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Journal of Physics A Mathematical and General

Dynamical phase transitions in the two-dimensional XY model

O. Golinelli, Bernard Derrida

Published Oct 7, 1989• 29 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/22/19/004

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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 study the time evolution of the distance between two configurations submitted to the same thermal noise for the 3 d ± J Ising spin glass. We observe three temperature regimes: a high-temperature regime where the distances vanishes in the long-time limit. An intermediate-temperature regime where the distance has a nonzero limit independent of the initial distance. A low-temperature regime where the distance in the long time limit seems to depend upon the initial distance. For the sake of comparison, we have repeated our simulations for the ferromagnetic case.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
The authors have performed a Monte Carlo study of the dynamical phase diagram of the two-dimensional p-state clock models for 4<or=p<or=10, by comparing the time evolution of two configurations subjected to the same thermal noise. For p=4, they find the expected two phase structure, similar to the Ising case, with a transition temperature which coincides with the known static one within the error bars. The dynamical critical exponents are measured. For p>or=5 they observe, between the low-temperature frozen phase and the high-temperature disordered one, two new phases, which are not present for p<or=4. The first is connected to the expected emergence of a spin-wave phase, and the other may be a purely dynamical effect although its connection to the soft phase might suggest the existence of a related equilibrium property. The transition temperatures are determined and are found to satisfy a phenomenological duality-like relation.
We study the time evolution of two configurations submitted to the same thermal noise for several two dimensional models (Ising ferromagnet, symmetric spin glass, non symmetric spin glass). For all these models, we find a non zero critical temperature above which the two configurations always meet. Using finite size scaling ideas, we determine for these three models this dynamical phase transition and some of the critical exponents. For the ferromagnet, the transition Tc ≃ 2.25 coincides with the Curie temperature whereas for the two spin glass models ( ± J distribution of bonds) we obtain Tc ≃ 1.5-1.7.
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.