable paramagnetic phase. The process involves several time-scales. Without being much affected, A first acts on S, whose state collapses in a very brief time. The mechanism differs from the usual decoherenc…
In this paper we solve the inverse problem for a class of mean field models (Curie-Weiss model and its multi-species version) when multiple thermodynamic states are present, as in the low temperature phase where the phase space is clustered. The inverse problem consists in reconstructing the model parameters starting from configuration data generated according to the distribution of the model. We show that the application of the inversion procedure without taking into account the presence of many states produces very poor inference results. This problem is overcomed using the clustering algori…
At low temperatures, a spin ice enters a Coulomb phase - a state with algebraic correlations and topologically constrained spin configurations. In Ho2Ti2O7, we have observed experimentally that this process is accompanied by a non-standard temperature evolution of the wave vector dependent magnetic susceptibility, as measured by neutron scattering. Analytical and numerical approaches reveal signatures of a crossover between two Curie laws, one characterizing the high temperature paramagnetic regime, and the other the low temperature topologically constrained regime, which we call the spin liqu…
We consider the zeros of the partition function of the Ising model with ferromagnetic pair interactions and complex external field. Under the assumption that the graph with strictly positive interactions is connected, we vary the interaction (denoted by $t$) at a fixed edge. It is already known that each zero is monotonic (either increasing or decreasing) in $t$; we prove that its motion is local: the entire trajectories of any two distinct zeros are disjoint. If the underlying graph is a complete graph and all interactions take the same value $t\geq 0$ (i.e., the Curie-Weiss model), we prove …
We introduce a mean-field model for analysing the dynamics of human consciousness. In particular, inspired by the Giulio Tononi's Integrated Information Theory and by the Max Tegmark's representation of consciousness, we study order-disorder phase transitions on Curie-Weiss models generated by processing EEG signals. The latter have been recorded on healthy individuals undergoing deep sedation. Then, we implement a machine learning tool for classifying mental states using, as input, the critical temperatures computed in the Curie-Weiss models. Results show that, by the proposed method, it is p…
