3 Phase Transitions and Critical Phenomena

Transcription

1 3 Phase Transitions and Critical Phenomena 3.1 The Ising Model Ferromagnetism is an important phenomenon in the study of solids. Certain metals, including iron, spontaneously develop a finite magnetization at ordinary temperatures. Above the so-called Curie temperature, however, these systems exhibit randomly oriented spins. Only below this temperature is a permanent, spontaneous magnet formed. It is also found that the heat capacity of such magnetic systems diverges near the Curie temperature. The Ising model is a seemingly very simple model that was developed to understand such behavior. Although developed specifically in connection with ferromagnetism, the Ising model has proven to be a very model that can also be applied to such diverse systems as interacting gases, simple binary liquid mixtures, and alloys. It also turns out to be far from simple to solve! We ll discuss one approach to solve the Ising model, as well as to understand phase transitions in general. The following Hamiltonian represents a simple model for a paramagnet, a material that does not exhibit a spontaneous magnetization in the absence of a magnetic field, but which responds by developing a magnetization in the presence of a field: N H = h s i, (3.1) where s i = ±1 for up and down magnetic spins. The number of microstates with a given magnetization M = N N is then i Ω = N! N!N!. (3.2) The entropy is then S = Nk {log2 12 (1+s)log(1+s) 12 } (1 s)log(1 s), (3.3) where we have used a version of Sterling s approximation logn! N logn N, (3.4) and the fact that N, = N (1±s)/2. Here, s = M/N is the average spin. Foragivenmagneticfieldh, theenergyofasingleup/downspinisǫ, = h. Since these are the only two possibilities, the probability that a single spin is up/down is given by e ±βh /Q, (3.5) where Q = e βh +e βh, and β = 1/(kT). Thus, the average spin s = eβh e βh Q = tanh(βh). (3.6) 3-1

2 The important physics that this simple model leaves out, of course, is the fact that the spins do interact. One must take interactions into account in order to understand transitions between phases. The interactions in a ferromagnet tend to align the a single spin with its neighbors. In the Ising model, we simply add to the Hamiltonian above a term that favors alignment of neighboring spins: H = h i s i J i,j s i s j. (3.7) Here, we have assigned an energy ±J for each pair of neighboring spins, depending on alignment. For J > 0, the energy is lower for spins that are aligned. This is what we expect for ferromagnets. The sum in this expression is over all pairs of neighboring spins ( bonds ). Note that each pair of spins that interact should only be counted once. For a regular square lattice in two dimensions, there are four nearest neighbor spins, corresponding to spins to the left and right, as well as above and below a single spin. In d dimensions, there will be z = 2d neighbors. One approach to solving the Ising model is to consider the average or mean effect of the z spins neighboring a single spin s i. This approach is known as mean field theory. We can write the Hamiltonian in the following way, H = h s i 1 2 J ( ) s i s nn, (3.8) i i nn where nn refers to the z nearest neighbors of spin s i. The factor of 1/2 here is introduced to correct for double counting pairs of spins, since each spin that is included in the first sum will also appear in the second, although each pair of spins should only be counted once. So far, no approximation has been made. The idea behind mean field theory is to treat the neighboring spins only in an average way. The part of the energy above that involves an individual spin s i can be written ( ) H i = hs i Js i s nn. (3.9) nn 3-2

3 Note that H H i, due to double counting of spin pairs. We approximate the term nn s nn by z s, since there are z nearest neighbors. This is a serious approximation, but we might expect that it is not a bad approximation if the number of neighbors is large, so the number of terms in nn s nn is large. The result looks just like the model for a paramagnet, except with an effective magnetic field h eff = h+zj s. In other words, the effect of the surrounding spins is similar to a molecular magnetic field acting on spin s i due its neighbors. With this approach, we obtain the average spin self-consistently using Eq. (3.6). s = tanh(β[h+zj s ]). (3.10) This can be solved graphically, with the result that the only solution for h = 0 and βzj < 1 is s = 0. In other words, there is no spontaneous magnetization. For kt < zj, however, nonzero solutions exist. Physically, these solutions correspond to states for which there is a spontaneous magnetization. We can solve this model in another way, that is more instructive. Within the mean field approximation, the energy per spin is approximately u = hs 1 2 zjs2, (3.11) where we have dropped the brackets denoting the average spin. What about the entropy? This we know from our solution of the paramagnet above. The free energy per spin in this approximation is therefore approximately given by f = hs+ kt zj s 2 + kt 2 12 s4, (3.12) for small s. In the absence of a magnetic field h, the minimum of this free energy with respect to the average spin s occurs for s = 0 when kt > zj. For kt < zj = kt c, there are two degenerate minima for s > 0 and s < 0. These states correspond to spontaneous magnetization up and down. These minima occur for f s = 0. Furthermore, near the transition, s 2 3(T c T)/T. (3.13) 3-3

