 And in particular, the example of the ferromagnetic phase transition, the onset of ferromagnetism at a critical temperature, we considered the magnetization in response to an applied magnetic field. When the temperature is high, the magnetization looks like that. It's linear when the field is weak in the field, and it saturates at high field. But when the temperature is below a critical temperature, there's spontaneous magnetization. So depending on whether we reduce the field from positive values or negative values to zero, there's a magnetization at zero field, which is either positive or negative. I'm actually considering the case of a uniaxial magnet. The magnet consists of spins. It can point either up or down. There's an applied field. I'll say that it's positive when it points down. The spins like to align with the field. So the field is down, and the spins prefer to point down. This is what the isotherm looks like when the temperature is less than the critical temperature, and we have spontaneous magnetization. So if I consider the phase diagram as a function of applied field or the magnitude of the applied field and temperature, we have spontaneous magnetization when the temperature is below the critical temperature. There is a line of first order phase transitions where we have phase coexistence, and the line terminates at a second order phase transition at the critical temperature. In contrast to the line of first order phase transitions that we discussed in the case of the liquid gas transition, there's no latent heat. Correspondingly, as we know from the Clausius-Clapeyron relation, the slope of the coexistence curve is zero. So the entropy per unit volume, in other words, is the same in the two phases, the spin up phase and the spin down phase, which is actually obvious from symmetry. There's no difference in the physics of the two phases. I can turn my head upside down and turn the spin up phase to the spin down phase. But in principle, we can have at zero applied field a boundary between the phase in which the spins are mostly up and the phase in which the spins are mostly down. That's what I mean by phase coexistence, like the surface separating a liquid from a gas. If I consider the magnetization, the magnetic moment per unit volume of the material or the absolute value of the magnetization as a function of temperature, add zero field, then there's no spontaneous magnetization when the temperature is greater than the critical temperature, when it's below the critical temperature, the magnetization turns on. Actually, OK, I'll say the absolute value of the magnetization. So it turns on when the temperature is slightly below the critical temperature, and then it goes to some zero temperature value. So the magnetization as a function of temperature looks kind of like that. And we'd like to understand how the magnetization turns on in the vicinity of the critical temperature as we cool from above the critical temperature to just below. That's the analog of what we studied in the gas-liquid transition when we cool just below the critical temperature and first see the appearance of separation of phases into liquid and gas with some volume discontinuity. So we talked about a model of a magnet, first of all, a model in which we ignore the interactions among the spins. We can think of the spins as being independent, each one interacting with the applied magnetic field. That's a good model of paramagnetic behavior. And a paramagnet, well, that's like the way a ferromagnet behaves at high temperature. The magnetization goes to zero when we turn off the applied field in a paramagnet. So I can consider the case where I have spins, which have a magnetic moment. So there's an energy difference between a spin down, aligned with the applied field, and spin up, an energy splitting between the two, which is twice the magnetic moment times the applied field. That's called that B. And that means then in thermal equilibrium the probability of being spin up or the fraction of all the spins which are spin up compared to the probability of being spin down is suppressed by a Boltzmann factor, e to the minus delta e over tau, the energy splitting divided by tau, or in other words, e to the minus 2 mu B over tau. So we saw that if I consider the magnetization then, which is the magnetic moment per unit volume, each spin has a magnetic moment of mu. The total magnetization is the number of downspins minus the number of upspins times the magnetization, sorry, times the magnetic moment of each spin and then divided by the volume. That can be expressed as n, the number of spins per unit volume times mu, the magnetic moment of each spin, times the function of the applied field, which is the hyperbolic tangent of mu times B divided by tau. The hyperbolic tangent is just, well, it looks like the function I drew there for my tau greater than tau c isotherm. It doesn't really look like that, does it? More like that. So it asymptotically goes to 1. That's where saturation occurs. And it has some non-zero slope at the origin. When we turn off the applied field, the magnetization goes to 0. Now if I'm going to have a phase transition, if I'm going to have coexistence of phases, then I have to include interaction somehow. Like in the van der Waals model, we included the effects of interactions among the molecules by adding the effect of a long range potential and also a short distance, hardcore interaction. So what we want to do in order to understand ferromagnetism is, boy, is this for us? Pretty cool. That we want to include an interaction which makes the spins want to align. So if one spin is up, its neighbors want to be up too. So spins prefer to align rather than anti-align. Two spins being both up is favored, even a zero field, over pointing in opposite directions. And a simple way to do that, kind of a crude model, is to take this field B that each spin is experiencing as a combination of the applied field H, plus some contribution to the field, which is the average field applied at that point due to all the other spins. That average field is going to be proportional to the magnetization. If the spins are pointing up, then that's going to give rise to a field which is going to want to make the spin at a particular position point up also. So our model would be to take B to be an effective magnetic field experienced by each one of the spins, which is the applied field H, which I control by turning a knob in the laboratory, and something proportional to the magnetization. And it'll just be a