Landau TheoryOverviewPreviously, we've found that the thermodynamic behavior of the meanfield Ising Model is entirely captured by its Landau Free Energy . On this page, we're going to analyze this function in its full gory detail. Starting firstprinciples from the microscopic Hamiltonian, we'll find the Landau Free Energy of the meanfield Ising Model, and then zoom in around the critical point to observe qualitative features of its phase transition (all in a meanfield treatment). It will be enlightening and beautiful! But the full power of the Landau Theory doesn't lie just here – it lies in its remarkable ability to generalize for all sorts of different phase transitions, such as superfluids, liquid crystals, shapeshifting transitions, orderdisorder transitions, and more. We'll see examples of other Landau Theories in the next section, when we'll also generalize to spatiallyvarying order parameters. And finally, we'll see how the behavior of the Landau Free Energy near a critical point is actually completely determined by symmetries. This sort of universality is the dream of theoretical physicists, and it's incredibly powerful: it says that the way a thermodynamic system behaves doesn't depend too much on all the minute micrsocopic details of what it's made out of; the emergent behavior has a sort of universal behavior. I hope this is enough motivation to see why it's well worth our time to understand Landau Theory. Here's the game plan: OutlineDeriving F(m) for the Ising ModelThe true power of Landau Theory is its generality, but I find it pretty tough to understand this generality without first grasping a concrete examples. For this reason, we'll first turn to our favorite model system: the Ising model. Let's start off by deriving explicitly from the microscopic Hamiltonian of the Ising Model. Remark
It's rather rare that we can calculate the explicit form of the Landau Free energy starting from a microscopic description of the system (the individual spins and their interactions). Rather, we typically start with symmetry arguments to figure out the shape that the Landau Free Energy must have; that is, we take a topdown rather than a bottomup approach. We'll discuss this philosophical point later when we talk about symmetry arguments. As before, we'll proceed with a variational approach. Rather than finding the exact free energy (hard!), we will find the variational free energy where is the exact Hamiltonian of the Ising Model and is our noninteracting variational guess of (Remember, our variational free energy is our best guess for the exact free energy.) Thankfully, we solved this problem a few weeks ago. With a bit of work, we found that the variational free energy was where the magnetization is given by , and the molecular field is determined by the selfconsistent relation , where is the number of nearest neighbors. (We derived the expressions for and while discussing noninteracting spins, and we derived the selfconsistency relation while finding the ‘‘best’’ variational guess for ). To proceed further, let us simplify our expression for . Notice that two of the terms are sums over sites , so it would be nice to also express the sum over bonds as a sum over sites. Well, one way to do this is to sum over all the sites, and then for each site, sum over the neighbors of each of the sites, but if you think about this a bit, you realize that you actually count each bond twice when you do this. (Think about it….carefully….) In mathematical symbols, we can write this as where is the number of nearest neighbors next to each site. With this observation, the (variational) free energy simplifies to We can interpret the quantity inside the brackets as a free energy density that we sum up over sites to find the total free energy: . To simplify the expression further, let's employ the relation to write everything in terms of rather than . (Remember, the whole name of the game here is to write everything in terms of the physical quantity . The ‘‘local field’’ isn't exactly a physically observable quantity, so we'd like to get rid of it.) We're left with Hooray – we've arrived at our expression for the free energy, and we've got rid of all the 's.