4 Thus, near the transition the magnetization is given approximately by provided that T < T c. M ±N 3(1 T/T c ), (3.14) Near the transition, the heat capacity can also be calculated, since U N 2 zjs2 3N 2 zj (1 T/T c). (3.15) The heat capacity is finite, with a value C = 3Nk/2 at the transition. The numerical solution of Eqs.(3.10) and (3.15) yields a heat capacity that decreases to zero for T < T c. 3-4

5 It is also instructive to calculate the susceptibility within the mean field approximation as well. Above the transition (i.e., for T > T c ), we find for small applied magnetic fields h that s = h kt zj. (3.16) This comes from minimizing f in Eq. (3.12) with respect to s for small h. Then, the susceptibility diverges above the transition as where γ = 1. χ M h = N kt zj T T c γ, (3.17) Although we may, and should question the validity of the mean-field approximation above, the results derived above are illustrative of rather general 3-5

6 behavior of systems near critical points. In particular, the continuous power-law growth of magnetization below the transition, as well as the power-law divergence of the susceptibility are characteristic of critical points. Nevertheless, it is worth thinking ahead about what important physics this kind of approach leavesout. Byassumingthat the environmentofeachspins i canbe treatedbya single number representing the same average field, we are clearly leaving out any possibility of fluctuations. This can be an important effect, since a fluctuation that gives more up spins in one region will tend be self-reinforcing by making neighboring spins also flip up. In fact, we might expect that this reinforcing effect becomes most important near the critical point, since the susceptibility χ, which measures how easy it is to change the magnetization, diverges! More subtly, we will also find that precisely near the critical point, the spatial extent of correlations in the magnetization gets very large. I.e., the fluctuations involve more and more spins the closer one gets to the critical point. With hindsight, it is, in fact, surprising that mean-field theory works as well as it does. It practice, it is still the first approach that most people take in dealing with a new problem in phase transitions. 3.2 Landau Theory Arguably the first reasonably successful and general approach to understanding phase transitions goes back to Landau in the 1930 s. The basic idea here is simple enough; one assumes that that one can write the free energy (e.g., f above) as a smooth function of a measurable parameter (the so-called order parameter) describing the transition. In the case of the Ising model, the natural order parameter is the magnetization M, or equivalently, the average spin s. This is a particularly convenient choice, since it is zero in the disordered phase (the paramagnet) and non-zero in the ordered phase (the ferromagnet). By assuming that the free energy is smooth, we expect to be able to expand this free energy in a Taylor series in the order parameter, which we will denote as m. So, for instance, for a magnet at temperature T in the absence of a magnetic field, we expect to have f(t,m) a(t)+ 1 2 b(t)m c(t)m4 +, (3.18) were we have used the fact that the Ising model is symmetric under flipping all spins (changing the sign of m), even if the equilibrium phase may develop a spontaneous magnetization that breaks this symmetry. The free energy of any state should be the same if m m, unless we have a non-zero magnetic field that breaks this symmetry. This is, of course, the (approximate!) form of the free energy we derived above for the Ising model. Here, the various coefficients in the Taylor series are unknown functions of the remaining variable T. We do not know what they are, but by taking a cue from the Ising model above, we see that if c(t) > 0, then the transition will occur when b(t) changes sign. So, if we now imagine that b(t) can also be expressed as a Taylor series about T c, 3-6

