# I'm getting weird autocorrelations when simulating an Ising model below the critical temperature

So I'm simulating an Ising model using Monte Carlo and the Metropolis algorithm. After letting it reach equilibrium, I try to calculate the autocorrelation of the magnetization. As long as the system is above the critical temperature (about 2.4), I get the expect results. But when it's below the critical point, I get a weird autocorrelation result:

That straight line is completely bizarre. Now, at this point I am below the critical temperature, so it's supposed to be different anyway, but I'm not sure. It doesn't feel right.

Is this result expected?

• I don't know anything about the subject matter or the methods but this is usually a valid question to pose when facing problems like this: are your assumptions/techniques valid below the critical temperature? It's critical for a reason, something must change there, so does your simulation, numerical method or governing equation assume something that is no longer true below that point? Or the post-processor that computes the autocorrelation even? Just a thought that might help you diagnose the problem, if there is a problem. – tpg2114 May 12 '14 at 19:34
• Alternatively, do you need to assume something in addition to what is already assumed in the models below the critical temperature? Maybe there is an additional constraint or something that needs to be considered. – tpg2114 May 12 '14 at 19:34

• I think this is more or less right. This is why, in general, you end up measuring the correlations of fluctuations around the mean, $\langle \delta m(t) \delta m(0) \rangle$ where $\delta m = m - \langle m \rangle$. – AJK May 13 '14 at 5:32