Self-Consistency

We come now to the heart of the mean-field method – the self-consistent requirement. We’ve assumed that the magnetization of all the spins surrounding are equal to m. The bonds connecting these spins to tend to align it in the same direction as the other spins on the lattice, assuming, as we do, that J is positive. This means that has a magnetization – and since it is like any other spin on the lattice, its magnetization must also be equal to m. This condition, namely that < > = m, makes the approximation self-consistent.

To calculate the average value of , we need the probability that is equal to +1 or –1. Probabilities are proportional to the Boltzmann factor, , and the factor that converts the proportionality to a specific probability is the partition function. The partition function for is

The first term in the sum corresponds to = +1, and the second term corresponds to = –1. It follows that the probability that the spin is equal to +1 is

Similarly, the probability that is equal to –1 is

Therefore, the average value of is given by the following:

Now, applying the self-consistency requirement, < > = m, we obtain the following relation:

(6.1)

The solution to this equation for a given temperature T is the magnetization m that characterizes every spin on the lattice.

General Behavior and Critical Temperature

Equation 6.1 cannot be solved analytically – and for good reason, since the behavior of m is inherently nonanalytic. Instead, we can find numerical solutions for m by graphing both sides of Equation 6.1 and looking for intersection points, which is where the two sides are equal. This is illustrated in Figure 6.2. In this plot, the straight line passing through the origin is the left-hand side of Equation 6.1; namely, y = m. The three sigmoidal-shaped curves represent the tanh function on the right-hand side of Equation 6.1 for a variety of temperatures. The intersection of y = m with one of the tanh curves gives a point – a value of m – where Equation 6.1 is satisfied for a specific temperature.

The first notable feature of Figure 6.2 is that there is always a solution at m = 0, no matter what the temperature. This is easily verified by substituting m = 0 into Equation 6.1. It is also clear from physical considerations – after all, if the bulk of the spins on the lattice have m = 0, then they will not induce a magnetization in , and hence its average value will be zero as well. At high temperatures m = 0 is the only solution, and it corresponds to a completely disordered lattice.

At low temperatures two additional solutions appear, one positive and one negative, as shown by the upper curve on the right-hand side of Figure 6.2. These two solutions are symmetric about m = 0, and correspond to the fact that the zero-field Ising model is unchanged by flipping all the spins. In addition, the finite-m solutions have a lower free energy than the m = 0 solutions at any given temperature, as we shall see later in this chapter, and hence they are the solutions that characterize the physical system. As the temperature is decreased toward zero, the finite-m solutions approach +1 and –1, indicating that the lattice is completely ordered with all spins up or all spins down. We will focus on the positive-m solutions.

Separating the regions of finite-m and zero-m solutions is the critical temperature , which corresponds to the middle curve in Figure 6.2. At this temperature, the initial slope of the tanh curve is equal to 1, the same as the slope of the line y = m. To find an expression for the critical temperature, we note that the series expansion for tanh(ax) for small x and constant a is

(6.2)

Applying this expansion to the right-hand side of Equation 6.1 yields

Thus, the initial slope of the tanh function is , and setting this equal to 1 yields the critical temperature . To be specific, we find

(6.3)

Rearranging gives

(6.4)

Thus, the key factor in determining the critical temperature, and the factor that distinguishes one lattice from another, is the coordination number, q.

For example, on the 2-D square lattice, where q = 4, we have = . For comparison, the exact result is = . On the triangular lattice, with q = 6, the mean-field critical temperature is = . This is to be compared with the exact result on the triangular lattice, which is = . For the simple cubic lattice, the mean-field critical temperature is the same as it is for the triangular lattice, since these lattices have the same coordination number. The best estimate for the critical temperature on the simple cubic lattice is = , which is considerably closer to the mean-field result. We collect these results, along with other critical properties, later in this section. In general, mean-field results are better the higher the dimension, where fluctuations about the mean value – which are ignored in this approach – are less important.

Now that we’ve determined the critical temperature, it’s useful to rewrite the self-consistency relation in terms of the dimensionless temperature variable . Starting with Equation 6.1, let’s multiply and divide by in the argument of the tanh to find the following:

Substituting the expression for the critical temperature, Equation 6.3, this simplifies to

(6.5)

The three curves in Figure 6.2, from top to bottom, are plots of the right-hand side of this equation for the values = 0.5, 1.0, and 1.5, respectively.

