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arXiv:0707.1959 · Journal of Statistical Mechanics Theory and Experiment

Logarithmic corrections in the ageing of the fully frustrated Ising model

Jean‐Charles Walter, Christophe Chatelain

Published Jul 4, 2008• 6 citations
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Abstract

We study the dynamics of the critical two-dimensional fully frustrated Ising model by means of Monte Carlo simulations. The dynamical exponent is estimated at equilibrium and is shown to be compatible with the value z c = 2. In a second step, the system is prepared in the paramagnetic phase and then quenched at its critical temperature T c = 0. Numerical evidence for the existence of logarithmic corrections in the ageing regime is presented. These corrections may be related to the topological defects observed in other fully frustrated models. The autocorrelation exponent is estimated to be λ = d as for the Ising chain quenched at T c = 0.

DOI
10.1088/1742-5468/2008/07/p07005

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We study the out-of-equilibrium dynamics of the fully frustrated XY model. At equilibrium, this model undergoes two phase transitions at two very close temperatures: a Kosterlitz–Thouless topological transition and a second-order phase transition between a paramagnetic phase and a low-temperature phase where the chiralities of the lattice plaquettes are antiferromagnetically ordered. We compute by means of Monte Carlo simulations two-time spin–spin and chirality–chirality autocorrelation and response functions. From the dynamics of the spin waves in the low-temperature phase, we extract the temperature-dependent exponent η. We provide evidence for logarithmic corrections above the Kosterlitz–Thouless temperature and interpret them as a manifestation of free topological defects. Our estimates of the autocorrelation exponent and the fluctuation-dissipation ratio differ from the XY values, while η( T KT ) lies at the boundary of the error bar. Indications for logarithmic corrections at the second-order critical temperature are presented. However, the coupling between angles and chiralities is still strong and explains why the autocorrelation exponent and fluctuation-dissipation ratio are far from the Ising values and seem stable.
With a large-scale Monte Carlo simulation, the aging of the two-dimensional dynamic Ising and XY models is numerically studied. Starting from a disordered state, the systems are quenched to critical temperature Tc and then evolve according to dynamics of model A. By calculating the autocorrelation function A(t,t′) we observe the aging phenomenon for both spin systems. Then, relatively accurate values of the dynamic critical exponent λc and z are estimated. Especially, we confirm that for the two-dimensional XY model, there is a strong logarithmic correction to scaling in the dynamic relaxation starting from a completely disordered.
We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength h = HTk B , as well as the Villain model on the square lattice. After a quench from a disordered initial state to T = 0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, L , we define and measure sample-dependent ‘first passage time’, t r , which is the number of Monte Carlo steps until the energy is relaxed to the ground state value. The distribution of t r , in particular its mean value, ⟨ t r ( L )⟩, is shown to obey the scaling relation, ⟨ t r ( L )⟩∼ L 2 ln( L / L 0 ), for both models. Scaling of the autocorrelation function of the antiferromagnetic triangular model is shown to involve logarithmic corrections, both at H = 0 and at the field-induced Kosterlitz–Thouless transition: however, the autocorrelation exponent is found to be H -dependent.
We study the dynamical scaling properties of the two-dimensional fully frustrated Ising model by simulated annealing. The model has a highly degenerate, critical ground state with well known equilibrium properties. To study its out-of-equilibrium behavior, we develop an annealing protocol that is convenient for extending the Kibble-Zurek mechanism to the case of an exponentially divergent correlation length in the limit of the temperature T→Tc=0T \to T_c = 0T→Tc​=0. We find excellent agreement with the scaling forms derived, including a dependence on the initial temperature that is not present in the conventional Kibble-Zurek mechanism. The dynamic exponent zzz is consistent among all quantities studied; fits of its dependence on the initial temperature give z=1.99(2)z = 1.99(2)z=1.99(2) (susceptibility), z=2.072(8)z = 2.072(8)z=2.072(8) (excess energy), and z=2.09(3)z = 2.09(3)z=2.09(3) (Edwards-Anderson order parameter). These values are close to z=2z=2z=2, which is consistent with previous studies of relaxation after instantaneous quenches and of time dependent correlations in equilibrium, including the estimate z=2.004(9)z = 2.004(9)z=2.004(9) from equilibrium autocorrelation functions.
With dynamic Monte Carlo simulations,aging phenomena of two-dimensional Ising model and dynamic XY model were studied numerically.Systems were quenched to their critical point from a disordered initial state.The autocorrelation function was calculated by heat-bath algorithm.Critical exponents were also calculated.It was confirmed that for the 2D XY model,there is a logarithmic correction to scaling.The universality proposal and their scale relation in aging phenomena were predicted.The scale relation agreed with renormalization group calculations.