The Free Energy

The reduced free energy per site, , is the fundamental relation that contains all of the thermodynamic behavior of a system. For the two-dimensional square lattice with nearest-neighbor couplings K, Onsager obtained the following result:

(5.1)

In this expression, the quantity u is given by

(5.2)

There is no closed-form, analytic expression for the integral in Equation 5.1. It can be evaluated in various limits, like for large or small values of K, for example, but in general a numerical integration is required.

Figure 5.2 shows , represented by the solid curve, as a function of the nearest-neighbor coupling K. The first thing we notice about is how simple it is – it starts at a finite value for small K and gently curves upward to approach a straight line for large K. There would be no way to know – just by looking at this curve – that it represents a system with a phase transition.

Let’s look at the small and large K limits more carefully. At high temperature ( ) the system is all entropy, as we’ve seen numerous times before. Thus, the reduced free energy per site in this limit is . At low temperature ( ) the system is all energy. Noting that there are two couplings per site, one vertical and one horizontal, this limit is .

The vertical dashed line in Figure 5.2 is of particular interest – it represents the location of the phase transition (or critical point) of the Ising model. As we shall see later in this chapter, the location of the critical point, , is given by the following expression:

(5.3)

You would never know that this is a special value of K by just looking at the free energy. The significance of the critical point becomes increasingly evident, however, as we consider the first and second derivatives of the free energy; that is, as we consider the internal energy and specific heat, respectively.

Internal Energy

The reduced energy per site for this system, which we designate by the symbol e, is defined as follows:

Note that the average value of the sum over nearest-neighbor spins is equal to the derivative of the free energy with respect to the corresponding conjugate variable, which in this case is K. Thus,

This quantity can be evaluated by taking the derivative of the integrand in the expression for the free energy. We find

(5.4)

The quantity u is given by Equation 5.2, as before. This integral must be evaluated numerically to yield results suitable for plotting.

Carrying out the numerical integration, we obtain the plot shown in Figure 5.3. Notice that the energy is zero when the coupling K is zero, as expected, and that it approaches 2K in the limit of large coupling – that is, in the limit of low temperature.

As before, the vertical dashed line shows the location of the critical point, which is more apparent in this case. In fact, the slope of the curve for e diverges at the single point .

Specific Heat

We turn now to the most dramatic illustration of the critical point in the two-dimensional Ising model – the infinite peak in the reduced specific heat per site, c. To obtain this result, we use the expression first derived in Equation 2.16; namely,

We apply this to the Onsager solution by taking the second derivative with respect to K of the free energy in Equation 5.1, and then multiplying by . The result is

(5.5)

As before, the quantity u is given in Equation 5.2.

To plot the specific heat, we simply carry out the numerical integration indicated in Equation 5.5 using basic mathematics software. The result is shown in Figure 5.4. Notice that the peak of the specific heat diverges to infinity at the location of the critical point, indicated by the vertical dashed line.

Recall from Chapter 2 that the specific heat is a measure of the fluctuation in the energy of the system. The average energy itself is perfectly finite at the critical point, as shown in Figure 5.3, but the fluctuations about that average tend to grow as the transition is approached.

Here’s a way to see how this can come about. Imagine flipping 20 coins and assigning +1 to heads and –1 to tails. The average value obtained by adding up the coins after each one is flipped is zero. The fluctuations about this value are fairly small, on the order of roughly plus or minus 2 or 4, because it’s unlikely to flip the coins and have 15 or 20 coins come up heads.

Now, imagine a coupling between the coins, like the coupling between spins in the Ising model. To do this, suppose we take 10 coins and lay them side-by-side, with the same face upward. Now glue them together to form a long strip. This couples them to one another. Do the same to the other 10 coins. Now, when we flip the coins, we can have the following results: 20 heads (sum = +20); 10 heads and 10 tails (sum = 0); or 20 tails (sum = –20). The average value is still zero, as before, but now the fluctuations are as large as the system itself – ranging from +20 to –20.

Something very much like this happens in the Ising system, as illustrated in Figure 5.5. As the critical point is approached from the small K side, shown on the left half of Figure 5.5, clusters of spins pointing in the same direction develop and grow in size as the coupling strengthens. This is indicated by the dark patches on the light (disordered) background. The typical size of the clusters is measured by the correlation length . When large patches of coupled spins flip from one sign to the other, the fluctuation in energy can grow infinitely large as the correlation length goes to infinity. On the other side of the critical point, the dark background indicates spins in the majority direction, and the light patches are clusters of spins pointing in the opposite direction. As before, the fluctuations can be unbounded near .

