Scaling arguments in random systems
Introduction
Here we continue with our discussion of scaling arguments for predicting the presence or absence of a phase transition in a statistical physics model. However, we will now discuss two famous examples of such arguments in models with quenched disorder. These models are meant to describe materials with some nonzero concentration of impurities. These impurities are unavoidable in experimental systems, and can arise e.g. from foreign atoms embedded in a crystal, breaking the otherwise-perfect discrete translation symmetry. One way of building these impurities into our statistical physics models is by including a source of randomness that is drawn from some a priori distribution, and does not evolve in time. These random variables are called quenched because their values are frozen in time, and not allowed to thermally equilibrate; they therefore behave as degrees of freedom that have been locked in place by a sudden quench down to lower temperatures rather than a slow cooling.
One might well ask whether studying models of this type will give us any insight into real systems, which clearly do not have their impurities distributed according to any idealized distribution, much less with a particular realization of the disorder that any instantiation of a theoretical model requires. This is where the important concept of self averaging arises. In the thermodynamic limit, a model with quenched randomness can be decomposed into many copies of the model, each with different disorder realizations. Therefore, in most cases, observable quantities will automatically be averaged over the disorder distribution, since their measured values will be an average over the local values in many smaller domains, each with different realizations of the disorder.
Random Field Ising Model: Imry and Ma
One prototypical example of a disordered statistical physics model is the random field Ising model. This model describes a ferromagnet with a finite concentration of magnetic impurities, which exert local fields on the spins. Its Hamiltonian is
\[H = -J \sum_{\langle ij\rangle} \sigma_i\sigma_j + \sum_i h_i \sigma_i\]where the sum over \(\langle ij\rangle\) is over bonds in a \(d\) dimensional lattice, and the \(h_i\) are random fields that act independently on each of the spins. For concreteness, we can take the \(h_i\) to be independent identically distributed gaussian random variables with mean \(0\) and variance \(b^2\). This Hamiltonian describes the tendency of neighboring spins to align, under the additional influence of random fields that may try to pin certain spins in the positive or negative direction. For now we will concern ourselves with the behavior of this model with no external field. The \(h_i\) provide local fields, but their average vanishes along with the global field. We would like to know: in what spatial dimension does the random field Ising model exhibit a second order phase transition between ordered and disordered states, for \(b>0\)? In other words, does the ordered state at low temperatures survive when we turn on the quenched disorder in the local fields?
We know that the pure Ising model (with \(b=0\)) has a phase transition with temperature from a disordered to ordered state, in dimensions \(2\) and above. How can we adapt our understanding of this pure model to include the case of random field disorder?
This argument (Imry, Ma, 1975) is of a similar flavor to the Peierls argument for the pure Ising model. However, here we will focus on the ground state, and ask whether it remains ordered when we introduce random field disorder. If it does not remain ordered, i.e. if the ground state has \(0\) net magnetization, then we know that the random field Ising model will fail to order at any positive temperature as well. Recal that in the Peierls argument we were interested in entropy because we wanted to know whether the Ising model exhibited a transition at finite temperature. However here only energy is important for our purposes. We note at the outset that this argument is valid for random field disorder but not necessarity random bond disorder, which is more subtle. The key aspect about random fields is that they couple linearly to the magnetization, and so locally they push the magnetization either positive or negative. The same is not true for random bond disorder, which we will not cover below.
Let us say that we have an ordered state (one of the ground states) of our Ising model without random fields, with all spins pointing in the same direction. We now apply a small random field (\(b\ll J\)). What happens? As we have seen previously, it is fruitful to think about flipping domains (or droplets) of spins, which in the case \(b=0\), incur an energy cost proportional to the length of the domain wall. However, in the case where we have random fields, it is possible that flipping a droplet of spins could actually lower the energy of the uniformly magnetized ground state, if the random fields conspire in the right way. In particular, the change in energy incurred by flipping a droplet of linear dimension \(L\) has a positive contribution proportional to \(L^{d-1}\) from the surface area of the droplet, and another random contribution with magnitude \(\sim L^{d/2}\), which comes from the addition of the \(\sim L^d\) independently distributed random fields in the interior of the droplet.
If \(L^{d-1}\) scales faster than \(L^{d/2}\), then flipping large domains becomes energetically unfavorable. This happens for \(d> 2\), indicating that for dimensions \(3\) and above, the ground state will not be disordered, and will have nonzero spontaneous magnetization. However for \(d-1<d/2\), the random fields can win out and flip arbitrarily large droplets, causing the disordering of the ordered ground state. [As an aside, note that for systems with continuous symmetry, the energy cost of a spin wave over a domain of linear size \(L\) scales as \(L^{d-2}\), which modifies the Imry Ma argument, giving a critical dimension of \(4\) instead of \(2\).]
What about the marginal case, \(d=2\), where \(d-1=d/2\)? Here the two contributions to the energy of flipping a droplet scale the same way with \(L\). This means that for any particular droplet, regardless of its linear size \(L\), the probability that it can be flipped and reduce the energy of the spin configuration is independent of \(L\), and given by the probability of drawing a gaussian random variable (the contribution from the random fields) of at least size \(\sim L\) (the contribution from the boundary) from a distribution with width \(\sim L\). The fact that this probability of an energy-lowering flip does not scale with \(L\) means that if we consider a large number of droplets, some finite fraction of them will favorably flip, including some droplets of arbitrarily large size. This indicates that the ground state will disorder upon introduction of random fields.
Another way to see this breakdown of long range order that happens with random fields is to imagine a box with positive spins on the boundary, in the ordered phase of the pure Ising model. When we turn on a random field, any particular spin on the interior of the box will flip with large probability if the box is big enough — allowing us to consider a large enough number of droplets containing the spin of interest.
