Introduction

Here we will discuss two famous arguments from statistical physics, that I think can lend a lot of intuition about how to think about what kinds of thermal excitations are and aren’t allowed by geometrical constraints. The two arguments we will discuss are the Peierls argument for why the classical Ising model has a phase transition in dimension 2 or higher, and the Hohenberg-Wagner-Mermin theorem, for why statistical mechanics models with a continuous symmetry have a phase transition in dimension 3 or higher — or equivalently, why they do not exhibit a conventional phase transition on dimension 2 or lower. These amount to predictions of the lower critical dimension of each model: the smallest spatial dimension in which the model exhibits a phase transition.

To me, both arguments have a similar flavor, and both can be mathematically rigorized and turned into proofs. In the end, I hope they will give us intuition about the importance of low energy, long wavelength excitations in determining the critical behavior of a system.

Peierls Argument

We will start with the Peierls argument concerning the existence of a phase transition in the Ising model. The Ising model is perhaps the simplest possible model of magnetism, which posits that each spin \(\sigma_i\) takes a value \(\sigma_i\in \{-1,1\}\) which we refer to as “down” and “up”, and the spins have a tendency to align with one another parameterized by coupling strength \(J\). The energy of the system is then \(H = -J\sum_{\langle ij\rangle} \sigma_i\sigma_j\) where the sum over \(\langle ij\rangle\) runs over all sites between which a bond exists. The dimension in which we embed our lattice changes the meaning of this sum: in 1d each spin is connected to only two neighbors, while in 2d there are connections with multiple neighbors (4 for a square lattice, 6 for a triangular lattice, et cetera).

After Ising showed the lack of phase transition at nonzero temperature in the 1 dimensional model, it was thought that there was perhaps no phase transition at nonzero temperature in any dimension. However Rudolf Peierls was able to show otherwise with an argument we will discuss here. Of central importance to the argument is the assumption of the “thermodynamic limit,” which takes the limit that the number of spins goes to \(\infty\), which is a reasonable assumption for any macroscopic magnetic material. We will be able to show that the Ising model must exhibit a phase transition in any dimension \(d\geq 2\).

When we say that the model exhibits a phase transition, it means that there is some temperature \(T_c>0\) such that for temperature \(T<T_c\), the system exhibits nonzero mean magnetization \(\langle M\rangle = \frac{1}{N}\sum_i \langle\sigma_i\rangle = 0\), while for \(T>T_c\) we have \(\langle M\rangle=0\). Note that angle brackets here denote thermal expectation over the Boltzmann distribution, which should preside at equilibrium.

As with many phase transitions, we can benefit from thinking about this case as a battle between energy and entropy. We know that at equilibrium, the free energy \(F = H-TS\) will be minimized, where \(H\) is the energy, \(T\) is temperature and \(S\) is entropy. The change in free energy, \(\Delta F\), associated with a change in the state of the system, will tell us whether this change can happen spontaneously.

We will consider a single spin, and ask whether, starting from an ordered configuration of all \(\sigma_i=1\), the focal spin can flip to \(-1\) spontaneously. Let us first consider \(d=1\). The easiest way for the focal spin to flip, which costs the least energy, is for it to be part of a cluster of length \(L\), which flips to spin up. This flipped cluster will incur an energy cost at its boundary, and in \(1\) dimension this cost is just \(4 J.\) However, the presence of the cluster increases the entropy of the system: there are \(L\) ways to position the cluster such that the focal spin is flipped, resulting in an entropy of \(\log L\). Therefore the free energy change resulting from the appearance of a cluster containing the focal spin is

\[\Delta F = 4J - T \log L.\]

We therefore see that no matter how small \(T\) is, we can always make \(L\) large enough to make the flipping of a cluster entropically favorable. Therefore the system will never fully order except at \(T=0\), at which point entropy does not contribute to \(\Delta F\).

How are things different in 2 dimensions? Now consider a cluster of boundary length \(L\) that contains the focal spin. The energy penalty of flipping such a cluster is proportional to its boundary length, and is given by \(2J L\). What about the entropy of such a flipped cluster? The exact entropy will be difficult to calculate, but we will make do with an upper bound for the entropy of such a cluster. In other words, we want to estimate how many clusters of boundary length \(L\) can be drawn while still containing a focal spin. Why do we use an upper bound? This is because if the system shows an ordering transition even for the largest possible entropy of a cluster, then it must show and ordering transition for the real entropy, since entropy quantifies the tendency not to order.