parameter in the model, which I'll call lambda, what that parameter is. So this is the external applied field, and this is the contribution to the field due to other spins. So in reality, the magnetic field is going to be varying from point to point in the sample. But we're going to ignore that. Like we did in the Van der Waals model, where we just assumed we had a uniform density, here we're going to assume that each spin sees the same average contribution to the field coming from the other spins. And so we call this mean field theory. And the important assumption of mean field theory is that we don't have to worry about the fluctuations in the magnetic field from point to point and from time to time. There's just some contribution to the average field that comes from the effect of all the other spins and is proportional to the magnetization. Now it's not a very good model. I mean, like the Van der Waals model, it works pretty well in a regime, which is the regime of high temperature, where the effects of the interactions aren't very important. But we want to apply it when there is spontaneous magnetization and phase coexistence where the interactions are important and we won't get quantitatively correct predictions about what happens. But we'll try to understand qualitatively what happens and extract some predictions that might be of wider applicability, like we did in the Van der Waals model of the gas liquid transition. So what our model says is that the magnetization can be the number of spins per unit volume times the magnetic moment of each spin and now the hyperbolic tangent has an argument. I just plug in the effective value of the field, which is mu over tau times h plus lambda m. So it'll be convenient to define little m to be the magnetization normalized by m mu. And in particular then, let's consider the case where there's no applied field, so h is equal to 0. And so this becomes a nonlinear equation for m, little m. Little m is equal to hyperbolic tangent of n mu squared over tau times lambda m. That is, I wrote capital M as n mu times little m. So I got an n mu multiplying the mu over tau and a lambda times m in the argument of the hyperbolic tangent. So that equation then we can solve for m and determine the spontaneous magnetization. Is this the equation that determines the magnetization when there's no applied field? Spontaneous just means what's left when we turn off the applied field, h equals 0. So there's a parameter in the equation, the coefficient of m in the hyperbolic tangent. And depending on its value, we can get either one or three solutions to the equation. So if the coefficient of m in the hyperbolic tangent, lambda n mu squared over tau is less than 1, then when I plot the hyperbolic tangent as a function of m, it has a slope less than 1 at the origin. So let me draw the line corresponding to m. And the hyperbolic tangent looks like that, no slope less than 1 at the origin. So the solution is where those two curves cross, and there's just one solution at m equals 0. So there's no spontaneous magnetization. When we turn off the applied field, the magnetization goes to 0. But if the parameter lambda n mu squared over tau is greater than 1, that means that here's line corresponding to function equal to m. The slope is greater than 1 at the origin. And then it turns over like hyperbolic tangent does. And so there are three solutions now where the two curves, the dotted line and the solid line cross, one of them at m equals 0 is before and two at non-zero values of m, which are equal in absolute value and have opposite sign. Well, this is just like in the Van der Waals model, where we had three solutions. But only two of them corresponded to stable phases. And in fact, if we did consider turning on a non-zero applied field, and we plotted the function h as a function of m, except I'm going to draw it as m as a function of h, even though m is not really a function, it's multivalued, it would look like this. So these are the three solutions that occur at h equals 0. And then when we extend away from the h equals 0 axis by turning on a non-zero value of h, it looks like this. And the interpretation is like before, we should find the stable phase by using a Maxwell construction. But here it's obvious that because of the symmetry between spin up and spin down, the coexistence is going to occur at h equals 0. That means that these two areas are equal. And that's the analog of what we had in the gas liquid transition when the temperature was below the critical temperature. And by an equal area construction, we found the pressure at which coexistence phases occurred. This is the condition that the temperature is less than some critical value, temperature less than what we'll call t-critical, which is lambda n mu squared. And this is the situation when the temperature is greater than t-critical. And so we really get all the features that I've described here. The magnetization as a function of the temperature when the temperature is less than t-critical when we do the Maxwell construction, it makes a jump at zero field. And when the magnetization is in between, the value of the magnetization at zero field for the two homogeneous phases, that means we have a mixed phase. There are two homogeneous phases that are favored. One has the spins mostly up. One with the spins mostly down. When the magnetization is in between, that means the sample has divided into two parts. One the spin up phase, the other the spin down phase. So if we consider the magnetization as a function of magnetic field when the temperature is above the critical temperature, and then we approach the critical temperature from above, we'll reach the case where we're crossing over from having a single solution to three solutions. Now, this is the critical isotherm. The critical isotherm has the feature that the magnetization as a function of the applied field on the critical isotherm when the applied field is zero has infinite slope. So in other words, the susceptibility, the magnetic isothermal susceptibility, the derivative of magnetization with respect to applied field on the isotherm, that is, with the temperature fixed when the temperature is equal to the critical temperature, and the applied field is zero is infinite, is diverging. That's the second order phase transition in this system. Now, it's becoming very soft, very susceptible to an applied magnetic field. And the magnetization is very sensitive to the applied field when we approach the critical temperature. So as we did in our model of