I'm considering the Ising model in a field on a square lattice in $d$ dimensions: $$H = -\sum_{\langle i j \rangle} \sigma_i \sigma_j - h \sum_{i} \sigma_i$$ As usual, $\langle i j\rangle$ refers to nearest neighbor couplings.

When $h>0$ and $0< T < \infty$, it follows that there is a finite magnetization density of $\mathbb{E}[\sigma_i] = m(h, T)>0$ in the thermodynamic limit.

One can imagine this might occur because there are more or larger finite domains of up spins than down spins, and that either of the following situations could arise:

  1. The up spins percolate in the system, so that there is a nonzero probability that a given spin is part of an infinite domain of up spins.
  2. The up spins do not percolate in the system, so that there is no infinite cluster of up spins containing a finite density of spins, but the fact that there are more or larger finite domains of up spins than down spins is sufficient to give a nonzero magnetization.

In the $1$d model, the latter mechanism without an infinite cluster occurs. In the $2$d model, I believe that the former mechanism with an infinite cluster occurs, at least for low temperatures. However, it's not clear to me that this holds at high enough temperatures. I go a little bit back and forth - I'm tempted that high temperatures (say large relative to $T_c$) should have only finite domains of either size, but I feel this would necessitate a phase transition on changing $T$ at nonzero $h$ which should not occur.

In $d > 1$, $h>0$, do the up spins percolate at all $0<T<\infty$? I believe they do at low temperature, but am less sure about high temperature (high relative to $T_c$).


1 Answer 1


Of course, when $h$ is large enough, the $+$ spins will percolate and not the $-$ spins (for instance, one can show that the $+$ spins dominate a very high density Bernoulli site percolation). Therefore, I'll mostly discuss what should happen when $h>0$ is small.

Subcritical temperatures ($\beta>\beta_c$)

In any dimension $d\geq 2$, it is known (see this paper) that the $+$ spins percolate when $h=0$. By monotonicity, this still holds for any $h>0$.

Moderately subcritical temperatures and small $h>0$

  • In any $d\geq 3$, as long as $h>0$ is small enough, there should exist $\tilde\beta>\beta_c$ such that both the $+$ and the $-$ spins should percolate when $\beta\in (\beta_c,\tilde\beta)$. As far as I know, this has only been shown rigorously when $h=0$ and $d\gg 3$, see this paper. However, this should still happen when $h>0$ is sufficiently small, since this would only very slightly change the density of $+$ and $-$ spins.
  • The case $d=2$ is very different, as there can never be coexistence of infinite $+$ and $-$ clusters, see again this paper. In particular, $+$ spins percolate in this case and not $-$ spins.

Very high temperatures ($\beta\ll 1$) and $h>0$ small enough

In this case, the spins are nearly independent, so that the Ising model is a small perturbation of Bernoulli site percolation with parameter very slightly larger than $1/2$ (since $h>0$ is taken very small).

  • When $d=2$, the percolation threshold is approximately $0.59$, so that there should not be percolation of $+$ spins (neither $-$ spins of course).
  • When $d\geq 3$, the percolation threshold is much lower than $1/2$, so that there should be percolation of both $+$ and $-$ spins.

I don't know a reference, but this does not seem very hard to prove (but I may be wrong, of course).

There is another, substantially more relevant, type of percolation that has been investigated: the percolation of Fortuin-Kasteleyn clusters (keywords: random cluster model, FK-percolation; this is the representation on which the Swendsen-Wang algorithm is based). For this notion of percolation, the onset of percolation exactly coincides with the usual phase transition in the Ising model when $h=0$, in any dimension. When $h\neq 0$, there is a nontrivial curve in the $\beta/h$ plane separating the percolative and non-percolative regimes, even though there is no phase transition in this regime (all thermodynamics quantities and all correlation functions are analytic, by the Lee-Yang theorem and related works). This curve is know as the Kertész line. There are quite a few works on that (both rigorous and numerical). You should not have too much trouble finding them using this terminology.

  • $\begingroup$ Thank you, this is an eye-opening answer. I agree that at high $T$ and small $h>0$ in three and more dimensions on the hypercubic lattice, it's similar to site percolation with a probability of up slightly larger than $.5$, and hence both ups and downs have site percolation in this regime at large $T$. I agree that in $2$d on the square lattice that neither need to site percolate at high $T$. I agree it's also tempting that for those $T<T_c$ for which the spontaneous magnetization is not large and $h$ is small that both ups and down will site percolate, again for three or more dimensions. $\endgroup$
    – user196574
    Jul 9, 2023 at 19:42
  • $\begingroup$ I agree that for $T<T_c$ and $h>0$, the ups must site percolate in $d \geq 2$. I'm still trying to fully understand how there can be singular behavior in geometric properties, while correlation functions seem insensitive to such behavior. It's good to see that the Kertész line exists, even if it's characterizing a somewhat different geometric quantity (trying to fully understand this still), where the geometric behavior is singular but the thermodynamic behavior is not. $\endgroup$
    – user196574
    Jul 9, 2023 at 19:47
  • $\begingroup$ Do you think there's any chance there are quantities in terms of the spin variables that can pick up on the non-analytic geometric behavior despite the analyticity of the partition function? For example, quantities like the product of all spins within a radius $R$, and seeing how the average of that quantity shrinks with the growth of $R$? I'm not sure if such quantities that depend on a growing number of spins might be able to get around the analytic partition function and show non-analytic behavior themselves. $\endgroup$
    – user196574
    Jul 9, 2023 at 19:52
  • $\begingroup$ +1, I'm going to wait to accept for a couple days so I have some time to think on this more. I think the discussion of moderately subcritical temperatures might conflict with the onset of site percolation being the Ising transition when $h=0$ or implies a rather complicated phase diagram in the $h,T$ plane. Would you mind drawing the phase diagram you expect? I'll also try to construct a diagram from your answer. $\endgroup$
    – user196574
    Jul 10, 2023 at 1:35
  • $\begingroup$ Here is how I interpret 2d: i.stack.imgur.com/qhnhr.png, and here is how I interpret 3d: i.stack.imgur.com/cRgC3.png. Please let me know if this accurately captures the behaviors mentioned in your post, and I'll mark as accepted $\endgroup$
    – user196574
    Jul 10, 2023 at 2:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.