In recent years, static and dynamic properties of non-180^{∘} domain walls in magnetic materials have attracted a great deal of interest. In this paper, spin-reorientation critical dynamics in the two-dimensional XY model is investigated with Monte Carlo simulations and theoretical analyses based on the Langevin equation. At the Kosterlitz-Thouless phase transition, the dynamic scaling behaviors of the magnetization and the two-time correlation function are carefully analyzed, and critical exponents are accurately determined. When the initial value of the angle between adjacent domains is slightly lower than π, a critical exponent is introduced to characterize the abnormal power-law increase of the magnetization in the horizontal direction inside the domain interface, which is measured to be ψ=0.0568(8). In addition, the relation ψ=η/2z is analytically deduced from the Langevin dynamics in the long-wavelength approximation, well consistent with numerical results.
Non-equilibrium critical dynamics of the two-dimensional XY model is investigated with Hamiltonian equations of motion. Critical relaxation starting from both ordered and random states is carefully analyzed, and the short- time dynamic scaling behavior is revealed. Logarithmic corrections to scaling are detected for relaxation with a random initial state, while power-law corrections to scaling are observed for relaxation with an ordered initial state. The static exponent η and dynamic exponent z are determined around and below the Kosterlitz-Thouless phase transition temperature. Our results show that the deterministic dynamics described by Hamiltonian equations is in the same universality class as the stochastic dynamics described by Monte Carlo algorithms and Langevin equations.
With a large-scale Monte Carlo simulation, non-equilibrium dynamics of the two-dimensional fully frustrated XY model is investigated. We tackle the Kosterlitz-Thouless phase transition. Starting from an ordered initial state, we study the dynamic evolution of the magnetization as well as a specifically defined Binder cumulant. From the dynamic scaling ansatz, we extract the correlating time of the dynamics and the spatial correlation length of the equilibrium state. The dynamic exponent z is determined with relatively high accuracy. Especially, we suggest and demonstrate how one may directly measure the dynamic exponent z above TKT from the scaling fit of the Binder cumulant. These results indicate that the dynamic exponent z fluctuates around z=2, and this is consistent with that observed at temperatures below the transition temperature TKT.
We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat
Cv for the XY model. The simulations are done on
642,
1282,
2562, and
5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate,
Tc=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
Using Monte Carlo and spin-dynamics methods, we have investigated the dynamic behavior of the classical, antiferromagnetic XY model on a triangular lattice with linear sizes
L< 300. The temporal evolutions of spin configurations were obtained by solving numerically the coupled equations of motion for each spin using fourth-order Suzuki-Trotter decompositions of exponential operators. From space- and time-displaced spin-spin correlation functions and their space-time Fourier transforms we obtained the dynamic structure factor
S(q,w) for momentum
q and frequency $\ensuremath{\omega}.$ Below
TKT (Kosterlitz-Thouless transition), both the in-plane
(Sxx) and out-of-plane
(Szz) components of $S(\mathbf{q},\ensuremath{\omega})$ exhibit very strong and sharp spin-wave peaks. Well above
TKT, Sxx and
Szz apparently display a central peak, and spin-wave signatures are still seen in
Szz. In addition, we also observed an almost dispersionless domain-wall peak at high $\ensuremath{\omega}$ below
Tc (Ising transition), where long-range order appears in the staggered chirality. Above
Tc, the domain-wall peak disappears for all q. The line shape of these peaks is captured reasonably well by a Lorentzian form. Using a dynamic finite-size scaling theory, we determined the dynamic critical exponent
z=1.002(3). We found that our results demonstrate the consistency of the dynamic finite-size scaling theory for the characteristic frequency ${\ensuremath{\omega}}_{m}$ and the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$ itself.