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Logarithmic corrections in the aging of the fully-frustrated Potts model

Christophe Chatelain

Published Jul 13, 2007• 0 citations
OpenAlex

Abstract

We study the aging of the two-dimensional fully-frustrated Ising and threestate Potts models by means of Monte Carlo simulations. The system is initially prepared at infinite temperature and then quenched either at its critical temperature or in the low-temperature phase for the three-state Potts model. Autocorrelation and response functions are shown to satisfy the scaling behavior conjectured for aging ferromagnets if one assumes the existence of logarithmic corrections. The dynamical and autocorrelation exponents are compatible with � = d and z = 2 for the Ising model and probably the threestate Potts model both at and below Tc. The fluctuation-dissipation ratio goes asymptotically to different non-trivial values at Tc and does not vanish below Tc.

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We study the aging of the two-dimensional fully-frustrated Ising and threestate Potts models by means of Monte Carlo simulations. The system is initially prepared at infinite temperature and then quenched either at its critical temperature or in the low-temperature phase for the three-state Potts model. Autocorrelation and response functions are shown to satisfy the scaling behavior conjectured for aging ferromagnets if one assumes the existence of logarithmic corrections. The dynamical and autocorrelation exponents are compatible with � = d and z = 2 for the Ising model and probably the threestate Potts model both at and below Tc. The fluctuation-dissipation ratio goes asymptotically to different non-trivial values at Tc and does not vanish below Tc.
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.
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 study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We perform an exact analysis of the spherical model in any dimension D >2 and numerical simulations on the two-dimensional Ising model. These systems are shown to be ageing in the long-time regime, i.e. their two-time autocorrelation and response functions, and associated fluctuation-dissipation ratio, are non-trivial scaling functions of both time variables. We show in particular that, for 1<< s (waiting time) << t (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value X ∞ , which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain X ∞ = 1-2/ D for 2< D <4 and X ∞ = ½ for D >4 (mean-field regime). For the two-dimensional Ising model we measure X ∞ ≈0.26±0.01.
We investigate the non-equilibrium critical evolution of statistical systems and describe of its some features, such as aging and violation of the fluctuation-dissipation theorem. We consider some theoretical results of computations for universal quantities that have been obtained in recent years, such as the exponents determining the scaling behavior of dynamic response and correlation functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics. We derive our original Monte Carlo simulation results for 3D pure and diluted Ising models with Glauber and Metropolis dynamics. For these models, we analyze the influence of critical fluctuations, different non-equilibrium initial states, and the presence of nonmagnetic impurities on the two-time dependence of correlation and response functions on characteristic time variables, such as waiting time |twt_{\rm w}tw​| and time of observation |t−twt - t_{\rm w}t−tw​| with |t>twt>t_{\rm w}t>tw​|⁠. We discuss the obtained values of the non-equilibrium exponents for the autocorrelation and response functions and values of the universal long-time limit of the fluctuation-dissipation ratio |X∞X^{\infty }X∞|⁠. Our simulation results demonstrate that the insertion of disorder leads to new values of |X∞X^{\infty }X∞| with |Xdiluted∞>Xpure∞X_{\rm diluted}^{\infty }>X_{\rm pure}^{\infty }Xdiluted∞​>Xpure∞​|⁠.
This work is a contribution to the study of universality in out-of-equilibrium lattice models undergoing a second-order phase transition at equilibrium. The experimental protocol that we have chosen is the following: the system is prepared in its high-temperature phase and then quenched at the critical temperature T c . We investigated by means of Monte Carlo simulations two quantities that are believed to take universal values: the exponent λ/ z obtained from the decay of autocorrelation functions, and the asymptotic value of the fluctuation–dissipation ratio X ( t , s ). This protocol was applied to the Ising model, the 3-state clock model and the 4-state Potts model on square, triangular and honeycomb lattices and to the Ashkin–Teller model at the point belonging at equilibrium to the 3-state Potts model universality class and to a multispin Ising model and the Baxter–Wu model both belonging to the 4-state Potts model universality class at equilibrium.