Another point of interest regarding the peak in the specific heat is the following: The height of the peak is infinite, but the area under the peak is finite. That is, the peak grows higher and higher, but at the same time it becomes narrower and narrower, resulting in a finite area. The significance of this is seen when one considers that the energy required to raise the temperature of a system from one temperature to another is equal to the corresponding area under the specific heat. Thus, even though the specific heat is infinitely large at the critical point, only a finite amount of energy is required to cross from one side of the critical temperature to the other. If the area under the specific heat had been infinite, as one might at first suppose it to be, then it would be impossible to raise the temperature from below to above the critical value.

Spontaneous Magnetization

The final thermodynamic function we consider is the magnetization with zero magnetic field, m – also referred to as the spontaneous magnetization because it arises spontaneously, without a magnetic field to produce it. This result was obtained by Onsager and presented at a conference in 1944, but was first published by C. N. Yang in 1952. The derivation is quite complicated, as one might expect, but remarkable cancellations occur near the end, leading to the following surprisingly simple expression:

(5.6)

Note that this expression is valid only for , which means for temperatures below the critical temperature, . Above the critical temperature, the magnetization is zero; m = 0.

A convenient way to plot this result is as a function of temperature, T. To do this, we work with dimensionless quantities; that is, we let (which is similar to setting c = 1 in relativity, or setting in quantum mechanics). The result is

(5.7)

In this expression, we have introduced the critical temperature, , which is the inverse of the critical coupling in Equation 5.3. Thus,

A plot of m as a function of is given in Figure 5.6. As before, the vertical dashed line shows the location of the critical temperature. The magnetization is zero for temperatures above the critical temperature.

At the critical temperature, an infinite cluster forms in the system, leading to the infinite fluctuations indicated in the specific heat. This infinite cluster, however, accounts for only a vanishingly small fraction of the spins in the infinite system. As a result, the magnetization starts off at zero just at . Once the temperature is lowered below , the magnetization rapidly saturates to m ~ 1 as the infinite cluster spreads and overwhelms the system.

Specific Heat

Let’s do a quick comparison between our series expansion results so far and the Onsager solution. The most dramatic feature in the two-dimensional Ising model, and the best place to test our results, is the reduced specific heat per site,

Using our series-expansion result for the reduced free energy, we find

The next term in the expansion, just for the sake of comparison, is .

We show these results on the large-K side of Figure 5.8. In particular, the lowest dashed curve is the fourth-order term, the dashed curve in the middle is the fourth-order term plus the sixth-order term, and the highest dashed curve is the sum of the fourth-, sixth-, and eighth-order terms. The solid curve is the exact Onsager solution.

It’s clear that our approximations improve with each additional term in the series expansion. It’s also clear, however, that the singular behavior at the critical point will be seen only with an infinite number of terms.

Higher Dimensions

To this point, we’ve considered only the two-dimensional, square lattice, but the same approach can be applied to other lattices as well. For example, suppose we would like to do an expansion on a d-dimensional, hypercubic lattice – that is, a lattice that generalizes from the square lattice (d = 2) and simple cubic lattice (d = 3) to arbitrary dimension d. The result is

You can check that this reproduces our previous result for the case of d = 2. If we now apply it to the 3-D simple cubic lattice, d = 3, we have

The changes in critical behavior as a function of dimension will be considered in the last section of this chapter.

One-Dimensional Behavior

The low-temperature approach can shed additional light on the behavior of an Ising model on a one-dimensional lattice, and in particular on the reason for the absence of a phase transition in that case. The same argument suggests that there is a phase transition in the two-dimensional lattice.

The spin-up ground state for a one-dimensional lattice with free boundary conditions is shown in Figure 5.9 (a). Notice that all spins point in the same direction, and that all bonds are “satisfied” – as indicated by the solid connecting lines. The first excited state, with one broken bond (dashed line), is shown in Figure 5.9 (b). The break can happen anywhere on the lattice, so if there are N sites on the lattice, it follows that there are N – 1 excited states.

Let’s calculate the change in free energy that occurs because of this excitation. First, recall that the free energy is . In this case, the excitation increases the energy E by the amount 2J. In addition, it occurs in locations, and hence it leads to an increase in entropy S of . As a result, the change in free energy is

Notice that the energy term increases the free energy, while the entropy term decreases it.