liquid gas transition, we can ask, how does the magnetization turn on as a function of temperature when the temperature is just below the critical temperature, close to but below? So how does magnetization as a function of temperature turn on, or temperature less than but comparable to critical temperature? So we can analyze that, like we did before, by doing a power series expansion in our equation determining the magnetization when the magnetization is small. So just to write it down again, our normalized magnetization, little m, is equal to hyperbolic tangent of n mu squared lambda over tau. And then plus a applied field over tau h. And now we know we can identify this as the critical temperature divided by the temperature. So it's close to 1 when the temperature is close to the critical temperature. So I'll call that 1 over tau hat since I had defined a tau hat to be temperature divided by critical temperature before. And it'll be convenient to consider epsilon, a measure of how far we are from the critical temperature rescaled so it's dimensionless. The deviation from the critical temperature divided by the critical temperature. So epsilon is positive above Tc, negative below. So now if we're interested in the spontaneous magnetization, we consider turning off the applied field, looking at how the stable solution for the magnetization as a function of temperature behaves. So now our equation is m equals hyperbolic tangent. And then I have m over tau hat on the other side. When the temperature is close to the critical temperature, the magnetization is just starting to turn on. That means tau hat is close to 1, close to the critical temperature. m is small when we solve for the magnetization because it's just starting to turn on. So that means we can expand the hyperbolic tangent in a power series to get the leading behavior when the temperature is close to the critical temperature. The hyperbolic tangent, it'll only have to expand it out to third order and it's an odd function, so it has only odd powers of x. The linear term is just x. The cubic term is minus 1 third x cubed. Hyperbolic tangent starts to turn over, so the cubic term has a negative coefficient. And our equation, then, if we use this expansion carried out to this order, says that the rescaled magnetization little m is equal to little m over tau hat from the linear term in the expansion of the tangent and then the cubic term minus 1 third m over tau hat cubed. So I can write that as putting the terms linear and m on the same side as m times 1 over tau hat minus 1 equals 1 third m over tau hat cubed. So when we're very close to the critical temperature, this coefficient is close to 0. I'm interested in finding the leading behavior when epsilon is small. And so this can be approximated by minus epsilon. And on the other side of the equation, to get the leading approximation, I can just replace tau hat by 1, because 1 over tau hat, so let's see, tau hat is equal to a 1 plus epsilon. So this is 1 over tau hat. And so 1 over tau hat, which is 1 over 1 plus epsilon, I'm just saying I can approximate by 1 minus epsilon. Then I subtract away the 1, so just minus epsilon plus higher order in epsilon, which I'm going to neglect, because I'm only interested in the leading behavior. So then the cubic equation that we have to solve for m is just m times minus epsilon equals 1 third m cubed in our leading approximation. So there's the unstable solution at m equals 0, but the stable solution corresponding to the spontaneous magnetization is m squared equals 3 times minus epsilon. So the magnetization turns on when the temperature is close to the critical temperature, like the square root of 3, tau critical minus tau over tau critical. So minus epsilon is 1 minus tau over tau critical. Well, the 3 in that expression, that depended on the details of our model, the equation of state that we wrote down that determined m. So 3 is model dependent. But the conclusion that the magnetization when the temperature is just below the critical temperature turns on, like the square root of tau critical minus tau, that really just depended on the equation being a cubic. It's the same behavior that we saw in the Van der Waals model and for the same reason. It follows from a cubic equation that determines m. So compare the Van der Waals density discontinuity, which is proportional to the square root of tau critical minus tau for the same reason. Now in the Van der Waals model, we also ask what happens to the compressibility, which diverges at the critical point. Here it's the magnetic susceptibility, which is the analogous quantity that diverges. And what's the nature of that divergence? OK, well, here's my equation. So the equation I want to solve when h is non-zero determines how the magnetization responds when I apply a weak field. So now we have little m equals hyperbolic tangent of m over tau hat, the term we had before, and then plus something proportional to the applied field. I'll call it little h. Little h means mu over tau times big h, the applied field. So now we want to solve this when the applied field is weak. And when the temperature is close to the critical temperature, so tau hat close to 1, we want to find the leading behavior where the susceptibility is blowing up at the critical point as we approach the critical point. And again, the idea is just to do the requisite power series expansion and being careful about what the leading terms are when tau is close to tau c, and the magnetization is therefore small. Well, we want to consider actually either coming from tau greater than tau c, in which case there is no magnetization, or tau less than tau c. But in either case, I can again use my power series expansion for the hyperbolic tangent like I did before. And since I'm interested in the case where you see around the problem. Is that all I need to do? Because maybe I can, you know, pushing the whole thing is too hard. Boy, this is kind of fun. I can build my own structure over here. It started, it's hard to stop, you know. I only want to have gone down far enough. Boy, at the end of the class, I'm going to kick these over. That's how you know I'm done. So, OK. So again, use power series expansion because we're interested in h and m small. And so I have m equals from linear term m over tau hat plus h. And then I have the cubic term m over tau hat plus h cube times minus a third. OK. So let's consider that to be an equation for h by putting everything but this over on the other side. So h is equal to m 1 minus 1 over tau hat