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Progress of Theoretical Physics

Dynamics of the Ising Model near the Transition Point

Ryuzo Abe

Published Apr 1, 1968• 35 citations
Publisher

Abstract

The Ising spin system is discussed in which the interaction of spins with heat bath causes the spontaneous flip of spins. It is assumed that the system is in thermal equilibrium up to some instant under a uniform and sufficiently small magnetic field and the magnetic field is switched off at that instant. The initial decay of magnetization in such a situation is studied. In the neighborhood of transition point, it is proved that a relaxation time associated with the initial decay is proportional to the susceptibility, and an exact proportionality constant is calculated for the two-dimensional square lattice with nearest-neighbor interaction.

DOI
10.1143/ptp.39.947

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The Ising spin system is discussed in which the interaction of spins with heat bath causes the spontaneous flip of spins. It is assumed that the system is in thermal equilibrium up to some instant under a uniform and sufficiently small magnetic field and the magnetic field is switched off at that instant. The initial decay of magnetization in such a situation is studied. In the neighborhood of transition point, it is proved that a relaxation time associated with the initial decay is proportional to the susceptibility, and an exact proportionality constant is calculated for the two-dimensional square lattice with nearest-neighbor interaction.
The Isin spin system is discussed in which the interaction of spins with a heat bath causes the spontaneous flip of spins. A treatment discussed in a previous paper for the square lattice is extended to the honeycomb and triangular lattices. In the case of honeycomb lattice, an additional term to the one studied in the previous paper is calculated explicitly at the transition point and an exact expansion is derived for the dynamic susceptibililty χ(ω) up to the term of the order of 1/ω2. Some discussions are given of the low frequency behavior of χ(ω) on the basis of variational principle.
Time-dependent statistics of the Ising model proposed by Glauber for the one-dimensional chain are extended to the example of a two-dimensional square lattice. Each spin is assumed to change its state through the interaction with a heat bath. The equations of motion for both the single spin and the spin correlation functions are solved approximately by using a decoupling procedure where the many-body correlation functions are taken as sums of products of pair correlation functions. As a special case, our theory allows the approximate calculation of the equilibrium properties of the system and it turns out that, in this case, our result is an improvement over the Bethe approximation. Both the frequency-dependent magnetic susceptibility and the decay of the magnetic moment to the equilibrium state are calculated above and below the Curie temperature. The fluctuation–dissipation theorem developed by Glauber for the linear chain is shown to hold in the two-dimensional case also.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.
We consider nonequilibrium critical dynamics of the two-dimensional Ising model for which the initial state is prepared by switching on random fields with zero mean and variance H . In the initial state there is no magnetic order but the clusters of parallel spins have a percolation transition for small enough H . Using heat-bath dynamics we measure the relaxation of the magnetization which shows a re-entrance in time. Due to cluster dissolution in the early time regime there is a decrease of the magnetization, followed by an increase due to nonequilibrium domain growth which itself turns to a decrease due to equilibrium relaxation. The power law decay of the nonequilibrium autocorrelation function is not influenced by the percolation in the initial state.
We consider the time evolution of entanglement in a finite two-dimensional transverse Ising model. The model consists of a set of seven localized spin-12\frac{1}{2}21​ particles in a two-dimensional triangular lattice coupled through nearest-neighbor exchange interaction in the presence of an external time-dependent magnetic field. The magnetic field is applied in different function forms: step, exponential, hyperbolic, and periodic. We found that the time evolution of the entanglement shows an ergodic behavior under the effect of the time-dependent magnetic fields. Also, we found that while the step magnetic field causes great disturbance to the system, creating rapid oscillations, the system shows great controllability under the effects of the other magnetic fields where the entanglement profile follows closely the shape of the applied field even with the same frequency for periodic fields. This follow-up trend breaks down as the strength of the field, the transition constant for the exponential and hyperbolic forms, or the frequency for periodic field increase leading to rapid oscillations. We observed that the entanglement is very sensitive to the initial value of the applied periodic field: the smaller the initial value is, the less distorted the entanglement profile is. Furthermore, the effect of thermal fluctuations is very devastating to the entanglement, which decays very rapidly as the temperature increases. Interestingly, although a large value of the magnetic field strength may yield a small entanglement, the magnetic field strength was found to be more persistent against thermal fluctuations than the small field strengths.