There's one last thing to do: we need to take the Legendre transform to find the free energy as a natural function of the magnetization . Remember, I spent a while on the probe fields page arguing why we wanted to express everything in terms of rather than unphysical probe fields, and why this meant that we needed to take a Legendre Transform from to . Well no recourse than to plug into . It turns out that the algebra works out the easiest if we start from a lesssimplified form of . So here we go: Our goal here is to get rid of all the 's and express everything in terms of the one true physical magnetization . Well, there's two 's to take care of here. The second term is pretty straight forward; we can invert the formula for magnetization and say that . For the first term, we have to perform a bit of clever hyperbolic trig massaging: which means that the first term becomes Triumphantly, we declare the Landau Free Energy of the meanfield Ising model: Let's recap what we've done. We wanted to see how the Ising Model behaved under a variational meanfield treatment. To proceed, we calculated the variational free energy , took a Legendre Transform, and then ended up with the (‘‘Gibbs’’) free energy as a natural function of the mean magnetization . The behavior of the Landau Free Energy will give us useful physical information. In particular, the most interesting thing about the meanfield Ising model is its phase transition at , so let's go ahead and see how the Landau Free Energy behaves around there. Expanding about critical point(Notational note: I'm gonna be sloppy and call this newly derived free energy density rather than . In class, we called this function . Same concept, different notation.) If you stare at the explicit formula for , you'll find that it's remarkably unenlightening: I have very little sense of how an inverse hyperbolic tangent behaves, much less this nasty combinations of terms. Thankfully, we can draw upon a little physical insight about the behavior around the phase transition at . Why can we Taylor Expand?
Thinking back to the meanfield solution to the Ising model, you'll remember that something remarkable and surprising happens when we cool the system below : the equilibrium magnetization suddenly goes from to . And in particular, it went from zero to nonzero continuously, which means that it's pretty close to zero around the critical temperature. So we're actually in luck: as long as we're close enough to , we can Taylorexpand the free energy in terms of small . The resulting polynomial will help us understand all the interesting correlations and relations that happen near the phase transition. Now, of course, once gets too big, there's now reason for us to trust our loworderexpansion, but the crucial point is that near the critical point, the only thing that matters are the lowestorderterms in . So if we want to figure out how things behave near the critical point, it's sufficient to expand the Landau Free Energy in small . Expanding the free energyNow….I'm a bit too lazy to do the algebra. It's just a Taylor expansion, so it shouldn't be too bad. I'll just quote our answer from class: where the dots represent higher order terms in . We can write this a bit more nicely as where the coefficients have been defined as There's a number of things we can say about this free energy function:
Looking at the minimaThe reason we cared about the Landau Free Energy was that its minima tells us the equilibrium value of the magnetization. With our earlier full expression for , it was hard to find the minima, because it had ugly functions like inverse hyperbolic tangents, but now that we've expanded it as a loworder polynomial, it's actually remarkably easy to find its minima. I like to think of it as highschool math with a graduatephysics interpretation. Before we plug in through the algebra (and trust me, it's very simple algebra!), it's informative to think about how we expect the solutions to behave, from the knowledge that we already know about the Ising Model. Great Expectations
We expect that at high temperatures (), the magnetization will be zero because the magnet is disordered, and thermal fluctuations are energetic enough to destroy any longrange order. And we expect that below the critical temperature, the magnetization will spontaneously become nonzero – and in fact, we expect there to be two equivalent solutions at , because of the spinflip symmetry that we talked about earlier. In terms of the Landau Free energy, the question ‘‘are we above or below the critical temperature?’’ is answered by ‘‘is positive or negative?’’, so we expect that the minima of will depend on the sign of . With this physical picture in mind, let us proceed to find the minimia of the Landau Free Energy. Take the derivative, set it equal to zero, and solve for : Either , or , a.k.a . The nonzero solution only makes sense if the thing in the square root is positive; i.e., if and we're below the critical temperature. So the equilibrium value of magnetization (the one that minimizes ) is given by
APPLETS! APPLETS!
