2D Ising model

The Ising model has been solved exactly in one and two dimensions Onsager s solution of the model in two dimensions is only at zero field. Infomiation about the Ising model in tliree dunensions comes from high- and low-temperature expansions pioneered by Domb and Sykes [104] and others. We will discuss tire solution to the 1D Ising model in the presence of a magnetic field and the results of the solution to the 2D Ising model at zero field. 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

Figure 1. Flipping a cluster in a simulation of the 2D Ising model. The solid and open circles represent the up- and down-pointing spins in the model.

In Figure 4 we have plotted on logarithmic scales the correlation time of the Wolff algorithm for the 2D Ising model at the critical temperature, over a range of different system sizes. The slope of the line gives us an estimate of the dynamic exponent. Our best fit, given the errors on the data points is... 

Figure 4. The correlation time for the 2D Ising model simulated using the Wolff algorithm. The measurements deviate from a straight line for small system sizes L, but a fit to the larger sizes, indicated by the dashed line, gives a reasonable figure of z = 0.25 0.02 for the dynamic exponent of the algorithm.

In fact, in studies of the Wolff algorithm for the 2D Ising model, one does not usually bother to make use of Eq. (2.5) to calculate r. If we measure time in Monte Carlo steps (i.e., simple cluster flips), we can define the corresponding dynamic exponent zsteps in terms of the correlation time rsteps of Eq. (2.5) thus ... 

Kornyshevand Vdfan [272-274] suggested the simplest way of describing the transitions between 1x1 and 1x2 states, based on an anisotropic 2D Ising model, and derived a closed expression for the temperature-surface charge phase diagram. 

Fig. 11. Temperature dependence of the sublattice magnetization for ErBa2Cu307. The solid curve is the fit to the data of Onsager s exact solution for the 5= 5, 2D Ising model (Lynn et al. 1989).

Fig. 7. Comparison of a specific heat anomaly calculated for Dyl237 with pure dipolar interaction and one calculated with a anisotropic exchange interaction (for both calculations a square of lOx lo magnetic ions was used). Since the dipolar interaction and the 2D Ising model belong to the same universality class it is not swprising to find exchange values which exactly reproduce the dipolar data.

Phase transitions of confined fluids were extensively studied by various theoretical approaches and by computer simulations (see Refs. [28, 278] for review). The modification of the fluid phase diagrams in confinement was extensively studied theoretically for two main classes of porous media single pores (stit-Uke and cylindrical) and disordered porous systems. In a slit-like pore, there are true phase transitions that assume coexistence of infinite phases. Accordingly, the liquid-vapor critical point is a true critical point, which belongs to the universality class of 2D Ising model. Asymptotically close to the pore critical point, the coexistence curve in slit pore is characterized by the critical exponent of the order parameter = 0.125. The crossover from 3D critical behavior at low temperature to the 2D critical behavior near the critical point occurs when the 3D correlation length becomes comparable with the pore width i/p. 

In order to get a sense of the large numbers, consider the 2D Ising model of locally interacting spins on a... 

---