[MUSIC] Let's consider the Ising model in one dimension. Consider a line of spins in one dimension that has periodic boundary conditions. What does periodic boundary conditions mean? This means that the last spin as N interacts with the first spin as 1. For the Ising model in one dimension, the energy can be written as. Now a useful way to rewrite the partition function is to use a new quantity termed as the transfer function, T, which depends on spin, S i, and S i plus 1. This is simply defined as the exponential of the average energy at location i and i plus 1. Now, with this definition, we can re-write the partition function as a product of these transfer functions. Now, let's construct a transfer matrix. The transfer matrix is simply a matrix of all possible outcomes for the two spins. The overall partition function now turns out to be the trace of the transfer matrix. The trace of a matrix is simply the sum of its eigen values. Now, here we're interested in the trace of the matrix T raised to the power n. It turns out that the trace of the eigenvalues of T raised to the power n are simply the eigenvalues based to the power n of the matrix T. In this case, the two eigen values, lambda plus and lambda minus, are given by. Therefore, now the exact solution for the partition function is simply given by the following expression. An important test of the expression that you've derived is whether we can recover back the solution for the non-interacting case. How do we impose the non-interacting case? Well, of course we impose this by setting the J value to 0. Now, setting the J value to 0, we do recover the non-interacting partition function. Now, what happens when we turn off the field? How do we do this? We do this by setting H, the field term to 0, and the partition function simply becomes. Now, the eigen value lambda plus is greater than lambda minus. And therefore, for large N we can approximate the partition function simply as the dominant eigenvalue lambda plus raised to the power of N. In this case, the average magnetization is given by. Note that this expression tells us that in the limit of turning off the field, that is as h turns to zero, the overall magnetization approaches zero. Hence this exact solution for the one-dimensional Ising model does not exhibit a phase transition from a disordered state to an ordered state. Now why is this the case? The physical justification for this lies in the fact that forming a disordered phase from an ordered phase occurs through a constant energy change. We only have to pay the energy penalty for flipping the end to spins. Therefore the overall energy penalty paid delta e is simply 4 times the coupling constant j in one dimension in dependent on the size of the system m. However in higher dimensions when dimension d is greater than or equal to 2. The energy cost actually scales with the system size N. And this leads to a finite temperature phase transition. For instance, in two dimensions, if you consider a square domain, the energy penalty paid depends on the length of the square. The length of the square depends on the square root of N, and hence the dimensionality plays a roll in determining the energy penalty. So, we've not yet seen a phase transition. So, what do we do? Well, let's go about solving the two dimensionalizing model. It turns out that the solution for the two dimensionalizing model is not as easily found. Actually in 1944, Lars Osonger, a famous physicist, found an exact solution for the to dimensional Ising model for h equals zero using a set of very sophisticated mathematical techniques. It turns out that the two dimensional ising model exhibits a spontaneous magnetization for sufficiently large values of the coupling constant, j. However, at present there is no exact solution for the Ising model for higher dimensions than two. The exact solution is highly complicated. Here, we'll introduce a simpler way of thinking about it under what is popularly known as the mean field approximation. The key idea behind the mean field approximation involves approximating the detailed interactions with the average of these interactions. Let's define the average magnetization per site using the following relation. Note that here, we've invoked spatial invariance which implies that the average of asi is simply given as the average of s. Let's define a coordination number, zed. Defined to be the number of nearest neighbors, to any given site divided by 2. Let's look at some examples for the coordination number. For a two dimensional square lattice, that turns out to be two. And for a three dimensional square lattice, that turns out to be three. The mean field approximation replaces the nearest neighbor interactions by the average interaction. That is, substituting the value for spin s j by the average of s j. Now, this gives the following approximation to the energy. Now, what happens as a result of this mean field approximation? It turns out that this reduces the problem to an effective non-interacting model. From this we can easily write the average magnetization per site using the expression that we derived earlier. Now, the solution here turns out to be an implicit relation. That is, the average magnetization is found to be dependent on a function of the average magnetization itself. Now for the case of turning off the external feet. That is, h = 0. The magnetization satisfies the equation. The magnetization m is equal to the hyperbolic tangent of K times m. Note that the coupling constant now is absorbed into the new constant, K. Now this equation is a self consistent mean field equation. What does that mean? It means that local order is dictated by the surrounding order, that is itself dictated by the local order leading to a self-consistency condition that must be met. The two scenarios that exist from this, for k less than 1, there is only one solution for the average magnetization that corresponds to the trivial solution of m = 0. Now what happens when k is greater than one? When k is greater than one, there exists three solutions that correspond to m equals zero and m equals plus or minus m-star. Turns out that the case m equals zero is an unphysical solution as it corresponds to a free energy maximum. From this analysis, we can define a critical K value set at 1. Now, from this emerges a critical temperature, and this gives us a way to rewrite the constant K as the critical temperature divided by the actual temperature. Now what happens close to the critical temperature? Close to the critical temperature we can expand the self consistent equation near the magnetization value M=0 to get the following expression. This expression tells us that the average magnetization scales as the square root of the difference between the critical temperature and the actual temperature. Now generally, the external field H acts as an intensive variable that is conjugate to the extensive variable, the magnetization capital M. Now, thermodynamic stability dictates that a derivative offered intensive variable with respect to its conjugate extensive variable is a positive quantity. We've seen many examples of this earlier. Similarly, thermodynamic stability for the icing model, dictates that the derivative of the external fill would respect to the magnetization is a positive quantity. Now let's plot the phase diagram for the mean filled solution that we have derived. This equation of state can be likened to the vapor liquid equation of state that is the equation of state that we learned from phase equilibrium. Now, the phase diagram for the mean field approximation for the Ising model exhibits very similar characteristics as the liquid vapor phase diagram. Now, what is the correspondence between the magnetic system and the vapor liquid system? The MH curve in the magnetic system is analogous to the PV curve in the liquid vapor system. This reflects universal physical issues that are addressed with the Ising model and why it can be used to understand a variety of phenomena. To summarize, we discussed the phase behavior of a magnetic system by analyzing the exact solution to the one dimensional Ising model and a mean field approximation to the two dimensional Ising model. We found that there is no phase transition that occurs in the one dimensional Ising model. However, the Ising model exhibits a true phase transition for dimensions D greater than or equal to two. This was argued by physical arguments through the mean field approximation. The h-M phase diagram in a magnetic system was found to behave akin to the p-V phase diagram a vapor liquid equilibrium. In both systems, a critical point exists that marks the onset of phase co-existence in the system. That is for temperatures greater than the critical temperature, the system does not exhibit a phase transition. And for temperatures below the critical temperature phase transitions exist.
