Wolff algorithm Ising models

Figure 2. Consecutive states of a 100 x 100 Ising model in equilibrium simulated using the Wolff algorithm. The top row of four states are at a temperature kT = 2.8 J, which is well above the critical temperature the middle row is close to the critical temperature at kT = 2.3 J-, and the bottom row is well below the critical temperature at kT = 1.8 J.

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 ... 

Figure 6. The correlation time T,tepl of the 20 Ising model simulated using the Wolff algorithm, measured in units of Monte Carlo steps (i.e., cluster flips). The fit gives us a value of zstep, = 0.50 0.01 for the corresponding dynamic exponent.

Potts models suffer from critical slowing down in just the same way as the Ising model. All the algorithms discussed in this chapter can be generalized to Potts models. Here we discuss the example of the Wolff algorithm, whose appropriate generalization is as follows ... 

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