7 at which it changes sign, then we find and similarly, b(t) b 0 (T T c )+, (3.19) c(t) = c 0 +c 1 (T T c )+. (3.20) Near the critical point, provided that c 0 > 0, we can find the equilibrium by minimizing the free energy to find which gives f m = 0 = bm+cm3 +, (3.21) m = ± b 0 /c 0 T T c β, (3.22) where β = 1/2. This is valid for temperatures just slightly below T c. (Apologies fortheuseofβ fortwothings. But, thereissimplynowayaroundthis: bothuses are too deeply imbedded in the statistical physics literature to do otherwise.) We could also calculate the behavior of both the heat capacity C and the susceptibility χ within this model (assuming an additional term hm in the free energy for small external magnetic fields). We would find essentially the same results as before, in that C remains finite with a discontinuity, and χ diverges with the same exponent γ as above. Since all we have really assumed here is that the free energy is smooth (analytic) as a function of the order parameter (here, m), this suggests that all phase transitions (with the same symmetry as the magnet) should have the same behavior, e.g., in m, C, and χ, at least near a critical point, where the order parameter is small, allowing for Taylor series expansions similar to those above. You will probably not be surprised to hear that this is false, in that β 1/2 and γ 1; we have, after all, made very strong assumptions/approximations above. What you probably will be more surprised to hear is that experiments on phase transitions in many different materials show THE SAME values of β and γ. So, while Landau theory may fail in detail (e.g., in predicting values of β and γ), it seems to get one very important qualitative feature right: critical behavior is remarkably universal from one system to another. This universality in itself begs an explanation. We will begin to see why this is true. And, in the process, we will also learn something very deep about the nature of condensed phases of matter with many degrees of freedom: most of the microscopic details, such as the precise way in which molecules interact with each other, become largely irrelevant in determining the way systems behave at the macroscopic scale. In the development of the statistical mechanics approach to try to derive thermodynamic properties, this is simply something that we hope for. Critical phenomena provide a wonderful window into the origins of the emergence of macroscopic simplicity out of microscopic complexity i.e., the fact that only a few thermodynamic parameters are enough to describe the macroscopic behavior of systems that are horribly complicated in microscopic detail. This is one of the reasons we will spend so much time studying relatively rare critical points! 3-7

8 Before we try to go beyond Landau theory, it is worth noting a few more things about it. First of all, it is a sort of mean-field theory, in that we have only accounted for a single, average value of the order parameter m, and have not allowed for this order parameter to fluctuate, either from point to point within the sample, or from time to time at the same point. We can improve on this limitation by allowing for a spatially varying order parameter m(x). This will not, in the end, cure the above failures of Landau theory, although we will learn something useful about spatial correlations. There is another important success of Landau theory. It shows us that symmetry alone can determine whether a phase transition is first-order (like the melting of ice, in which there is a latent heat) or second order (like the development of ferromagnetism below the Curie temperature). Notice that if we do not have the m m symmetry of the magnet, we could not assume all odd terms in the Taylor series expansion of f vanish above. You can check for yourselves, for instance, that with a free energy of the form f(t,m) a(t)+ 1 2 b(t)m c(t)m d(t)m4, (3.23) the order parameter jumps discontinuously from one value to another as b(t) decreases below the value 2c 2 /(9d). (Here, d is assumed to be positive for stability.) While this may seem like a curiosity, it explains why the transition between isotropic (I) and nematic (N) phases of liquid crystals is first order: the symmetry of liquid crystals is different from magnets, and permits odd terms in the Landau theory (Chaikin and Lubensky 2000). Even when the symmetry of the problem does not permit odd terms in the free energy, it is still possible to have a first-order transition. An example of such a situation is what happens in the model above when the 4th order term is not positive: f(t,m) a(t)+ 1 2 b(t)m c(t)m d(t)m6 +. (3.24) Here, we assume that d > 0, which ensures that the model is stable (m does not diverge.) By plotting this function for various values of the parameters b, c, and d > 0, you can convince yourself that there can be a transition with a discontinuous jump in m (sign of a first-order phase transition). The other interesting thing about this model is that it exhibits a line of critical points that becomes a first-order phase boundary. This so-called, tricritical behavior is seen in the superfluid transition of 4 He- 3 He mixtures. As 3 He is added to 4 He, the continuous (critical) transition to a superfluid still occurs, but at a lower temperature. As more 3 He is added, however, this transition becomes first-order (discontinuous), with coexisting 4 He-rich superfluid (S) and 3 He-rich normal fluid (N). The dotted line indicates a line of critical points. 3-8