plus 1 over 1 over 1 over 1 over 1 over 1 over 1 over 1 over 1 over 1. And then I have the third m over tau hat plus h cube. Well, so it's actually a cubic equation for h, but since I'm only interested in the leading behavior, it's a lot simpler than that. Because again, expanding 1 over tau hat, like I did over here, as 1 minus epsilon, and then because h is systematically smaller than m, because epsilon is small, h is smaller than m in this leading term by a factor which is going to 0 as epsilon goes to 0, as the temperature goes to the critical temperature. And that means to get the leading behavior, I can neglect this h compared to this m over tau hat, which is essentially m, tau hat is approximately 1. And just write this as 1 third m cube, plus things that are higher order than what I've kept, down by additional powers of epsilon. So now I'm interested in the susceptibility, the relationship between the applied field and the magnetization. Well, let's consider dH dm, so there's a leading term which is just epsilon differentiating 1 third m cube, gives me m squared. Now when the temperature is just above the critical temperature, there's no magnetization. So I'm interested in the limit of weak field when the magnetization is 0. So this is equal to epsilon when epsilon is positive, or temperature above critical temperature, since in that case m is equal to 0. But when the temperature is just below the critical temperature, then I have to use that m squared is equal to minus 3 epsilon. So it's epsilon minus 3 epsilon, which is minus 2 epsilon, when the temperature is less than the critical temperature. So that's the case in which m squared is equal to minus 3 epsilon. Now epsilon is negative here, so dH dm for non-zero epsilon is positive on either side. If I'm interested in the susceptibility without these renormalization factors, keeping in mind that I define little h to be mu over tau times big h and little m to be big m divided by n mu, that means the magnetic susceptibility, the derivative of big m with respect to big h at constant temperature has a factor of n mu squared over tau times d little m dh, and the little m dh is just the reciprocal of what we just derived dH dm. Now when we're close to the critical temperature, where our approximations are valid, that is small epsilon, then I can say the susceptibility, derivative of magnetization with respect to applied field at constant temperature, well, I can replace t by tc in this pre-factor since t is close to the critical temperature. So n mu squared over critical temperature times either epsilon, which is tau minus tau critical, oh, wait a minute, no, that's good. Tau minus tau critical over tau critical to the minus one, since here we have dm dh and this was dH dm, so one over epsilon, or one half of tau critical minus tau over tau critical to the minus one, depending on whether tau is greater than tau critical or tau less than tau critical. So first of all, the susceptibility diverges as the temperature goes to the critical temperature. It goes like absolute value of epsilon to a power, usually that power is called minus gamma, and we found gamma is equal to one in this model. We've also found that there's a difference in slope as the inverse susceptibility approaches the critical temperature depending on whether we approach from below or above, so that inverse susceptibility as a function of temperature goes to zero at the critical temperature and has a negative slope for tau less than tau c, positive slope for tau greater than tau c, and here the slope is twice as big as here. I probably exaggerated that a little bit, but slope twice as big. And we have this pre-factor and mu squared over tau c. Well, of course, that depends on the model. The pre-factor is model dependent, but the prediction that gamma is one is a more universal prediction. It applies to a whole class of models that are qualitatively like this one, models in which the relationship between m and h is given by a cubic equation. So it's the same kind of behavior we had in the Van der Waals model, including the factor of one half difference in slope question. Yeah, so this is the inverse of tau minus tau c, and this is the inverse susceptibility as a function of temperature, and that's where the factor of two slope is. Equivalently, there's a factor of two in the difference in the coefficient of the divergence going like one over tau minus tau c. Okay, so as you might expect, there's sort of a more general way of looking at these things, which is kind of powerful and therefore useful. Let's talk about that for a minute. It's called the Landau theory of phase transitions, which encompasses the types of predictions that we got from both models. The, our simple model of ferromagnetism in our Van der Waals model of the gas-liquid transition. And what the Landau theory does is it clarifies really what the underlying assumptions are that we need in order to get these predictions. So we're going to consider, take the following point of view. The free energy, we can think of as a function of the magnetization if we constrain the magnetization to have a particular value. And then the preferred magnetization, the most probable configuration, the equilibrium configuration will be determined by minimizing that function, okay? So magnetization as a function of temperature determined by minimizing a function of magnetization and temperature, the Helmholtz free energy, okay? Now we've seen that the kind of characteristic thing that happens in phase transitions and second order phase transitions is that quantities, observables, that you can measure behave in some singular way as a function of the temperature or a function of other parameters we can control. So in particular, the, I almost kicked it over before the class was done. The magnetization turns on like a square root. It's not an analytic function, add t equals tc, okay? It has a square root branch cut there. And the susceptibility has a divergence as t goes to tc. So those are both examples of singular behavior. We wanna understand the origin of that singular behavior. So this has a square root singularity in particular in the models we've studied. But the idea of Landau theory is, yeah, well, it's often the case that unless there's some reason to the contrary, that functions that arise in physics have a power series expansion, a Taylor series expansion. And if we consider expanding the magnetization as a function of the temperature around t equals tc, it doesn't. So that's what I mean when I say it's