Now, in general, a thermodynamic system seeks to lower its free energy. In this case, we can see that for any finite temperature, T, the entropy term will dominate in the limit of , and hence the free energy is lowered as a result of the excitation. This means that the excitation is favored at finite T and occurs freely throughout the system. But note that the excitation flips a finite fraction of the spins, destroying any possible ordered phase; that is, there is no magnetization for any finite T. As a result, there is no finite-temperature phase transition.

This same argument can be applied to the N × N two-dimensional lattice, but with the opposite conclusion. Once again, the first excitation, as shown in Figure 5.7 (b), will proliferate throughout the system at any finite temperature. That is the same. What is different, however, is that this excitation is just a single spin flipped out of an infinite lattice – and this has a vanishingly small effect on the magnetization. To flip half of the lattice and destroy the magnetization, as in one dimension, would require breaking at least N bonds – but this would make the energy cost significantly greater than the entropy gain. Using arguments along these lines, Peierls was able to show rigorously that the magnetization persists to a finite temperature in the two-dimensional lattice. The actual value of that finite temperature was obtained several years later using duality, which we cover in Section 5.4.

Evaluating the Partition Function: Graphical Analysis

Returning to the partition function, we can now write it as follows:

(5.12)

The term in front of the summation simply represents one factor of for each of the 2N nearest-neighbor pairs. The product over nearest neighbors, on the other hand, is best evaluated in terms of graphs on the lattice.

To see how this can be done, notice that each term in the product has either a 1 or a v for each nearest-neighbor pair. Graphically, we associate each v with a line, or bond, connecting the corresponding nearest neighbors. If a nearest-neighbor pair has the term 1 instead, we represent that by the absence of a bond.

For example, in Figure 5.10 (a) we show the “empty” graph (no bonds), which has a 1 associated with each nearest neighbor pair. The contribution this graph makes to the partition function is

Notice that the sum over all configurations of N spins yields , as expected.

Figure 5.10 (b) shows the graph with v for one nearest neighbor pair, and 1 for all the others. The contribution corresponding to this graph is

The fact that this sum is equal to zero follows because for any spin .

The simplest nontrivial graph that has a nonzero contribution is an elementary square of four bonds, as shown in Figure 5.10 (c). The contribution to the partition function in this case is

This is the same contribution as for the empty graph, except for the inclusion of the term , which represents the four bonds.

Looking at these results, it becomes clear that the only way a graph can give a nonzero contribution is if it has an even number of bonds (0, 2, 4) associated with each site. Any site with an odd number of bonds will have raised to an odd power, and that will sum to zero.

The contributing graphs to order and are shown in Figure 5.11. Notice that there is one graph per site (N) for order . Similarly, there are two graphs per site (2N) for order , one vertical and one horizontal.

Combining these results, we can write the partition function to order as follows:

The corresponding reduced free energy per site is

The specific heat is easily evaluated with , and the results for the three nontrivial terms just given are plotted on the small-K side of Figure 5.8. We see again that the approximation becomes better with each additional term in the series, and that the singularity will only be seen in the limit of an infinite number of terms.

Graphs in One Dimension

The graphical analysis of this section can be applied to any dimension – one, two, three, four, and so on – as well as to any type of lattice structure – square, triangular, honeycomb, cubic, face-centered cubic, and so on. The application to one dimension is particularly enlightening, since it leads to the exact solution.

Let’s start with a one-dimensional chain (free boundary conditions) of N spins. The lattice is shown in Figures 5.12 (a) and (b). The empty graph is shown in Figure 5.12 (a), and the contribution this graph makes to the partition function is

(5.13)

There are no other contributing graphs. As an example, consider the graph in Figure 5.12 (b). This graph, and all others you could draw, will have sites with an odd number of bonds – and hence zero contribution. Thus, the exact partition function for this system is the result given in Equation 5.13, in agreement with Equation 4.2. Quite simple.

Next, let’s consider a one-dimensional Ising system with periodic boundary conditions; that is, a ring lattice, as in Figures 5.12 (c) and (d). The analysis here is almost as simple as it is for the chain. Again, we have the empty graph, as in Figure 5.12 (c). This has the contribution given in Equation 5.13, except that N – 1 (the number of bonds in the chain lattice) is replaced by N (the number of bonds in the ring lattice) in the exponent of the (cosh K) term. We have another contributing graph, however – the full graph; that is, the graph with N bonds connecting all N sites, as in Figure 5.12 (d). The contribution from this graph is . It follows that the complete partition function for the ring is

(5.14)

This result agrees with Equation 4.34, as expected.