Noting that the boundary of the flipped cluster is a self-avoiding path starting at some point within distance \(L\) of the focal spin, we can conclude that number of such flipped clusters must be less than \(L^2\) times the number of self avoiding paths of length \(L\), which is in turn less than \(L^2\) times the number of different paths of length \(L\). This is a very crude estimate, but good enough for our purposes. The number of different paths of length \(L\) is exponential in \(L\), and so we have that the entropy of the flipped cluster scales no faster than \(\log(L^2e^L) \sim L\), where we drop the logarithmic correction. Therefore the free energy associated with this cluster excitation is

\[\Delta F = 2JL - T L = L(2J-T).\]

Therefore we see that depending on the temperature \(T\), this excitation can be favorable or unfavorable. In other words, there is a phase transition! The argument is similar for higher dimensions — the important thing is to show that the number of surfaces of codimension \(1\) enclosing the focal spin is exponential (up to polynomial factors) in the area of the surface.

Hohenberg Wagner Mermin Theorem

We will now make a similar argument for in a different type of model, in dimension \(d\leq 2\). We will consider for concreteness the XY model, where spins are unit length vectors parameterizes by an angle \(\theta_i\) at each site. The Hamiltonian is now given by \(H = J\sum_{\langle ij\rangle} \cos(\theta_i-\theta_j)\). In the ordered state of this model, the spins spontaneously choose a direction in which to point, and the average direction of the spins, given by \(\frac{1}{N}\sum_i\langle\cos\theta_i\rangle \hat x + \frac{1}{N}\sum_i\langle\sin\theta_i\rangle \hat y\), is nonzero. The important difference between the XY model and the Ising model is that here the degrees of freedom are continuous, whereas in the Ising model they are discrete. This drastically changes the types of excitations we can get disrupting the ordered state. In the discrete case, the excitations are localized, in the form of domain walls that separate a flipped cluster from the rest of the system. But now we can get collective modes that span the whole size of the system.

These modes can be lower energy than the domain walls of the Ising model. In particular, consider an excitation that spans a system of linear size \(L\) with periodic boundary conditions. The angles \(\theta_i\) make a full twist from \(0\) to \(2\pi\), so that the different between neighboring angles is \(2\pi/L\). From the Hamiltonian of the model we can see that the energy of interaction between two neighbors is \(\cos(2\pi/L) \sim L^{-2}\), and so the energy of the whole excitation scales as \(L\times L^{-2} = 1/L\). This energy vanishes as the system size gets very large, and so we see that for any positive entropy of this excitation (which it certainly has), this system-wide excitation will happen spontaenously down to arbitrarily low temperatures. In fact it is natural to expect the entropy to be logarithmic in \(L\) for arbitrary spatial dimension, solely due the translational degree of freedom of the excitation across the length of the system.

What about in higher dimensions? The argument translates very similarly, except that now we see the energy of the excitation is \(L^d\times L^{-2}\). Therefore for \(d>2\) the energy diverges with system size and an ordering transition will occur at sufficiently low temperatures, since the energetic disfavorability of excitations will dwarf the entropic favorability. In \(d=2\) the crude estimate above gives that the energy of a system-spanning excitation does not scale with \(L\), while the entropy is logarithmic in \(L\), indicating the failure of spontaneous long range ordering. While this is strictly true, the full picture in \(d=2\) is rather more subtle, and involves the onset of “quasi long range order,” as worked out in detail by Kosterlitz, Thouless and many others. However, our crude estimates have told us to expect an ordering transition at nonzero \(T\) in dimension greater than 2.

Conclusions

What have we learned from the preceding two arguments? The Ising model has a phase transition in dimension 2 and above, and the XY model has a second order phase transition in dimension 3 and above. All this has come from the rough estimation of the energetic and entropic contributions of excitation of the system that prevent it from ordering. Hopefully it has served to illustrate how scaling arguments can go a long way toward rather nontrivial results.

References

A nice explanation of the Peierls argument

Understanding Hohenberg Wagner Mermin from field theory

Wikipedia