singular. Okay, so just emphasize the generality of the predictions. I'm gonna consider m to not necessarily be the magnetization, but just some quantity on which the free energy depends. But what we'd like to understand is how the singularity arises. But the assumption of Landau theory is that f itself is an analytic function. Analytic just means we can expand it in power series. i.e. has power series expansion, either temperature equals critical temperature, or about temperature equals critical temperature, and m equals zero. And so there's nothing non-singular about the Helmholtz free energy as a function of temperature or magnetization. But singularities can arise because the most probable configuration is found by minimizing this function, f. So I'm going to consider, in place of the magnetization, something called psi, but you can think of it as the magnetization if you want to. I'm gonna call it the order parameter of our system. It could be equals m equals magnetization. It could be the difference between density of liquid and gas, for example, in our Van der Waals model, the gas-liquid transition. The characteristic thing about the so-called order parameter psi is that when the temperature is greater than the critical temperature and there's no applied field, our applied field means something that we can turn on to control the system, like the magnetic field in our model of ferromagnetism. The minimum of the free energy as a function of psi and tau occurs at, I'll call it, psi zero equals zero. The minimum occurs at zero value of psi, the order parameter. And on the other hand, when the temperature is less than the critical temperature, then free energy is minimized at some non-zero, while psi equals a non-zero value. So you can think of it as the magnetization. There's either spontaneous magnetization or there is not. The reason we call it the order parameter is that the low temperature phase is more ordered than high temperature phase. So again, in the case of magnetization, what we mean by order is that there's spontaneous magnetization at zero applied field and the spins tend to point in the same direction. We call that long range order. What's long range about it is that when the system is magnetized, you can look at any small region of the magnet and if the spin is up in this small region, then it's very likely to be up in this region as well, even when they're far apart from one another. Because in the spontaneously magnetized phase, the spins globally all want to line up in the same direction. When the temperature is below, that's what happens when it's below the critical temperature when it's above, then the spins tend to choose random directions. Nearby spins might be correlated, but as you go farther away from one particular spin, it's like the system forgets whether this spin were up or not. So when these two are far apart, when this spin is up, it doesn't give you much information about what to expect for this spin. As they get arbitrarily far apart, they become uncorrelated. So when temperature is greater than critical temperature, spins uncorrelated at large distance, can we say that's the disordered phase? When the temperature is below the critical temperature, spins correlated even at large separation. And so we speak of the ordered phase. In fact, I'm going to simplify things by assuming there's a symmetry between taking the order parameter to minus itself, so that f of psi tau equals f of minus psi tau, or in other words, psi goes to minus psi symmetry. So if you flip over all the spins, that leaves the free energy unchanged. Well, that's what happens in our ferromagnetic example. The free energy doesn't really care about the difference between up and down when we turn off the applied field. There's a symmetry in that sense. This symmetry isn't really needed to reach the conclusions that we're looking for. But it makes the story simpler. So that's what we'll do. And so when we're in the low temperature phase, we say that the symmetry is spontaneously broken, or symmetry is broken. Physics doesn't seem to care if we flip the spins over. But if you look at one of the homogeneous phases, it's either going to have spins up or spins down. So if, say, it's a spin-up phase, the spin-up phase doesn't have the symmetry, right? It's not unchanged when you flip the spin over. It becomes the other homogeneous phase. So if you look at one phase or the other, the symmetry seems not to be present. There's a difference between up and down. The spontaneously magnetized phase with spin-up doesn't respect the symmetry between spin-up and spin-down. What happens instead is that there are two homogenous phases and they're mapped to one another under the symmetry. So that's what we mean when we say the symmetry is broken in either one of the homogeneous phases. Whereas if oxides zero equals zero, then we say symmetry is manifest. There's just one state, and if you turn all the spins over, it looks the same. There's a homogeneous phase in which there's no spontaneous magnetization. And so if you flip all the spins over, and so if you flip all the spins, you're still in the same phase, okay? Okay, so what we wanna do is really quite simple. We just wanna take the Helmholtz free energy, expand it into power series, and see what the consequences are when the temperature is close to the critical temperature. So first of all, I consider free energy as a function of the order parameter in the temperature and expand it around xi equals zero. So there's a constant term, which depends on the temperature. Because of the symmetry, I only have even terms, not odd terms. So there's no linear term. There's a quadratic term. It has some coefficient that depends on temperature. And there's no cubic term because of the symmetry that there's a quartic term with some coefficient that depends on temperature. And so on, higher powers of xi, the order parameter. Now, to understand the second order phase transition, let us just consider, yeah. So is that for a... Well, it's for either one because I haven't told you what the function is, okay? So for any value of the temperature, I can imagine expanding the free energy as a function of xi about xi equals zero. Then I'll have coefficients that depend on the temperature in that power series expansion. This is always symmetric. That's right. So the free energy itself is a function which has manifest symmetry. The breaking of the symmetry occurs when we find its minimum. As there'll be two different minimum when we're in the low temperature phase, neither