Monte Carlo Method

As we have seen, to study the statistical mechanical behavior of the Ising model, we must be able to measure observables, such as $m$ as a function of temperature (i.e. $\beta$). That requires carrying out the sum over all configurations, which, as we have seen, for any modest-sized system looks impossible. Fortunately, Monte Carlo methods come to our rescue.

The aim of the Monte Carlo method is to generate a random sequence (Markov chain) of configurations of spins $\{s\}_i$ so that the probability of encountering the configuration $\{s\}$ is given by $P(\{s\})$ in Eq (1) above. If we do that, the mean value of any observable is simply its average value on the Markov chain. That is for $N$ configurations in the random sequence we have

$$\langle {\cal O} \rangle = \sum_{i=1}^N {\cal O}_i,$$ (2)

where ${\cal O}_i$ is the value of the observable on the configuration $\{s\}_i$.

How do we get such a random sequence? There are two methods in common use: ``heatbath'' and ``Metropolis''. Both methods sweep through the lattice in a systematic way, updating the spin on each site once. At the end of the sweep all sites are updated and we have a new random configuration. The process is repeated, generating a new random configuration after each complete sweep. So the key question is how to update the spin on a given site to get the desired overall probability. This is done by examining how the orientation of the single spin we are updating affects the overall probability, assuming all the other spins are kept constant. A single spin interacts with all of its neighbors and with the external magnetic field. Suppose the sum of the four neighbor spins is $S_n$. Then the single spin $s$ contributes a factor

$$f(s) = e^{-\beta(S_n+h)s}$$

to the total probability $P(\{s\})$. Both updating methods are designed to reset the spin $s$ at random with probability proportional to this factor.

The heatbath method simply resets the spin to $+1$ with probability $f(+1)/[f(+1) + f(-1)]$ and to $-1$ otherwise. One can imagine that the single spin is being placed in contact with a heatbath with inverse temperature $\beta$ and allowed to equilibrate according to the laws of Boltzmann statistics, while holding all the other spins constant.

The Metropolis method is only slightly more involved. In this case we are asked whether to flip the spin or leave it the same. The decision is based on the ratio of $P(\{s\})$ after and before flipping the spin we are updating, while keeping all the other spins untouched. This ratio is simply $f(-s)/f(s)$. If the ratio is greater than one, flipping the spin increases $P(\{s\})$, and the Metropolis method instructs us to flip in that case. If the ratio is less than one, we are decreasing $P(\{s\})$. But we allow that to happen at random occasionally. To be precise, we allow the flip to happen with a probability equal to the ratio $f(-s)/f(s)$. That policy has the consequence of frequently allowing small decreases in $P(\{s\})$ but only infrequently allowing large decreases. Notice that the ratio $f(-s)/f(s)$ depends on the temperature. As the temperature is decreased to zero, i.e. $\beta$ gets large, it becomes less and less likely to accept a flip that decreases $P(\{s\})$.

It can be shown that both methods lead to the desired result, namely, that the Markov chain converges to a sequence in which the probability of encountering the spin configuration $\{s\}$ is given by $P(\{s\})$.