The time evolution of the spin distribution for the XY model in mean-field theory is described by the Fokker–Planck transport equation. This equation is Fourier transformed to generate an infinite-dimensional nonlinear dynamical system. The otherwise well-known behavior of the XY model is then explained using a fully dynamical-systems approach. The evolution of the phase-space trajectories of this dynamical system from initial nonequilibrium states is obtained. Regardless of the initial spin distribution, its attractors (all are point attractors) determine the equilibrium magnetic state of the model. The phase transition at a critical temperature is accompanied by a supercritical bifurcation. Numerical results for the time dependence of the phase-space trajectories and the order parameter are presented.
The time evolution of the spin distribution for the XY model in mean-field theory is described by the Fokker–Planck transport equation. This equation is Fourier transformed to generate an infinite-dimensional nonlinear dynamical system. The otherwise well-known behavior of the XY model is then explained using a fully dynamical-systems approach. The evolution of the phase-space trajectories of this dynamical system from initial nonequilibrium states is obtained. Regardless of the initial spin distribution, its attractors (all are point attractors) determine the equilibrium magnetic state of the model. The phase transition at a critical temperature is accompanied by a supercritical bifurcation. Numerical results for the time dependence of the phase-space trajectories and the order parameter are presented.
Non-equilibrium critical dynamics of the two-dimensional XY model is investigated with Hamiltonian equations of motion. Critical relaxation starting from both ordered and random states is carefully analyzed, and the short- time dynamic scaling behavior is revealed. Logarithmic corrections to scaling are detected for relaxation with a random initial state, while power-law corrections to scaling are observed for relaxation with an ordered initial state. The static exponent η and dynamic exponent z are determined around and below the Kosterlitz-Thouless phase transition temperature. Our results show that the deterministic dynamics described by Hamiltonian equations is in the same universality class as the stochastic dynamics described by Monte Carlo algorithms and Langevin equations.
A critical point is associated with a diverging relaxation time which always makes the dynamics across a critical point fascinating. For example, when the temperature of a classical system undergoing a classical phase transition (CPT) at a finite temperature T c is suddenly changed from a value higher than the critical temperature to a lower value across the critical point, the system does not equilibrate instantaneously. Domains of ordered regions are formed which grow following a phase ordering dynamics which leads to a dynamical scaling [104]. In this section, we will discuss the recent studies of non-equilibrium dynamics of quantum systems driven across QCPs where the dynamics is unitary unlike the dynamics across a finite temperature critical point. The non-equilibrium dynamics of a transverse XY spin chain was first investigated in a series of papers [48, 46, 47] where the time evolution of the model was studied in the presence of various time-dependent magnetic fields and the nonergodic behavior of the magnetization was pointed out. A similar result was also obtained in [505]. There is a recent upsurge in studies of non-equilibrium dynamics of a quantum system swept across a QCP. These studies are important for exploring the universality associated with quantum critical dynamics. Moreover, recent experiments with ultracold atomic gases [343, 663, 478, 87] have stimulated numerous theoretical studies. The main properties of these atomic gases are low dissipation rates and phase coherence over a long time so that the dynamics is well described by the usual quantum evolution of a closed system. In the subsequent sections, we shall discuss that when a quantum system initially prepared in its ground state is driven across a QCP, the dynamics fails to be adiabatic however slow the rate of change in the parameters of the Hamiltonian may be. This is due to the divergence of the characteristic time scale of the quantum system, namely, the relaxation time close to the QCP. This non-adiabaticity results in the occurrence of defects in the final state of the quantum Hamiltonian.
We have analyzed the dynamic phase transition of the kinetic Ising model in mean-field approximation by means of an analytical approach. Specifically, we study the evolution of the system under the simultaneous influence of time-dependent and time-independent magnetic fields. We demonstrate that within the approximate analytical treatment of our approach, the dynamic phase transition exhibits power-law dependencies for the order parameter that have the same critical exponents as the mean-field equilibrium case. Moreover we have obtained an equation of state, with which we can prove that the time-independent field component is effectively the conjugate field of the order parameter. Our analysis is limited to the parameter range, in which only second-order phase transitions occur, i.e., for small applied field amplitudes and temperatures close to the Curie point. In order to ensure the reliability of our analytical results we have corroborated them by comparison to numerical evaluations of the same model.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to 1922, for several temperatures below, at, and above TKT. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function Sxx also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in Sxx, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}TKT, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.
We obtain the steady-state phase diagram of a transverse-field XY spin chain coupled at its ends to magnetic reservoirs held at different magnetic potentials. In the long-time limit, the magnetization bias across the system generates a current-carrying nonequilibrium steady state. We characterize the different nonequilibrium phases as functions of the chain's parameters and magnetic potentials, in terms of their correlation functions and entanglement content. The mixed-order transition, previously observed for the case of a transverse-field Ising chain, is established to emerge as a generic feature of a wider class of out-of-equilibrium problems. The critical exponents associated with this universality class are determined analytically. Results are also contrasted with those obtained in the limit of Markovian reservoirs. Our findings should prove helpful in establishing the properties of nonequilibrium phases and phase transitions of extended open quantum systems.
We investigate the dynamical phases and phase transitions arising in a classical two-dimensional anisotropic XY model under the influence of a periodically driven temporal external magnetic field in the form of a symmetric square wave. We use a combination of finite temperature classical Monte Carlo simulation, implemented within a CPU+GPU paradigm, utilizing local dynamics provided by the Glauber algorithm and a phenomenological equation-of-motion approach based on relaxational dynamics governed by the time-dependent free energy within a mean-field approximation to study the model. We investigate several parameter regimes of the variables (magnetic field, anisotropy, and the external drive frequency) that influence the anisotropic XY system. We identify four possible dynamical phases: Ising-SBO, Ising-SRO, XY-SBO, and XY-SRO. Both techniques indicate that only three of them (Ising-SRO, Ising-SBO, and XY-SRO) are stable dynamical phases in the thermodynamic sense. Within the Monte Carlo framework, a finite-size scaling analysis, shows that XY-SBO does not survive in the thermodynamic limit giving way to either an Ising-SBO or a XY-SRO regime. The finite-size scaling analysis further shows that the transitions between the three remaining dynamical phases either belong to the two-dimensional Ising universality class or are first-order in nature. Within the mean-field calculations yield three stable dynamical phases, i.e., Ising-SRO, Ising-SBO and XY-SRO, where the final steady state is independent of the initial condition chosen to evolve the equations of motion, as well as a region of bistability where the system flows to either Ising-SBO or XY-SRO (Ising-SRO) depending on the initial condition. Unlike the stable dynamical phases, the XY-SBO represents a transient feature that is eventually lost to either Ising-SBO or XY-SRO. Our mean-field analysis highlights the importance of the competition between switching of the stationary point(s) of the free energy after each half cycle of the external field and the two-dimensional nature of the phase space for the equations of motion.