of which has the symmetry by itself, rather the two are related by the symmetry. And there'll be a spin-up solution and a spin-down solution, positive and negative magnetization. Neither one by itself is invariant when we flip all the spins over. Instead, the two phases are mapped on another under the symmetry. That's what broken symmetry means. I mean, technically, maybe I cause confusion by not saying it this way. We usually speak of spontaneously broken symmetry. I mean, the symmetry is broken only by the state of the system, not by the free energy function itself. I could break the symmetry explicitly by adding terms which are not even in psi. Okay, but I'm not doing that. I'm gonna stick with the symmetry in the Helmholtz free energy function. And the breaking of the symmetry will arise only from minimizing it in the low temperature phase. So what corresponds to a second order phase transition according to Landau, is that the coefficient of xi squared, which I call g2 of tau has a zero. It's equal to zero at a special value of the temperature. We'll call that the critical temperature. Now, according to our analyticity assumption, we can expand in a power series in temperature or in the order parameter. So I can expand g2 in a power series around at zero at tau equals tau c. I can write g2 as some number alpha times tau minus tau c plus higher order in tau minus tau c. If it has a zero at tau equals tau c, there's no constant term, okay? Now, what, I mean in principle, alpha could be either positive or negative, and I'm just expanding some function. But in the cases that are physically relevant, alpha has a special sign, namely alpha is positive. In other words, g2 is greater than zero for temperature above critical temperature, and g2 is less than zero for temperature below critical temperature. Well, generically it will be. In principle, it might not be, but we're picking out a special value of tau where tau is equal to tau c, and so that will be one isolated point on the real line temperature because we're imposing one condition. When we impose that condition, then generically there's no reason for the second derivative to vanish as well, sorry, for the first derivative to vanish as well. We've set g2 equal to zero, alpha is its second derivative, and there's no reason in general why the function and its first derivatives should both vanish for the same value of tau. And for that reason, unless there's some additional condition, we don't expect alpha to be zero. In principle, it could have either sign. So in other words, if we consider the free energy as a function of order parameter, when the temperature is above the critical temperature, it's quadratic and positive for small psi, and then the quartic terms maybe kick in for higher values of psi. But when the temperature is less than the critical temperature, the quadratic term is negative. So it's negative for small psi, but then the quartic terms, which I'm all going to assume is positive, that's another thing that doesn't have to be true for a general function, but will be true in the typical cases of physical interest, not all cases. So g4 at tau c, positive. The quartic term is positive, and free energy as a function of temperature looks like this, when the temperature is less than the critical temperature. So we find the spontaneous magnetization, or the value of the order parameter, in other words, by minimizing this function. Because of the symmetry, there are two solutions, and they are related by order parameter goes to minus itself. That's the spontaneous breaking of the symmetry. There are two homogeneous phases, one that prefers positive values, the other prefers negative values of the order parameter. And right at the critical temperature, the free energy function is extremely flat. There's no quadratic term, only a quartic term, which is positive. So we can make a general statement under these analyticity assumptions, the assumptions that we can expand in power series, about how the magnetization, or how the value of the order parameter that's non-zero turns on when the temperature is just below the critical temperature. Well, how does non-zero value turn on for tau slightly below critical temperature? So I just write F for temperature near critical temperature. I can take G zero to just be a constant, evaluated at Tc, but I want to expand because the coefficient of the quadratic term vanishes at Tc to include the term that's linear in T in the coefficient of order parameter squared. And then I have one fourth G four side of the fourth plus still higher order inside. So if we then minimize the free energy, solving first derivative of F with respect to order parameter equals zero, that means I forgot the alpha, I guess. This was supposed to be the one half in or not, but of course it doesn't matter. What did I do in the notes? I guess I called this one half alpha minus tau c. So this should have a one half alpha. I put in the one half in the one quarter, so when we differentiate, those get canceled. So the condition that we're at a minimum when tau is less than tau c is alpha tau minus tau c or xi plus G four xi cubed equals zero. So there will be three solutions. X i equals zero, that's the unstable one, which is at the local maximum of our free energy potential. And the stable ones are the actual minima and they occur at values with the same absolute value, the same value of xi squared, namely alpha over G four times tau c minus tau or alpha is positive. And so the conclusion is that the minimum of the free energy occurs for value of the order parameter which turns on like minus epsilon to a power, beta, again epsilon is tau minus tau c over tau c and beta is just one half. The same thing we found in our ferromagnetic model and the Bander-Walds model. Just looked at from another point of view now. I should say plus or minus I guess because there are two solutions corresponding to the two homogeneous phases that can coexist. Now if I want to understand the susceptibility, I have to consider coupling some external field to our order parameter. Like we coupled the external magnetic field in the magnetic model, I'll call the external field lambda. And what we'll do is we'll just modify the free energy as a function of the order parameter by adding a linear term proportional to the applied field. So it's equal to what we would have in the absence. I don't know, maybe I should put a tilde on it. So this is the free energy when there isn't applied field without the tilde. It's the free energy when there isn't