9 These examples show why Landau theory is so useful, even if it is not rigorous or accurate in its predictions of critical behavior. We will later return to the origins of this failure, and why it is particularly apparent at the critical point. But, it is worth reflecting one of the key assumptions of Landau theory, that the free energy is analytic. Although this seems like a very innocent assumption, it is actually dangerous. Imagine building up a description of increasingly complicated systems beginning a particle at a time. The kinetic energy of a particle is certainly analytic as a function of the momentum. Let s assume that the interaction potential describing the way particles interact is also analytic. As we add more and more particles, it would seem that the Hamiltonian for the system will be analytic for any finite number of particles. It might get very hard oreven impossible to solve this problem in practicefor more than a few particles (like 3, even). But, how can the free energy not be analytic? It will turn out that this failure of analyticity occurs (in general) in the thermodynamic limit, N. In fact, in a sense, there IS not real critical behavior for any finite system. More on this later. Of course, in practice, N A is a big number! 3.3 Critical Phenomena and Scaling Inthemiddle ofthe 20thcentury, anumberofgroupswereperformingevermore precise measurements of the behavior of magnets and other systems exhibiting critical behavior. These experiments kept pushing to temperatures closer and closerto the critical temperature, from above (t (T T c )/T c 0 + ) and below (t 0 ). In fact, these experiments can go to very low reduced temperatures t of order 10 3, or even 10 5 in some cases. These experiments consistently showed behavior very similar to what mean- 3-9

10 field theory or Landau theory predicted. Specifically, it was seen that C h=0,t 0 + C h=0,t 0 m h=0,t 0 χ h 0,t 0 + t α t α t β t γ χ h 0,t 0 t γ m h 0,t=0 h 1/δ sign(h). These are written in terms of magnetic properties. But, this generalizes easily to other systems, such as the liquid-gas transition, where, for instance, the density difference between liquid and gas plays the role of the order parameter m, while pressure is analogous to the magnetic field. Then, for instance, δ expresses the degree of pressure as a function of volume along the critical isotherm. Mean-field theory predicts that that α = α = 0, β = 1/2, γ = γ = 1, and δ = 3. Experiments on a variety of systems shows that, indeed, the exponents above and below the transition seem to be the same, but that the values are very different from those predicted by mean-field theory. For instance, α 0.1 (i.e., the heat capacity diverges!), β 0.3, γ 1.3, and δ 4 5. In order to understand this, let s begin by trying to fix Landau theory. Let s write the free energy in a simple, canonical form: f(t,h) = hm+ 1 2 btm cm4, (3.25) where we shall treat b and c as constants. The equation of state can be obtained by differentiation with respect to m: h = btm+cm 3 = m(bt+cm 2 ). (3.26) As t 0 +, we find that m = h/(bt), which means that χ = χ + t γ, (3.27) where χ + = 1/b, and γ = 1. Also, as t 0, m 2 = b t /c, so that m = b/c t β, (3.28) where β = 1/2. Clearly, it is the exponent 2 in Eq. (3.15) that leads directly to β = 1/2. Perhaps we should try h = btm+cm 3 = m(bt+cm 1/β ) (3.29) as an equation of state. Obviously, unless 1/β happens to be an even integer, this cannot make any sense unless we replace m 1/β by m 1/β. Likewise, perhaps we could correct the exponent γ 1 by the equation of state h = m(bt γ +c m 1/β ). (3.30) 3-10

11 This would give us the right divergence of χ as t 0 +. But, again, if this is to make sense, we must take the absolute value of t here in order to describe thing below the transition. In that case, however, we need to also account for the sign of t. Hence, we presumably should try h = m(±b t γ +c m 1/β ). (3.31) Unfortunately, there are real problems with all these attempts to correct Landau theory. Not only do we have nonanalytic behavior at the critical point t = h = 0, but we even have nonanalytic behavior away from the critical point along the critical isotherm (t = 0)! While divergences and even nonanalytic behavior are expected AT the transition, there is NO experimental evidence of any singularities on the critical isotherm for finite h. There is another way to do this that DOES work. Beginning again with the original, mean-field equation of state, we can rescale the whole equation by b 3/2 t 3/2 /c 1/2 : ( D h t = m ( ) ) 1/β m B t β ±1+ B t β, (3.32) where B = b/c, D = c 1/2 /b 3/2, β = 1/2, and = 3/2. As strange as this may appear, this CAN be generalized to arbitrary values of the two exponents β and in a consistent way, without introducing any singularities away from the critical point. For general values of these exponents, as t 0 +, D h t m B t β. (3.33) Thus, the susceptibility scales according to χ t β, or γ = β. Equation (3.32) can be inverted to obtain the reduced or normalized magnetization ˆm = m as a function of the reduced field B t ĥ = D h β t. This can be done graphically for the mean-field case of β = 1/2, for which we can solve ĥ = ˆm ( ±1+ ˆm 2). 3-11