Before continuing, we should mention a particularly severe drawback of the mean-field approximation – namely, that it predicts a finite critical temperature for the 1-D Ising model. For example, the 1-D chain lattice has a coordination number given by q = 2, which implies a critical temperature of = . In fact, as we know from Chapter 4, there is no phase transition in one dimension, and hence the actual critical temperature is zero. The mean-field method predicts a phase transition in 1-D because, as we’ve stated, it ignores fluctuations, and fluctuations, as we saw in Section 5.2, destroy the ordered state in one dimension at any finite temperature.

Numerical Solutions and Cobwebbing

Finding numerical solutions to Equation 6.5 is fairly easy to do. Of course, one can always make a guess, and then refine the guess to get closer and closer to the desired solution. However, since the tanh curve on the right-hand side of Equation 6.5 crosses the line y = m with a slope less than 1, it follows that successive iterations of the tanh function will automatically converge to the desired solution. No guessing is required. It’s as if the equation solves itself.

To see how this works, we can think of Equation 6.5 as a relation that iterates from one approximation for the magnetization m to the next. Thus, if is the current approximation to the solution, the right-hand side of the equation gives the next approximation, , which is closer to the true solution. Specifically,

Iterating this relation gives the solution to any desired accuracy. The ultimate result – the solution we seek – can then be seen as a fixed point of this iteration, which we designate as and define as follows:

(6.6)

For each value of there is a corresponding finite, positive value of .

For example, suppose we would like to find the numerical value of for the case = 0.5. This corresponds to finding the intersection point of the tanh curve and the y = m straight line in Figure 6.3. For our initial guess, which we will call , let’s take 0.5 – we could make a much better guess by referring to Figure 6.3, but this crude choice is useful for illustrative purposes. The next iteration, , is given by

We show both of these values of m on the horizontal axis in Figure 6.3. The next value of the magnetization, , is given by

We also show in Figure 6.3, as well as the next iteration, .

We can continue with this process as many times as desired, and visualize it with cobwebbing between the tanh curve and the straight line – similar to the cobwebbing we did in Figure 3.4. Successive iterations converge to the intersection at the fixed-point magnetization, which in this case has the value = 0.9575 … . Carrying out the same procedure for a variety of values of gives the results listed in Table 6.1.

Table 6.1 Fixed point values m* for Equation 6.6 corresponding to various values of . Results are given to four decimal places.

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The data points in Table 6.1 are plotted in Figure 6.4, along with the overall solution, which is represented by the solid curve. Notice that the magnetization, m = m*, saturates to m = 1 as the temperature goes to zero, and goes to zero, m = 0, at . The magnetization remains zero for all higher temperatures.

The fact that the magnetization is zero for all temperatures above , but is nonzero below that temperature, means that the magnetization is a nonanalytic function. There is no analytic function, in fact, that can be zero over a finite range of temperatures and then abruptly become nonzero – there must be a singularity at the switchover point. This is precisely why there is no analytic solution to Equation 6.5. In addition, as we will show later in this section, the singularity at the critical temperature is a power-law singularity, characterized by a critical exponent. While this type of singularity is also predicted by Onsager’s exact solution for the 2-D Ising model, the mean-field prediction for the value of the exponent is different from the exact result, and is also different from the value seen in experiments.

Comparing Magnetizations

One might wonder whether the magnetization curve shown in Figure 6.4 gives results for the square lattice, the triangular lattice, or perhaps the simple cubic or face-centered cubic lattice. In fact, it represents all of them.

To see this, note that the temperature scale in Figure 6.4 is normalized to the critical temperature; that is, we plot m versus . Now, each lattice has its own specific value of , which in general are quite different. Once we normalize the temperature to this value of , however, the magnetization for each lattice vanishes at the same point, . All the other points on the curve are the same for all lattices as well. This can be seen by referring to Equation 6.5, where we see that m depends only on the variable ; all of the dependence on the lattice type has been encapsulated into the value of . In this respect, we can think of the magnetization in Figure 6.4 as the universal, mean-field magnetization for a single-spin cluster on any lattice.