one which is why it doesn't depend on lambda but rather on tau. And then the leading behavior when lambda is small is a linear term minus lambda psi. The applied field breaks the symmetry. We have an even function of psi in the absence of the applied field. In the presence of the applied field there's no preferred direction. So there's a term linear in psi. And we call this coefficient the applied field. Well, it's the generic behavior when I break the symmetry by turning something on. Again, when the, I'm just doing a power series expansion when psi is small, there's a linear term which is the leading symmetry breaking term. It has some coefficient. I'm gonna call it the applied field. And I'm going to suppose that this linear term is something that I can adjust in the lab by turning a knob. And I'd like to know what the response of the system is when I make such an adjustment that explicitly breaks the symmetry. So what that's gonna do is it's going to favor one or the other of the minima that are present when the applied field is off. If lambda is positive, that means that we'll get a lower value of the free energy when psi is positive than when it's negative. So f tilde as a function of order parameter will have its global minimum. It'll have two local minima when lambda is small, but the global minimum will occur when the order parameter is positive. So that will be favored. And we'll just have one favored phase for a non-zero value of lambda. It's only when we turn off lambda that the two become equally likely. That is, they both have the same free energy. So this is what happens when lambda is positive, when lambda is negative, it's the mirror image where the negative value of psi is the favored solution. So we cross over as we vary lambda, when lambda hits zero, then we can have a mixed phase when the value of the order parameter is in between the two global minima when lambda equals zero. This is f tilde or f when lambda is equal to zero. Then the way to minimize the free energy for values of the order parameter between these two values, plus psi zero and minus psi zero, is to make a mixture of the positive and negative as i zero phase, and that corresponds to the dotted line. That's the coexistence region. Okay, so the equation that determines the order parameter in the presence of the external field is found by minimizing f tilde, or in other words, lambda is equal to partial derivative with respect to free energy, partial derivative of free energy with respect to order parameter at constant temperature. The susceptibility we define as the response of the order parameter to the applied field, so that's psi d lambda at constant temperature. That's just a second derivative. Well, actually, it's inverse is a second derivative of the free energy. Since lambda is df d psi, I can say that inverse susceptibility is second derivative of Helmholtz free energy with respect to order parameter at constant temperature. And so again, we can ask, how does it behave when the temperature is close to the critical temperature? So with this expression for the free energy, and I differentiated now twice with respect to psi, I guess I had already differentiated it once over here, so I'll differentiate it again. Here I differentiated it once. So that means that the inverse susceptibility is alpha tau minus critical temperature, and then differentiating the G4 xi cube term plus three G4 psi squared. Okay. Well, I could have made it 112, and then I would have, I mean, it's just a rescaling of G4. So if I'd made it 112, then I would write G4 here instead of three G4. Yeah, well, you know, the G4 is just a parameter, and it's one of the things that's not universal about the function. Different materials will have different values of G4 at the critical temperature. So if I really wanna know exactly what the susceptibility is in some particular material, I wanna know what G4 is, but if I wanna know what the model independent predictions are, then it's not gonna matter. Right. So again, we play the same game that I described earlier for the magnetic model. We can consider approaching the critical temperature either from below or from above. Well, I actually meant F because I determined, yeah, I guess to leading order, it doesn't matter, but I said that the condition that F tilde is minimized in the presence of an external field, this follows from D F tilde d psi equals zero. So I minimize the free energy in the presence of the external field, and that tells me how lambda is determined by the Helmholtz free energy. I'm actually gonna be interested in the response when the applied field is very weak. So I'm really expanding around lambda equals zero. So for this susceptibility, I can just imagine evaluating this derivative at lambda equals zero, in which case I don't have to worry about the distinction between the F and F tilde. And so now when we approach the critical temperature from above and we turn off the field, so lambda is zero, inverse susceptibility when lambda equals zero, lambda equals zero means psi equals zero, and it's just alpha times tau minus tau C, but tau greater than tau C. When tau is less than tau C, then even when I turn off this susceptibility, psi squared is non-zero, it's given by alpha over G four tau C minus tau. So chi inverse at lambda equals zero is equal to alpha tau minus tau C, but then I have to include the additional term plus G four times alpha over G four tau C minus tau. So these are opposite in sign, but this is three times as big. So it's two alpha tau C minus tau, when tau is less than tau C. So it's the same story we saw earlier. The inverse susceptibility goes to zero, linear in tau minus tau C. The slope as tau approaches tau C of the susceptibility depends on whether we approach from above or below. There's a factor of two difference. So again, we can say in particular that the susceptibility or the, well, I guess I'll say the susceptibility at zero field approaches absolute value of epsilon, the dimensionless deviation from the critical temperature to the minus gamma, gamma is equal to one. So the main thing we've gained by this analysis that we, oh, I see I told you I was gonna kick it over at the end, that we gained in this analysis that we didn't have before, is we see that these predictions really just follow from this assumption of analyticity. Sounds like a very general assumption. We can expand in a power series innocent enough. But I already let the cat out of the bag. I lectured two ago. These predictions would seem to be very general or