In my opinion, the best way to visualize the minima of this quartic is by playing with interactive sliders. You can visually see how the unique minimum above the critical tempearture starts to ‘‘flatten out’’ as you approach criticality, and then the two new minima at appear once you've cooled below the critical point. You can also see the dilemma of ‘‘spontaneous symmetry breaking’’: if you're standing on top of the hill at below the critical temperature, both sides look ‘‘the same’’ to you, so how do you know which minimum to pick? To summarize, the Landau Free Energy lets us visualize the phase transition from to in an intuitive manner. It recapitulates the behavior we've found earlier, and in addition, it makes the quantitative prediction that the magnetization grows like once we cool below the critical temperature. There's other sorts of funny things that happen near the critical point, too. Many of these behaviors have to do with what happens when you poke the system with an external perturbation such as a magnetic field. We'll find that systems near criticality become so sensitive that the tiniest flicks can lead to longranging correlations and slow relaxations. External FieldsTo study what happens when we prod or poke a system, we need to introduce an external field into the Hamiltonian of the system. By enticing the spins with an energetic reward of (so alluring!), the external field makes the spins ‘‘want to point’’ in a particular direction. For you thermodynamic nerds out there (like myself), you can think of this field as a conjugate force to the order parameter in the free energy ;) Drawing Phase DiagramsWith this external field, we now have a total of two knobs that we can twiddle to mess with the system: there's the quadratic coefficient (which, again, you can think of as the temperature), as well as the external field , which makes the spins prefer one direction to the other and ‘‘breaks the symmetry’’. Since there's two different knobs to twiddle, we can summarize the behavior by drawing an phase diagram. Each point on the phase diagram correponds to a particular combination of and . An experimental protocol such as ‘‘heat the system to 400K, then turn on a strong magnetic field of 1T, then cool back to 300K’’ corresponds to a path on a phase diagram. In this picture, phase transitions correspond to funny behaviors of the equilibrium magnetization as you walk along different paths on the phase diagram. So for instance, if the magnetization suddenly jumps discontinuously, you've walked over a firstorder phase transition! Or if it ‘‘kinks’’ (i.e., its derivative is discontinuous), then you've walked over a secondorder phase transition. To figure out where phase transitions occur, we'll need to figure out the equilibrium value of at each point on the phase diagram. That is, we'll need to minimize for different values of and . When we plot the value of on the phase diagram, the discontinuous jumps correspond to firstorder transitions, and the places where the derivates jump correspond to secondorder transitions. Again, the algebra is relatively straightforward, but I always like to gain a qualitative picture of what's going on before diving into any calculations. So go have some fun with the applets before reading on. Trust me, they're pretty addictive… Features of Phase DiagramThe phase diagram is a nice way to visualize what happens as we vary the two parameters of our model (the temperature and the external field ). To summarize some of the findings from playing around with the applet:
Philosophy of Phases
Are spinupordered and spindownordered different phases of the Ising model? Well…..nope. You can undergo a phase transition without transitioning between phases (?). More precisely, it's possible to continuously twiddle the external knobs on the Ising system, and start from spinupordered, and end with spindownordered, without any sudden discontinuous jumps in the magnetization. All you have to do is to ‘‘walk around the line of firstorder transitions’’ by heating up the magnet, dialing the external field in the other direction, and then cooling the magnet again. So they're really the same phase… By a similar argument, the liquid and gas phases are really the same phases as well, because you can continuously get from one to the other by just walking around the critical point on the phase diagram. Of course, typically in our daytoday life, we don't reach pressures and temperatures high enough to walk around that point, so we're forced to cross the line of discontinuous density to boil water. This begs the natural question, ‘‘what counts as distinct phases of matter’’? As we see, it's a pretty subtle question. Well, we can be sure that two points on the phase diagram are in different phases if there's no way to get from one point to the other without passing through a line of discontinuities. I wish I could explain further, but we should move on to explicitly calculating critical exponents. Discovering More Critical ExponentsSo far, we've answered lots of fun qualitative questions about the phase diagram of a meanfield Ising model, but it's time to get a bit more quantitative. Pictures can only get you so far. Just by looking at the shape of , you can deduce that that the magnetization is zero above , but becomes nonzero below . However, to figure out the exact functional dependence of how depends on your distance below , you need to do a bit of algebra to find that . We're about to do the same thing here: now that we have some intuition about