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When I run my implementation of the Wolff algorithm on the square Ising model at the theoretical critical temperature I get subcritical behaviour. The lattice primarily just oscillates between mostly positive and mostly negative states. I find that I need to increase the temperature to get behaviour that looks critical.

At first I thought it was a bug in the program, but the behaviour actually makes sense. On an infinite lattice the Wolff algorithm should produce clusters of all sizes at $T_C$. This means that most of the clusters it tries to produce are larger than the lattice used in the simulation, and most clusters end up reaching the boundaries and filling most of the lattice. It also goes back to the point Kadanoff always made that true criticality is only possible in infinite systems.

I find that I do get critical looking behaviour at slightly higher temperatures than theory predicts. The required temperature increases with decreasing lattice size.

Is there any literature on this effect?

How do people compensate for it in practice?

Is there a formula for the temperature adjustments for different lattice sizes?

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    $\begingroup$ This is a well-understood finite-size effect, resulting from the fact that the correlation length in your system cannot become larger than the system size. It implies a shift of the (apparent) critical point of order $1/n$, if you're on an $n\times n$ torus. AFAIK, this was first studied by Ferdinand and Fisher in 1969 (Phys. Rev. 185, 832). $\endgroup$ – Yvan Velenik Aug 13 '15 at 7:50
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    $\begingroup$ @YvanVelenik: If you could elaborate a bit on that, that would make a good answer! $\endgroup$ – ACuriousMind Aug 13 '15 at 13:06
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    $\begingroup$ @ACuriousMind: Done (but it is a bit sketchy, as I don't have so much time now; in particular, it would be nice to provide a picture of the specific heat for the 2d Ising model on a finite torus, as well as the limit quantity). $\endgroup$ – Yvan Velenik Aug 13 '15 at 15:12
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The first thing to realize is that there are no "true" phase transitions (in the sense of non-analytic behaviour of thermodynamic potentials) in finite systems. This is the main difficulty one faces when analysing phase transitions using (most) computer simulation schemes.

In particular, such simulations are only reliable as long as the observed correlation length is significantly smaller than the system's linear size. However, when there is a second-order phase transition, the correlation length diverges at the critical point, which implies that close to the "true" critical temperature, the behavior observed in a finite system will be smoothed out (and, it turns out, the natural finite-volume analogue of the critical point is shifted, see below).

Now, of course, a large enough finite system will still display a behaviour that "resembles" a phase transition, but with its singularities smoothed out. To extrapolate results to infinite systems then requires (i) the determination of finite-volume analogues of the limiting quantities (in particular the critical temperature), (ii) examining how these finite-volume quantities change when the system's size is increased. In order to help with this extrapolation procedure, physicists have devised various finite-size scaling theories.

I assume that you are working on a torus (i.e., with periodic boundary conditions) of linear size $L$. This is the simplest case, as far as finite-size effects are concerned, since one then avoids the additional difficulties related to the presence of the system's boundary.

The first detailed finite-size scaling theory was developed by Ferdinand and Fisher in 1969 in a classical paper published in Phys. Rev. 185, 832. They used the exact results available for the two-dimensional Ising model to analyze finite-size effects on the free energy and specific heat.

The specific heat of a finite-volume Ising model does not diverge. However, it still displays a sharp increase in a narrow region around the "true" critical point $T_c$. Fisher and Ferdinand proposed to define the finite-volume analogue $T_c(L)$ of the critical temperature as the value of the temperature at which the specific heat is maximal. They then argued that $$ T_c - T_c(L) \sim L^{-1/\nu}\,, $$ where $\nu$ is the critical exponent associated to the specific heat, which is given by $\nu=1$ for the two-dimensional model.

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  • $\begingroup$ Thanks. That is exactly what I was looking for. The $T_c$ estimates I got give the proportionality constant $a^*$ almost exactly a factor of 4 from Ferdinand and Fishers -.3603. I was choosing the temperature by looking at the log-log plots of the cluster size distribution and picking those which gave the straightest line the possible sizes. Not very rigorous, but I get 3 significant figure agreement with their formula for lattice sizes 64, 127, 512 after the factor 4 adjustment. I wonder why the factor of 4. $\endgroup$ – Daniel Mahler Aug 13 '15 at 22:35

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