Therefore we have seen that the ground state of the random field Ising model is disordered for any nonzero random field in \(d=2\), but does have a second order phase transition at finite disorder strength in \(d=3\) and above.
An extension due to Aizenman and Wehr
It turns out the the Imry Ma argument also casts light on the fate of first order transitions in random fields. Recall that a first order transition is one in which the order parameter (in magnetic systems, usually the average magnetization) jumps discontinuously across the transition. For example, the ground state of the pure Ising model exhibits a first order transition as a function of external magnetic field: the average magnetization jumps from \(-1\) to \(1\) as the field is tuned from negative to positive. Typically phases that are separated by a first order transition will coexist at the transition point: this phase coexistence is a hallmark of first order transitions. We can consider the boundaries between coexisting phases as boundaries of droplets in the random field Ising model. We saw that in \(d > 2\) the energy cost of making a large droplet outweighs the energy gain, and so large droplets will not flip spontaneously, preserving phase coexistence. However in \(d\leq 2\) one can flip arbitrarily large droplets and the phase coexistence necessary for a first order transition is not possible. Therefore random field disorder eliminates first order transitions in \(d\leq 2\), as proved by Aizenman and Wehr in 1989.
Harris Criterion
Another famous argument concerning critical behavior of disordered systems is due to Harris (1974). Here the question is: under what circumstances will quenched disorder change the critical exponents of a phase transition? In other words, is the character of the phase transition stable to small amounts of disorder added to the system? We have intentionally been vague about how the disorder is added (in the bonds, or fields, etc.) — the hypothesis is that for small amount of disorder, for which is the Harris argument is relevant, the type of disorder added should not matter.
In order to make sense of this argument, we will benefit from thinking of a disordered magnetic sample as a large number of adjacent boxes, each with different realizations of the disorder. As a result of the different realizations of disorder, the critical properties of each sample are slightly different. In particular, they all have a slightly different \(T_c\). [Strictly speaking a critical temperature is only defined for an infinite system — but in the infinite system limit we can think of splitting up the system into a large number of large subsystems.] If we split our system into subsystems of linear dimension \(\xi\) where \(\xi\) is the correlation length of the spin-spin correlation function, then the spins in each box can be thought of as an independent system. The fluctuations in \(T_c\) from one box to another will have typical size \(\xi^{-d/2}\) due to the independent impurities in a box of size \(\xi\). Now as we approach the critical temperature, the correlation length \(\xi\) itself changes, and in particular it diverges as \(\lvert T-T_c\rvert^{-\nu}\) where \(\nu\) is an exponent characterizing the phase transition in the clean system (without disorder).
Since \(\xi\) diverges at criticality, the width of the distribution of \(T_c\)’s, given by \(\xi^{-d/2}\), vanishes as the critical point is approached. However the relevant question is whether it vanishes faster or slower than the distance (in temperature) to the critical point. In other words, does \(\xi^{-d/2}\) vanish faster or slower than \(\lvert T-T_c\rvert\)? If it vanishes faster, then it means that sufficiently close to the critical point, we can treat the distribution of \(T_c\)’s from the disorder as much narrower than the distance the system is from criticality. Therefore the system will not have a fraction of its “sub-boxes” at criticality until the whole system is at criticality. The critical point will not be affected by the spread in the \(T_c\)’s of its subsystems.
By contrast, if the \(T_c\) distribution width shrinks slower than \(T-T_c\) as \(T\to T_c\), then we will have a change in the transition, since as the system approches its critical temperature, more and more of its sub-boxes will start to experience a transition due to their disorder realizations. So we see that the condition for the critical point to be unaffected by a small amount of randomness is that, as \(T\to T_c\),
\[\xi^{-d/2}<|T-T_c| \implies |T-T_c|^{\nu d/2}<|T-T_c| \implies \frac12 \nu d > 1.\]This expression, \(\nu>2/d\), is what is known as the Harris criterion for the stability of a fixed point to small amounts of disorder. What can we make of this in light of our previous discussion of the Imry-Ma argument? Well, for the clean Ising model in \(d=2\), it is known that \(\nu=1\) — therefore we find that the Harris criterion is only marginally met — and we cannot say with certainty whether the critical point will be affected or not. The same is true for \(d=4\), when \(\nu=1/2\).
However, for \(d=3\), the correlation length exponent of the Ising model is known to be \(\nu\approx 0.63\). Note that this is smaller than \(2/d=2/3\). Therefore we can say that for the 3D Ising model, disorder is a relevant perturbation at criticality. This agrees with the Imry-Ma argument, and as work on the random field Ising model has shown: a small amount of randomness will cause the pure Ising 3D critical point to flow (in the renormalization group sense) toward the zero temperature random field Ising fixed point, which is governed by a different set of critical exponents.
We might well ask whether these new critical exponents are governed by a Harris-like criterion. It turns out that in fact at this disorder-governed fixed point, the correlation length exponent obeys \(\nu>2/d\), as was proved by Chayes et al, 1986.
One important thing to note about the Harris criterion is that it is essentially a perturbative statement, and may not be true for large amounts of disorder, which could change critical properties of a system even if \(\nu>2/d\). However, it is an important tool for assessing the relevance of disorder at criticality, with use cases throughout the study of critical phenomena in disordered systems.
References
The ‘Harris criterion’ lives on: an article by Harris himself, looking back on his 1974 work
This document covering much of the same material
Slava Rychkov’s introductory lecture on the random field Ising model
The Imry-Ma argument: Imry, Ma, 1975
A general reference: T. Vojta’s review with many more details