12 In general, we write ) ˆm = W ± (ĥ, (3.34) where ± refer to the t > 0 and t < 0 branches of these solutions of ˆm in terms of ĥ. I.e., ( m B t β = W ± D h ) t (3.35) This suggests that if we plot m/ t β versus h/ t, we should see all of the data for T > T c collapse onto a single curve defining the function W +, while all of the data for T < T c should collapse onto another curve defining W, provided that we choose the right exponents β and. In fact, this works remarkably well. Starting with raw data of the form shown below, rescaling the data in this way results in two curves similar to the following. You should try this yourself with classic data obtained by Weiss and Forrer in 1926 (next page). 3-12

13 T (K) h (oe) M (emu/g) , , , , , , , , , , , , , , , , , , , , , , , , , , , Data taken from Statistical Mechanics, K. Huang (Wiley, New York, 1987). Original data taken from P. Weiss and R. Forrer, Ann. Phys. (Paris) 5, 153 (1926). These data were reanalyzed by Kouvel and Fisher (J.S. Kouvel and M.E. Fisher, Phys. Rev. 136, A1626 (1964)), who found γ = 1.35 and δ = The transition temperature is T c = Unfortunately, this is not the full data set. These data have been analyzed and re-analyzed many times in the literature. One such example is the following plot of scaled field versus scaled magnetization (Arrott and Noakes, Phys. Rev. Lett. 19, 786 (1967)). Although these particular data were obtained for Nickel, when experiments are performed on other ferromagnets, the same exponents AND scaling functions W ± are found, apart from possibly different scaling factors B and D! If that is not surprising enough, when experiments are performed on fluids (liquid-liquid phase separation, or liguid-gas transitions), not only is similar scaling behavior found, but again, the exponents and even scaling functions W ± are the same as for magnets! 3-13

14 The fact that our scaling hypothesis (actually, due to Widom) works has important implications for other exponents, such as γ. Since we can write m in terms of h and one of two universal functions W ±, means that we can now calculate the susceptibility. For t > 0, in particular, ( ) m χ = t β W + (0). (3.36) h h 0 ± Here, W + (0) refers to the derivative of W +. Thus, γ must be β. In other words, = β +γ. (3.37) A similar argument for t < 0 tells us also that γ = γ. (3.38) What about the critical isotherm, where t = 0? As we noted above, it is well established that the magnetization and magnetic field are both well-defined and 3-14

15 smooth away from the critical point, along the critical isotherm. But here, both ˆm and ĥ appear to be ill-defined (infinite!) at t = 0. Thus, the critical isotherm corresponds to the asymptotic regime of very large arguments of the functions W ±. How do these functions behave for large arguments? What if W(x) itself exhibits a power-law here? If W(x) W x λ as x, then m t β W h λ / t λ. (3.39) But, if this is to make any sense as t 0, then the t-dependence in the numerator and denominator must just cancel. I.e., λ must be β/. We can then find the relationship between m and h along the critical isotherm: m h λ, so that δ = 1/λ = 1+γ/β. (3.40) Moreover, the fact that the data collapse for ˆm and ĥ for the same above and below the critical temperature, we must also have that the exponent λ, as well as the amplitude W must be the same for W + and W. In other words, the functions W + and W must converge for large arguments. What we learn from all of this is that basic thermodynamics imposes constraints on some of the various exponents. Specifically, there only appear to be two independent exponents so far. In fact, the exponent α is also completely determined by β and γ. In order to see this, we note that m and h are conjugate thermodynamic variables, like volume and pressure for a gas. Thus f h = m t β W ± ( Dh/ t ). (3.41) We can integrate this (with respect to h), to find that f = mdh t β+ ˆmdĥ t β+. (3.42) Since we find that meaning that It can also be shown that C = T 2 f T2, (3.43) C t β+ 2, (3.44) α+2β +γ = 2. (3.45) α = α. (3.46) Thus, there appear to be at most just two independent critical exponents, say β and γ, from which the other four exponents can be derived. In other words, although the various critical exponents are not so simple as mean-field would predict, there seems to be some order among these exponents. 3-15

