The Critical Dynamics for the Two-Dimensional Two-Spin Exchange Ising Model

G. A. T. Allan

Hans-Otto Heuer

Wei Zhong, Debabrata Panja, G. T. Barkema

We establish that the Fourier modes of the magnetization serve as the dynamical eigenmodes for the two-dimensional Ising model at the critical temperature with local spin-exchange moves, i.e., Kawasaki dynamics. We obtain the dynamical scaling properties for these modes and use them to calculate the time evolution of two dynamical quantities for the system, namely, the autocorrelation function and the mean-square deviation of the line magnetizations. At intermediate times 1≲t≲L^{z_{c}}, where z_{c}=4-η=15/4 is the dynamical critical exponent of the model, we find that the line magnetization undergoes anomalous diffusion. Following our recent work on anomalous diffusion in spin models, we demonstrate that the generalized Langevin equation with a memory kernel consistently describes the anomalous diffusion, verifying the corresponding fluctuation-dissipation theorem with the calculation of the force autocorrelation function.

Adriana G. Moreira, J. Kamphorst Leal da Silva, F. C. Sá Barreto

One-dimensional kinetic Ising models with long-range exchange of two spins with opposite signs are considered. The exchange probability is a power-law function of the distance between exchanged spins. Rigorous lower bounds for the dynamical exponent z are obtained from the initial response and the scaling hypothesis of the relaxation time. The dependence of the z exponent on the maximal exchange distance and on the exchange probability is studied. Nonuniversal behavior in the ferromagnetic model with two different alternating coupling constants is found. The z exponent is also obtained from the domain-wall argument.

Jean Christian Anglès d’Auriac, R. Maynard, R. Rammal

Ramgopal Agrawal, Leticia F. Cugliandolo, Lara Faoro +2 more

The ±J Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value -J with probability p and +J with probability 1-p. It is especially appealing due to its connection to quantum error correcting codes. Here, we investigate the nonequilibrium critical behavior of the two-dimensional ±J Ising model, after a quench from different initial conditions to a critical point T_{c}(p) on the paramagnetic-ferromagnetic (PF) transition line, especially above, below, and at the multicritical Nishimori point (NP). The dynamical critical exponent z_{c} seems to exhibit nonuniversal behavior for quenches above and below the NP, which is identified as a preasymptotic feature due to the repulsive fixed point at the NP, whereas for a quench directly to the NP, the dynamics reaches the asymptotic regime with z_{c}≃6.02(6). We also consider the geometrical spin clusters (of like spin signs) during the critical dynamics. Each universality class on the PF line is uniquely characterized by the stochastic Loewner evolution with corresponding parameter κ. Moreover, for the critical quenches from the paramagnetic phase, the model, irrespective of the frustration, exhibits an emergent critical percolation topology at the large length scales.