Recent works unraveled an intriguing finite-time dynamical phase transition in the thermal relaxation of the mean field Curie-Weiss model. The phase transition reflects a sudden switch in the dynamics. Its existence in systems with a finite range of interaction, however, remained unclear. Here we demonstrate the dynamical phase transition for nearest-neighbor Ising systems on the square and Bethe lattices through extensive computer simulations and by analytical results. Combining large-deviation techniques and Bethe-Guggenheim theory we prove the existence of the dynamical phase transition for arbitrary quenches, including those within the two-phase region. Strikingly, for any given initial condition we prove and explain the existence of non-trivial speed limits for the dynamical phase transition and the relaxation of magnetization, which are fully corroborated by simulations of the microscopic Ising model but are absent in the mean field setting. Pair correlations, which are neglected in mean field theory and trivial in the Curie-Weiss model, account for kinetic constraints due to frustrated local configurations that give rise to a global speed limit.
Recent works unraveled an intriguing finite-time dynamical phase transition in the thermal relaxation of the mean field Curie-Weiss model. The phase transition reflects a sudden switch in the dynamics. Its existence in systems with a finite range of interaction, however, remained unclear. Here we demonstrate the dynamical phase transition for nearest-neighbor Ising systems on the square and Bethe lattices through extensive computer simulations and by analytical results. Combining large-deviation techniques and Bethe-Guggenheim theory we prove the existence of the dynamical phase transition for arbitrary quenches, including those within the two-phase region. Strikingly, for any given initial condition we prove and explain the existence of non-trivial speed limits for the dynamical phase transition and the relaxation of magnetization, which are fully corroborated by simulations of the microscopic Ising model but are absent in the mean field setting. Pair correlations, which are neglected in mean field theory and trivial in the Curie-Weiss model, account for kinetic constraints due to frustrated local configurations that give rise to a global speed limit.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.
We uncover a finite-time dynamical phase transition in the thermal relaxation of a mean-field magnetic model. The phase transition manifests itself as a cusp singularity in the probability distribution of the magnetization that forms at a critical time. The transition is due to a sudden switch in the dynamics, characterized by a dynamical order parameter. We derive a dynamical Landau theory for the transition that applies to a range of systems with scalar, parity-invariant order parameters. Close to criticalilty, our theory reveals an exact mapping between the dynamical and equilibrium phase transitions of the magnetic model, and implies critical exponents of mean-field type. We argue that interactions between nearby saddle points, neglected at the mean-field level, may lead to critical, spatiotemporal fluctuations of the order parameter, and thus give rise to novel, dynamical critical phenomena.
The laws of nature encompass the small, the large, the few, and the many. In this thesis we are concerned with classical (i.e. not quantum) many-body systems, which refers to any microscopic or macroscopic system that contains a large number of interacting entities. The nearest-neighbor Ising model, originally developed in 1920 by Wilhelm Lenz, forms a cornerstone in our theoretical understanding of collective effects in classical many-body systems, and is to date a paradigm for statistical physics. Despite its elegant and simplistic description, exact analytical results in dimensions equal and larger than two are difficult to obtain. Therefore, much work has been done to con- struct methods that allow for approximate, yet accurate, analytical solutions. One of these methods is the Bethe-Guggenheim approximation, originally developed independently by Hans Bethe and Edward Guggenheim in 1935. This approximation goes beyond the well- known mean field approximation, and explicitly accounts for pair correlations between the spins in the Ising model. In this thesis, we embark on a journey to exploit the full capacity of the Bethe-Guggenheim approximation, in non-uniform and non-equilibrium settings. After we formally introduce the original Bethe-Guggenheim approximation for uniform systems, we will extend its scope to non-uniform systems, and derive a Cahn-Hilliard free energy functional. Here we find that the one-dimensional equilibrium concentration profile undergoes a delocalization-induced broadening transition at interaction strengths near and above the thermal energy. The broadening transition arises from a decreasing amplitude of capillary wave fluctuations, and is not accounted for in the mean field approximation. Finally, going beyond equilibrium properties, we also study the kinetics of the Ising model in and out of equilibrium. First, based on firm theoretical and experimental evidence, we con- struct an Ising-like minimal model for interacting cellular adhesion molecules. Here we find a dynamical critical point where adhesion cluster dissolution and formation are the fastest, and undergo a qualitative change in dynamics. Second, we demonstrate the existence of a finite-time dynamical phase transition for disordering quenches in the nearest-neighbor Ising model. Starkly different from the mean field approximation, the time at which the dynamical phase transition occurs, is bounded from below by a speed limit. Altogether, in this thesis, we unveil the non-trivial and a priori non-intuitive effects of pair correlations in the classical nearest-neighbor Ising model, which are taken into account in the Bethe-Guggenheim approximation, and neglected in the mean field approximation.
We propose two different macroscopic dynamics to describe the decay of metastable phases in many-particle systems with local interactions. These dynamics depend on the macroscopic order parameter m through the restricted free energy F(m) and are designed to give the correct equilibrium distribution for m. The connection between macroscopic dynamics and the underlying microscopic dynamic is considered in the context of a projection-operator formalism. Application to the square-lattice nearest-neighbor Ising ferromagnet gives good agreement with droplet theory and Monte Carlo simulations of the underlying microscopic dynamic. This includes quantitative agreement for the exponential dependence of the lifetime 〈\ensuremath{\tau}〉 on the inverse of the applied field H, and the observation of distinct field regions in which \ensuremath{\Lambda}\ensuremath{\equiv}d ln〈\ensuremath{\tau}〉/d\ensuremath{\Vert}H${\mathrm{\ensuremath{\Vert}}}^{1\mathrm{\ensuremath{-}}\mathit{d}}$ depends differently on \ensuremath{\Vert}H\ensuremath{\Vert}. In addition, at very low temperatures we observe oscillatory behavior of \ensuremath{\Lambda} with respect to \ensuremath{\Vert}H\ensuremath{\Vert}, which is due to the discreteness of the lattice and in agreement with rigorous results. Similarities and differences between this work and earlier works on finite Ising models in the fixed-magnetization ensemble are discussed.
We study the nonequilibrium aspects of the kinetics of a mixed spin-1 and spin-2 Ising system including the biquadratic nearest-neighbor pair interaction ( K ) and crystal-field interaction ( D ) under the presence of a time-dependent oscillating external magnetic field within a mean-field approach. In particular, we calculate the dynamic phase transition (DPT) temperatures and present dynamic phase diagrams. The set of mean-field dynamic equations is obtained by employing Glauber transition rates. We investigate the time variations of average order parameters to find the phases in the system. We also study the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the phase transitions and obtain the DPT points. The dynamic phase diagrams are presented in three different planes. Phase diagrams contain disordered (d), ferrimagnetic (i), antiquadrupolar or staggered (a) phases, and four coexistence or mixed phase regions, namely i+d, i+a, i+a+d and a+d mixed phases, that strongly depend on interaction parameters.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbor interactions. In order to generate the random magnetic fields, we have considered random variables { h } that change randomly with time according to a double-Gaussian probability distribution, which consists of two single-Gaussian distributions, centered at + h o and − h o , with the same width σ. This distribution is very general and can recover in appropriate limits the bimodal distribution ( ) and the single-Gaussian one ( h o = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32–512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurrence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities h o for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ.