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arXiv:2402.07505 · Physical review. E

Dynamical phase transitions in the X Y model: A Monte Carlo and mean-field-theory study

Mainak Pal, William Baez, Pushan Majumdar, Arnab Sen, Trinanjan Datta

Published Nov 12, 2024• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.110.054109

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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.
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We present a comprehensive numerical study of dynamic phase transitions in the two-dimensional kinetic Ising model under a nonantisymmetric time-dependent magnetic field including a sinusoidal term and a second harmonic component. We demonstrate that the expected antisymmetric property and the scaling behavior of the order parameter are maintained using the recently proposed generalized conjugate field approach. Via a detailed finite-size scaling analysis we compute, for zero-bias field, the set of critical exponents suggesting that the Ising universality class is conserved, even in the absence of half-wave antisymmetry in the time-dependent magnetic field. Our results verify up-to-date experimental observations and provide a deeper understanding of nonequilibrium phase transitions, establishing a broader framework for exploring symmetry-breaking phenomena in driven magnetic systems.
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The two-dimensional ferromagnetic Ising model in the presence of a propagating magnetic field wave (with a well-defined frequency and wavelength) was studied by Monte Carlo simulation. This study differs from similar earlier studies, because in earlier studies the oscillating magnetic field was considered to be uniform in space. The time averaged magnetization over a full cycle (the time period) of the propagating magnetic field acted as the dynamic order parameter. The dynamical phase transition was observed. The temperature variation of the dynamic order parameter, the mean square deviation of the dynamic order parameter, the dynamic specific heat and the derivative of the dynamic order parameter were studied. The mean square deviation of the dynamic order parameter and the dynamic specific heat show sharp maxima near the transition point. The derivative of the dynamic order parameter shows a sharp minimum near the transition point. The transition temperature is found to depend also on the speed of propagation of the magnetic field wave.
The 2D XY model is studied by Monte Carlo simulations where two configurations subjected to the same thermal noise are compared. The authors observe three temperature regimes: a high-temperature phase where the two configurations meet, an intermediate phase where they do not meet but their distance becomes independent of the initial distance and a low-temperature phase where the distance remembers for very long the initial distance. The two transition temperatures T 1 =1.61+or-0.01 and T 2 =0.95+or-0.05 are estimated by means of finite-size scaling.
We have studied the nonequilibrium dynamic phase transitions of both three-dimensional (3D) kinetic Ising and Heisenberg spin systems in the presence of a perturbative magnetic field by Monte Carlo simulation. The feature of the phase transition is characterized by studying the distribution of the dynamical order parameter. In the case of anisotropic Ising spin system (ISS), the dynamic transition is discontinuous and continuous under low and high temperatures respectively, which indicates the existence of a tri-critical point (TCP) on the phase \linebreak boundary separating low-temperature order phase and high-temperature disorder phase. The TCP shifts towards the higher temperature region with the decrease of frequency, i.e. T TCP =1.33 × exp(−ω/30.7). In the case of the isotropic Heisenberg spin system (HSS), however, the situation on dynamic phase transition of HSS is quite different from that of ISS in that no stable dynamical phase transition was observed in kinetic HSS after a threshold time. The evolution of magnetization in the HSS driven by a symmetrical external field after a certain duration always tends asymptotically to a disorder state no matter what an initial state the system starts with. The threshold time τ depends upon the amplitude H 0 , reduced temperature T / T C and the frequency ω as τ = C .ω α . H 0 −β .( T / T C ) −γ .