what happens in the phase diagram, it's time for us to do some specific calculations. Notice that our first critical relation () took the form of a power law – that is, the quantity we cared about varied as the distance from the critical point raised to some power. We'll find that these other relations will also hold powerlaw forms. Again, the name of the game is to minimize the free energy for different values of and on the phase diagram. This time, because we added an external field, the free energy looks like where the term breaks the symmetry by enticing one of the two states with an energetic reward. Now the derivative becomes which unfortunately is a cubic equation! Thankfully, we won't need to explicitly find the roots of this cubic. How magnetization varies with external fieldSince we introduced an external field, a natural question to ask is how the equilibrium value of changes as we twiddle the external field. That is, we want to walk leftright on the phase diagram, and see what happens to the magnetization. Evidently, the dependence of is going to change depending on what the temperature is. To make life the easiest, we'll just consider what happens at the critical temperature (or equivalently, ). This corresponds to walking leftright along the axis. Thankfully, if we set in our cubic formula for , then one of the terms disappears, and the cubic becomes really easy to solve: Again, we've uncovered another powerlaw relation: the magnetization varies as the cube root of the external field as we sit at the critical temperature. Above, I've plotted the behavior of this function in green. As you go left to right along the xaxis, it corresponds to twiddling the knob on the external field. Exactly when , we find that , as expected, and as you make the field stronger and stronger, the magnetization gets bigger and bigger. For comparison, I've also plotted the fielddependence of magnetization for two other temperature isotherms, at hotter and cooler temperatures than the critical temperature. (The blue and red curves correspond to walking lefttoright on the phase diagram slightly above or slightly below the axis.) Above the critical temperature, the magnetization doesn't depend quite so strongly on the external field – it doesn't change quite as much when you change the field a certain amount. And below the critical temperature, the magnetization undergoes a jump when you pass from to . This plot is a different way of visualizing the firstorder phase transition from earlier. Notice that the slope of the graph is vertical at when you sit at the critical temperature! This suggests that the magnetization is extremely sensitive to the external field at the critical point – the smallest little hint of an external field is sufficient to change the magnetization by a ton. We say that the system is highly susceptibile to external field perturbations near the critical point. SusceptibilityTo quantify just how sensitive the system is to external fields, we can calculate the zerofield susceptibility at different temperatures near the critical point. (The susceptibility is just another name for the slope of the graph when we pass through .) We want to figure out how this susceptibility changes as we vary the temperature around the critical point. In terms of the phase diagram from earlier, we want to walk on a vertical line at near the critical point , and find the susceptibility at different points along this line. To find , we're going to do a clever trick. Earlier when we set equal to 0, we ended up with the cubic equation Since our quantity of interest is a derivative with respect to , let's take the derivative of both sides of the equation above. We end up with Thankfully, we can find the suscpetibility without having to solve a cubic equation! If we shuffle terms around, we find that where in the final step, I plugged in our results from earlier that above and below . There's a few comments to make:
Moreover, the diverging susceptibility tells us an extremely wacky fact about physics near criticality: the tiniest nudges can cause the hugest changes. Fluctuations can play a very big role in the behaviors of such systems! Summarizing our critical exponentsLet's step back now to review the big picture: Summary
We want to understand what happens to an meanfield Ising model as we vary the external field and temperature .
The quantitative dependencies near the critical point typically had the form of power laws like . The critical exponents are particularly interesting because they don't depend on the microscopic details and because you can measure them in experiments. We encountered three different critical exponents:
Lots of very interesting physics. Nonetheless, the Landau Theory is still rather restrictive, for a number of reasons. We encountered one of the caveats when we found the susceptibility: we found that as we get closer to the critical point, the system becomes more and more sensitive to little perturbations, which means that thermal fluctuations about equilibrium become more important. In particular, this calls into questions our original assumption that we could treat the entire system as a uniform magnetization everywhere. We've neglected the spatial fluctuations at different sites on the lattice. So in the next section, we're going to learn how to generalize from a uniform magnetization everywhere to a spatially varying magnetization that can have interesting textures and spatial fluctuations. Leave a Comment Below!Comment Box is loading comments...