Let’s turn now to a comparison between our mean-field magnetization and the exact results for 2-D and 3-D lattices. In Figure 6.5, the mean-field magnetization is the lowest curve (solid). The uppermost curve (dashed) is the exact magnetization for the 2-D square lattice, as given by Equation 5.6. It differs significantly from the mean-field result. This isn’t particularly surprising, of course, given that the comparison between 1-D and mean-field isn’t even close. Better agreement is found with the results for the 3-D simple cubic lattice, shown by the middle curve (dotted). These results are from exact series expansions, and can be considered to be exact to within the precision of the graph. Notice that as the dimension increases, the results of mean-field theory generally tend to improve – a direct reflection of the fact that fluctuations have less impact the greater the dimension.

Finally, three different series of experimental data points are shown for comparison as well. The points shown with (X) are from experiments on iron, those with (O) are from nickel, and those with (∆) are from cobalt. The agreement between the mean-field theory and experiment is quite reasonable, especially given the simplicity of the single-spin, mean-field approximation.

Magnetization Critical Exponent: Single-Spin Mean Field Theory

As has been mentioned, the mean-field magnetization has a power-law singularity as the critical temperature is approached from below. We will now derive this result, as well as the precise value of the exponent in the power-law dependence. The exponent is of particular interest in critical phenomena because, as it turns out, it has the same (universal) value over a wide range of Ising models, depending mostly on the dimension of the system, but not at all on many other details.

To explore the behavior near the critical point, we expand the tanh function on the right-hand side of Equation 6.5 for small values of the magnetization m. We expanded this function in Equation 6.2 to find the critical temperature, but this time we carry the expansion to one higher order in small quantities. The result is

(6.7)

Applying this to Equation 6.5, we find

Rearranging, and canceling one power of m, yields

(6.8)

At this point it’s useful to introduce the normalized reduced temperature, t, defined as follows:

Notice that t vanishes linearly as T approaches from below. Substituting this definition into Equation 6.8 yields

Taking the square root of both sides gives the temperature dependence of m near :

(6.9)

We note that the quantity in parentheses, , approaches 1 at the critical temperature. As a result, it follows that the magnetization vanishes near as

In addition, the slope of m versus T goes to infinity as is approached as .

These results verify that m does indeed have a power-law singularity near . Recall that the magnetization critical exponent, , is defined as

Clearly then, the mean-field critical exponent for the magnetization is

As mentioned previously, this result applies to all mean-field approximations.

Magnetization Critical Exponent: Exact 2-D

By way of contrast, let’s take a look at the magnetization exponent from the exact result in 2-D. Referring to Equation 5.6, we can write the exact magnetization as follows:

Recall that the dimensionless critical temperature is , where, from Equation 5.3, we have

Noting that , we can rewrite as follows:

Now, when t = 0 in this expression – which corresponds to the critical point – the sinh function is equal to 1. That is,

It follows that when t = 0, as expected.

Next, consider a straightforward Taylor series expansion of the sinh function for small values of t. This gives

Notice that we’ve simplified the expression by introducing the following constant:

(6.10)

This constant results from two identities. First,

Second, noting that , we have

These identities combine to give the expression for a in Equation 6.10.

Substituting the expansion of sinh into , we find

Clearly, the exact magnetization goes to zero as t raised to the 1/8 power, and hence for 2-D the exact magnetization exponent is

This exponent applies to all ferromagnetic Ising models in 2-D, which includes the square lattice, triangular lattice, hexagonal lattice, and others.

Collected Critical Properties

We display the dimensionless critical temperature and magnetization exponent for various lattices in Table 6.2. The results listed as “Exact” come from explicit solutions for the 1-D and 2-D lattices, and from exact series expansion analyses for the 3-D lattices. The mean-field results come from the single-spin approximation of this section.

Table 6.2 Critical properties (dimensionless critical temperature and magnetization exponent ) of various lattices from both exact and single-spin mean-field calculations. The “Exact” results for 3-D are the best estimates from exact series expansions, and are considered to be correct to this number of decimal places.

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Notice that the value of the mean-field magnetization exponent is the same for all dimensions. In contrast, the exact magnetization exponent varies with dimension, but it is the same for different lattices with the same dimension, even if those lattices have different types of interactions. This is the basis of the concept of “universality” in critical phenomena – critical exponents depend on the global symmetries of a system, not on the details of the local structure and interactions. On the other hand, there is no “universality” in the critical temperature – instead, it is sensitive not only to the type of lattice and the dimensionality, but also to the type of interactions on the lattice.

