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Journal of Physics A Mathematical and General

Dynamical phase diagram of the two-dimensional p-state clock model

Y. Leroyer, K. Rouidi

Published Apr 21, 1991• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/24/8/031

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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.
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 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.
The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
We consider a two-dimensional antiferromagnet Ising system interacting with a heat bath at temperature T. The dynamics of the system is simulated by two competing stochastic processes: the two-spin-exchange Kawasaki kinetics at temperature T>0 and the one-spin-flip Glauber dynamics at T(G)-->0(-), which mimics the increase of the energy of the system. These two processes have probabilities 1-p and p, respectively. Monte Carlo simulations were employed to determine the phase diagram for the stationary states of the model and the corresponding critical exponents. Contrary to the ferromagnetic case, the phase diagram obtained does not exhibit the phenomenon of self-organization: for any nonzero value of the competing parameter p, and for any value of T, the only stationary phase which remains is the ferromagnetic one. At the phase transition between the antiferromagnetic and paramagnetic phases, at p=0, the values found for the critical exponents agree with those of the corresponding equilibrium Ising model.
We study the finite-temperature transition to the 1/2 magnetization plateau in a model of interacting S = 1/2 spins with longer range interactions and strong exchange anisotropy on the geometrically frustrated Shastry-Sutherland lattice. In previous studies, it was obtained from Monte Carlo calculations that the transition to the plateau state occurs via two successive transitions with the two-dimensional Ising universality class, when the quantum exchange interactions are finite, while a single phase transition takes place in the purely Ising limit[l]. To understand these behaviors, we introduce the generalized four-state chiral clock model and perform Monte Carlo calculations for this model. By comparing the phase diagrams of the two models, we find that the topology of the thermal phase diagram is the same each other - the criticality of the thermal phase transition in the original model can be well explained by that of the generalized four-state chiral clock model.
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.