Critical Exponents

If a thermodynamic quantity Q vanishes at the critical point, we write it in the limit of small t as follows:

In this expression, E is referred to as the critical exponent. It follows that as for positive E. Similarly, if Q diverges to infinity at the critical point, we will write it a little differently:

Notice that we’ve included a minus sign in the exponent this time, so that if E is positive, then as . In general, we always define critical exponents to have positive values by including negative signs where needed.

The most common critical exponents are those related to the specific heat, c ( ), the magnetization, m ( ), and the magnetic susceptibility, (exponent = ). Near the critical point, we can write these thermodynamic functions as follows:

Notice that we’ve included minus signs for the quantities (c and ) that diverge at the critical point.

Now, the critical exponent is a bit of a special case, so we’ll start with and come back to later. Referring to Figure 5.6, we see that m vanishes at the critical point. The expression in Equation 5.6 immediately indicates that the exponent in this case is (we shall give the details of this conclusion in the next chapter). In three dimensions, the corresponding value is . Table 5.1 collects these results, as well as all the other critical exponents discussed in this section.

Next, we consider , which is a measure of the fluctuation of the magnetization. This quantity diverges to infinity at the critical point. The exponent describing this divergence in two dimensions is . In three dimensions, the approximate value of the exponent is .

Finally, we return to the specific heat and its exponent, . The value of this exponent in two dimensions is . What does this mean? Well, the most obvious answer is that the specific heat approaches a constant value at the critical point, say with one constant value above the critical point and another constant value below it. This means a discontinuity in the specific heat, which is certainly a singularity. A discontinuity in the specific heat is seen in four dimensions, and in mean field theory, as indicated in Table 5.1.

There is another possibility, however. Consider the following definition for the natural log:

This shows that can also indicate a logarithmic divergence at the critical point; that is,

This, in fact, is the case for the two-dimensional Ising model. Thus, the specific heat in two dimensions diverges, though it does so more slowly than any positive power of t. In three dimensions, it is found that .

Universality

The results shown in Table 5.1 illustrate one of the key properties of critical exponents; namely, the concept of universality classes. For example, we show the values for the exponents in 2-D in the top row of Table 5.1, though we don’t specify which of the two-dimensional lattices we have in mind. That’s because it doesn’t matter – the critical exponents are the same on the square, triangular, hexagonal, and other 2-D lattices. The thermodynamic functions and the location of the critical points will vary from lattice to lattice, but close to the critical point each system behaves the same.

Thus, we can say that the top row in Table 5.1 represents the 2-D Ising universality class. In general, universality classes are determined by large-scale, global features of a system – like the dimension and the type of spin – and are unaffected by small-scale, local features – like the number of nearest neighbors or the detailed interactions. So, for example, adding a second-neighbor interaction in two dimensions will change the location of the critical point, but it won’t change the exponents.

The next row in Table 5.1 gives the critical exponents for the 3-D Ising universality class. Notice that these exponents differ from the corresponding two-dimensional values, but again they don’t depend on details like the specific structure of the 3-D lattice. All of these conclusions are the result of decades of hard work analyzing series expansions, and applying other techniques as well. The concept of universality classes was discovered as a result of the hard work done on these systems dating back to the 1960s.

The bottom row in Table 5.1 shows the results for a four-dimensional lattice. Now, of course, there are no four-dimensional crystals in nature. From a theoretical perspective, however, there’s no problem considering a four- or higher-dimensional lattice – in fact, one can gain valuable insight in this way. For example, we find that the critical exponents are the same for all dimensions equal to or greater than 4; thus, 4-D is a sort of boundary – below 4-D critical exponents vary with dimension, but above it they remain the same as the dimension is changed. It turns out that fluctuations are much less important in higher dimensions, and hence all higher dimensions behave the same near the critical point.

In addition, we shall see in the next chapter that the exponents in 4-D correspond exactly to the exponents found in the approximation methods referred to as mean-field theories. This makes sense, because mean-field theories ignore fluctuations – which is a good approximation in higher dimensions. We shall pursue these concepts in greater detail in the next chapter.