Zero-Field Properties

To begin, we note that near a critical point, the order parameter – which is m in the case of an Ising magnet – is arbitrarily small. Hence, it seems reasonable to consider an expansion of the free energy in powers of m. We know that this may be problematic, owing to singularities in the system, but it makes sense to start here.

In the case of zero magnetic field, where the system is unchanged by reversing all the spins, we can expand the reduced free energy per site, f, as follows:

Note that we include only even powers of m in our expansion – this ensures that f is invariant under a change in sign of m. In addition, we truncate the expansion at the fourth order since this is high enough to yield the critical behavior of interest. We will have more to say about the multiplicative terms a and b as we continue our discussion.

Now, the basic idea in this approach is that a system in thermodynamic equilibrium seeks to minimize its free energy. To look for the minimum values of the free energy, we plot f in Figure 6.7 for three different cases. In each case, we assume that b is greater than zero – this bounds the minima to small values of m and keeps the free energy from going to minus infinity for large m. The value of a can change sign, however. For a > 0 and a = 0 the free energy has a single minimum, which is at m = 0; for a < 0 the free energy has two symmetrically placed minima at positive and negative values of m. The value of the free energy at the nonzero-m solutions is always less than it is at m = 0, which is actually a local maximum for a < 0.

We’ve seen this kind of behavior before. In fact, when we studied the self-consistent condition for the single-spin mean-field calculation, Equation 6.5, we saw that m = 0 is always a solution. We also saw that for temperatures below the critical temperature there are two new solutions, with symmetrically placed positive and negative values of m. At the time, we mentioned that these finite-m solutions minimize the free energy, and we can see now that this is indeed the case in Figure 6.7.

In addition, it follows that we can identify the critical temperature in the Landau–Ginzburg approach with the value . After all, that is where the finite-m solutions first begin to appear. The simplest assumption for the dependence of a on temperature near the critical point, t = 0, is that it varies linearly. Hence, we can write

In this expression, is the negative of the rate of change of a near t = 0.

To explore the behavior of the magnetization m near the critical point, and to determine the magnetization critical exponent, we look for minima of the free energy function . To do this, we take the derivative of with respect to m and set the result equal to zero. This yields

Clearly, m = 0 is always a solution, as expected, and it is the physical solution for a ≥ 0; that is, for temperatures above the critical temperature. In addition, the finite-m solutions, which apply for temperatures below the critical temperature, are given by the following:

It follows that m vanishes like the square root of t:

As a result, the critical exponent for the Landau–Ginzburg approach is

This is exactly the same magnetization critical exponent as in other mean-field approaches. In general, all versions of mean-field theory predict all the same exponents.

Finite-Field Properties

Let’s take a look now at the Landau–Ginzburg free energy in the case that the system has a finite magnetic field, h. This adds the following term to the expression for f:

This term breaks the symmetry about m = 0, as one would expect, and follows from taking the term in the Hamiltonian (Equation 4.37) and replacing with its mean value, m.

In Figure 6.8 we show plots of the free energy f for the following cases: (a) h > 0; (b) h = 0; and (c) h < 0. When h > 0 there is only one lowest minimum in the free energy, and it occurs at a positive value of the magnetization, m > 0. It follows that the system responds to the finite magnetic field with a finite magnetization.

When the magnetic field is zero, h = 0, the free energy is again symmetric about m = 0. For temperatures below the critical temperature there are two finite-m minima, and they have the same value of the free energy. Thus, we expect the system to have a coexistence of a positive-m state and a negative-m state in zero field.

Finally, with h < 0 the free energy again has only a single lowest minimum, which this time occurs at m < 0. As h is lowered from positive values to zero, and then to negative values, the magnetization jumps discontinuously – a first-order, or discontinuous, phase transition – from the positive-m state to the negative-m state. There is no critical point when the magnetic field is finite.

Thus, the Landau–Ginzburg approach provides a useful way of visualizing the behavior of a system as it undergoes various phase transitions. It also gives predictions for the critical exponents, and these are in agreement with other mean-field theories.