wrong. It is true that for a big class of phase transitions we get exponents. Magnetization or xi turning on like minus epsilon to a power inverse susceptibility diverging like epsilon to the minus gamma where we get the same exponents, beta and gamma for lots of different phase transitions. But these universal exponents are not the ones we predicted for uniaxial magnets or gas-liquid transitions, beta is about 0.32. And for such transitions gamma is about 1.3. Whereas here we predicted from land-out theory beta equals one half, beta equals 0.5, and gamma equals one. So the analyticity assumption is actually wrong. The Helmholtz free energy is a singular function at the critical point. And a lot of work in modern theoretical physics has to do with understanding those singularities. It turns out that we can still make some fairly general predictions and that's what the homework is about. This week's homework is based on material that's not in the book. I tried to write the problem set so it would be kind of self-explanatory. But the main point of the homework problems is that we can make an assumption which can be true even when analyticity is violated about the scaling behavior of thermodynamic functions. And we can then express exponents like beta and gamma in terms of parameters that characterize that scaling behavior. And we can find relations among different observable quantities that are satisfied more generally even when land-out theory fails. And those predictions can be compared with experiment and they are successful. So actually let me just do this since I have arguably a minute or two. What I'm about to explain is actually also explained in the homework assignment but maybe it'll be useful to explain it in class two. I think I can do it in just a couple of minutes which sort of provides the prototype for how to solve these problems. Well, of course one half is also not an integer and land-out theory predicted 0.5 but we were able to understand that as arising from minimizing some function which was in effect a quadratic function. If we had gotten rational numbers we might have been able to explain them that way in some generalization of land-out theory. These aren't even necessarily rational numbers. And so the fact that they are weird numbers suggests that we can't easily understand what's going on by minimizing some function that you could easily guess. So the challenge of computing those functions is in some cases quite difficult. There are some successes and some failures in trying to compute them from first principles. I might say a little more about that next time. But the assumption that we can, in the end, justify and extract predictions from is the so-called scaling hypothesis. And it says something about how, if we're interested in considering the behavior as a function of the applied field and the deviation from the critical temperature, it's convenient to talk about the Gibbs free energy which is explicitly a function of the applied field. So I can consider, Gibbs free energy is a function of epsilon in effect the temperature and the applied field lambda. Consider the case where these are both small where we're considering a weak applied field and temperature close to the critical temperature. Well, I'll just say small. And then what we mean by scaling is that I can consider scaling the temperature, the deviation of the temperature from the critical temperature and the applied field by appropriate factors that are related to one another. Omega is just some dimensionless number which I'm using to keep track of how things scale. And this is what I mean by the scaling hypothesis that I can scale the Gibbs free energy by some factor, which I'm calling omega, by replacing the temperature and the applied field by new values which are obtained by multiplying the original values by powers of the scaling variable. And so suppose I wanna find the susceptibility. So just take it for granted that such a statement is true. And since we can write the order parameter as minus derivative of Gibbs free energy with respect to applied field, that just comes from looking at the, well, we've talked about it before how the Gibbs free energy changes when you make small changes. It's the same thing here. And that means that we can write the susceptibility which is the derivative of the order parameter with respect to the applied field as a second derivative of Gibbs free energy. Okay, so now all I gotta do is this. I look at this scaling equation, differentiate both sides with respect to lambda, set lambda equal to zero because I'm interested in the susceptibility at weak field. And so differentiate twice with respect to lambda. And this becomes, I get two powers of omega to the q. When omega to the q, each time I differentiate with respect to lambda, I'm then gonna set lambda equal to zero. So then I get this susceptibility which is the derivative of this function with respect to it's a second argument twice. Set lambda equal to zero, so it's a function only of the temperature, omega to the p times epsilon. That's what I get when I differentiate the left-hand side twice. If I differentiate the right-hand side twice, then, well, this just becomes the susceptibility as a function of deviation of temperature from critical temperature. So that tells me something about the susceptibility. How do I use it? Well, I can choose this scaling variable to behave any way I want. So when the temperature changes as I vary epsilon, no one can prevent me from varying capital omega as well. So I can choose capital omega so that omega to the p, epsilon is equal to a constant as I vary epsilon. So then this just becomes a constant. And then the statement becomes that this susceptibility scales, if I divide by omega, like omega to the 2q minus one. What good is that? Well, because I know how omega scales as a function of the temperature. Omega goes like one over epsilon to the one over p. So that means I can write the scaling of the susceptibility with temperature by substituting in omega goes like one over epsilon to the one over p as this goes like one over epsilon to the 2q minus one over p. So that's an example of how I can express the scaling behavior of some observable quantity, in this case, the susceptibility, in terms of the parameters p and q in the scaling hypothesis. We have two parameters p and q and they in principle could be anything but there are more than two things we can compute in terms of them. So they're non-trivial relations and so you'll work those out in the homework. Okay.