So, I'm trying to simulate the Ising Model on a 2D square lattice of spins. When exploring the auto correlation of the magnetisation: Where the auto covariance: $$A(T) = \langle(M(t)\ - \langle M\rangle)(M(t + T_L)\ - \langle M\rangle)\rangle$$ The auto correlation is then $A(T)/A(0)$. $\langle \quad\rangle$ Represents averaging over a long time. Defining a decorrelation time, $\tau_e$, to be the lag time, $T_L$, required for the auto correlation to fall below $e^{-1}$.

Plotting this out I get, in simulation units, for a 5x5 lattice:

A large range of temperatures

and, for far few temperatures around $T_C$:

Around the critical temperature

I have some ideas of the features of this plot.

  • Firstly, closer to, but below Onsager's analytic value for the critical temperature, $T_C$ (around 2.0 - 2.2), there's long range magnetic order in the system, the lines are roughly straight. This makes sense, for a roughly constant magnetisation, with small thermal fluctuations, you'd expect a straight line $a(t) = 1$. The reason the line slopes is because the time averaging was done over a finite time, and because there's still some thermal activity in the system. So, as temperature increases the gradient of these lines become more negative. (Is there a way of "pulling apart" the value of the slope into these contributions?)

  • Beyond the critical temperature the system is disordered, so the auto correlation is no longer a straight line. (Looks exponential-ish?). The higher the temperature, the faster the decline.

  • Something that confuses me a bit is that region of colder temperatures that seem to just drop to $0$ auto correlation almost instantly, below about T = 1.8. My reasoning is that there are so few thermal fluctuations that $M - <M>$ is very often $0$. I believe this is just a product of a finite length simulation. Is there anything physical going on here? The line at T = 1.48 is just the first time my program plotted a line that didn't go below $e^{-1}$ in the range of $T_L$ in an attempt to "catch" this transition; I don't think it's ended up being a physically meaningful thing. There seems to be a fair bit of randomness as to whether a low temperature line behaves like this.

Am I thinking on the right lines here or am I totally interpreting this wrong? Or is my simulation just busted?

Thanks for any assistance. :)

EDIT: After some more digging, the temperature range of the low $\tau_e$ zone below $T_C$ seems to depend on the size of the lattice. I'm quite stumped:

Te Vs N

EDIT 2: After reading this question I think I have a better understanding: The correlation I'm measuring is actually the correlation between fluctuations rather than the magnetisation itself. In that case, it makes sense as to why this would decay exponentially both above and below $T_C$. I've encountered the idea of "Critical Slowing Down" around $T_C$ which may explain the large $\tau_e$ values there. If this is the right way to be thinking about this, I still have two questions:

  • Why does critical slowing down occur near $T_C$ - I've heard that the correlation length (and presumably the correlation time) diverges there, but have yet to find anything to explain why. If I am indeed measuring the correlation between fluctuations, why would this persist especially long near $T_C$?

  • Why does the point at which $\tau_e$ suddenly diverge seem to decrease in temperature with decreasing lattice size, despite the fact that $T_C$ increases for smaller finite lattices?

My current understanding


For an infinite lattice:

  • $0 < T < T_C$: All spins are under a strong pressure to align with neighbours, thus any fluctuation is quickly neutralized without much chance to affect the future evolution of the system.

  • $T >> T_C$: The system is fluctuation dominated: There is lack of alignment between local spins, so any given configuration of the system at one point in time is less likely to have influence on its configuration at another point in time. I.e. the system is dominated by a stochastic process and hence less deterministic.

  • $T = T_C$: The above two effects exactly balance: The correlation length of the system diverges, clusters of all sizes form with equal probability. Any fluctuation in the system will then influence infinity many other spins and necessarily have some effect on all future states. Hence $\tau_e$ also diverges.

What we see in the above figures is a similar effect: However, as the finite lattice size becomes smaller lower temperature fluctuations have a greater chance of affecting the future evolution of the system, because a single spin constitutes a greater proportion of all spins in the system. $\tau_e$ also never truly diverges, due to the finite nature of the lattice. As the lattice size increases, the position of these "spikes" in $\tau_e$ will move asymptotically closer to $T_C$ for an infinite lattice. Computationally, this means that the time to bring a system to equilibrium, as well as the time needed to elapse to get an averaged value of a given accuracy, increases dramatically around these "spikes".

  • 2
    $\begingroup$ To answer your first question. If you are simulating the system with single spin-flip algorithms like the Glauber dynamics, then near $T_c$, these single flips are not sufficiently correlated to account for the divergence in correlation length. Cluster algorithms, like the SW algorithm, address this. $\endgroup$
    – PeaBrane
    Apr 7 '21 at 16:38
  • $\begingroup$ Exponential decay is better seen in a log-plot (as a straight line), wheres the power law dynamics is a straight line in a log-log graph. Then finding the decay time sis a matter fo doing simple linear regression. $\endgroup$ Apr 11 '21 at 17:33
  • $\begingroup$ Maybe this reference is interesting. They study what happens if you change the temperature across the critical point in finite system sizes. journals.aps.org/prb/abstract/10.1103/PhysRevB.89.054307 In essence I think that you are on the right track. The fact that correlation length diverges is because the whole system needs to get correlated at the critical point such that it can 'choose a polarization' together. But the correlation time diverges as well, leading to critical slowing down, (also known as the Kibble-Zurek effect). $\endgroup$
    – Wouter
    Apr 16 '21 at 11:09
  • $\begingroup$ In principle, phase transitions are only defined for infinite systems (the 'thermodynamic limit'). So studying finite systems only works in the sense that you extrapolate from them. Note that because the correlation length diverges, very close to the critical point, it will get larger than the system size and in this size-dependent temperature window around the critical point, large finite-size effects can be expected $\endgroup$
    – Wouter
    Apr 16 '21 at 11:12
  • $\begingroup$ @PeaBrane actually, the single spin-flips are more experimentally relevant than cluster algorithms $\endgroup$
    – Wouter
    Apr 16 '21 at 11:15

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