To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
We present a comprehensive comparison of spin and energy dynamics in quantum and classical spin models on different geometries, ranging from one-dimensional chains, over quasi-one-dimensional ladders, to two-dimensional square lattices. Focusing on dynamics at formally infinite temperature, we particularly consider the autocorrelation functions of local densities, where the time evolution is governed either by the linear Schr\"odinger equation in the quantum case or the nonlinear Hamiltonian equations of motion in the case of classical mechanics. While, in full generality, a quantitative agreement between quantum and classical dynamics can therefore not be expected, our large-scale numerical results for spin-
1/2 systems with up to
N=36 lattice sites in fact defy this expectation. Specifically, we observe a remarkably good agreement for all geometries, which is best for the nonintegrable quantum models in quasi-one or two dimensions, but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Our findings indicate that classical or semiclassical simulations provide a meaningful strategy to analyze the dynamics of quantum many-body models, even in cases where the spin quantum number
S=1/2 is small and far away from the classical limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to
1922, for several temperatures below, at, and above
TKT. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function
Sxx also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in
Sxx, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}
TKT, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.
A Hamiltonian dynamics is defined for the
XY model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte Carlo results in the explored temperature region. The behavior of the magnetization and energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature ${T}_{\mathrm{KT}}\ensuremath{\approx}0.855.$
Using tensor network methods, we simulate the real-time evolution of the lattice Thirring model quenched out of equilibrium in both the critical and massive phases and study the appearance of dynamical quantum phase transitions, as nonanalyticities in the Loschmidt rate. Although the presence of a dynamical quantum phase transition in the model does not correspond to quenches across the critical line of the equilibrium phase diagram at zero temperature, we identify a threshold in the energy density of the initial state, necessary for a dynamical quantum phase transition to be present. Moreover, in the case of the gapped quench Hamiltonian, we unveil a connection of this threshold to a transition between different regions in the finite-temperature phase diagram.