16 3.4 Why does scaling theory work? We are left with the question of why our basic scaling hypothesis works. One very basic observation we can make from the fact that we observe power-law behavior, e.g., the divergence of χ near t = 0, is that things look essentially the same as one moves closer to the critical point, i.e., as t decreases toward zero. This property is especially clear from Widom s approach to scaling: a magnet close to the critical point behaves just like one far from the critical point, apart from a simple rescaling of various physical quantities like h and M. This mathematical property, known as scale invariance, appears to be a key physical property of systems near critical points. In fact, scale invariance is a very important property of many physical systems, and scaling theories and approaches such as we have employed above can be applied rather widely. Consider a simple mathematical property of a function such as the susceptibility χasafunction oft. Thefact that weobservea power-lawdependence tells us that there is no intrinsic scale, e.g., for t. Contrast this with, for instance, an exponential dependence, for which there IS a characteristic scale. If χ were exponential in t, we would necessarily have a characteristic scale t 0 for t: χ e t/t0. Instead, χ is a homogeneous function of t of degree γ: χ(λt) = Λ γ χ(t), (3.47) for Λ > 0. Our magnetic system, however, depends on two thermodynamic variables, t and h. What if the free energy f itself is also homogenous in both of these variables? We ll examine the consequences of f(λ a t,λ b h) = Λf(t,h). (3.48) The reasonfor assumingthis form isthat, since ourcriticalpoint isat t = h = 0, for any finite t or h, Λ > 1 moves us farther away from the critical point (at least for a,b > 0). But, it may do so at different rates for the t and h. The assumption we are making here is that, apart from an overall scaling factor, the free energy is the unchanged. (By the way, our choice of exponent 1 for Λ on the right-hand side of this equation is not essential. If we replace this by another exponent c 0, we could redefine a and b so that the equation above is valid. There are, thus, only two independent exponents above, and we have chosen this particular form without any loss of generality.) The free energy can then be expressed as From this, we obtain the entropy f(t,h) = Λ 1 f(λ a t,λ b h). (3.49) S(t,h) = F T f t Λ 1+a f 1,0 (Λ a t,λ b h), (3.50) 3-16

17 where f i,j represents the i-th derivative of f with respect to its first argument, and j-th derivative with respect to its second argument. The heat capacity is then C(t,h) = T 2 F T 2 f 2 t 2 Λ 1+2a f 2,0 (Λ a t,λ b h). (3.51) The magnetization is given by m(t,h) = f h Λ 1+b f 0,1 (Λ a t,λ b h). (3.52) The susceptibility involves one more derivative with respect to h: χ(t,h) = m h Λ 1+2b f 0,2 (Λ a t,λ b h). (3.53) These relations can be used to find the various critical exponents as follows. For h = 0, if we let Λ = t 1/a, then where ± refer to t > 0 and t < 0. Thus, Similarly, for m and χ we find that and This means that and C(t,0) t ( 1+2a)/a f 2,0 (±1,0), (3.54) α = (2a 1)/a. (3.55) m(t,0) t (1 b)/a f 0,1 ( 1,0) for t < 0, (3.56) χ(t,0) t (1 2b)/a f 0,2 (±1,0). (3.57) Also, for t = 0, by considering Λ = h 1/b we find that Thus, β = (1 b)/a (3.58) γ = (2b 1)/a. (3.59) m(0,h) h (1 b)/b f 0,1 (0,±1). (3.60) δ = b/(1 b). (3.61) You should check that the above scaling relationships among these exponents (Eqs. 3.40, 3.45) are satisfied. 3-17

18 3.4.1 The story so far... So, we learn, yet again that there appear only to be two independent exponents, on which all the others depend. Beyond that, we now see that the number of these key independent parameters describing the critical behavior is connected to the fact that we have two (intensive) physical parameters describing the system. For the magnet, these are the temperature (t) and magnetic field (h). For liquid-gas systems, the corresponding parameters are the temperature and the pressure. So far, I have made things sound a bit simpler than they really are. In particular, I implied that the critical behavior of magnets is the same as for all other systems. This is not the case. For instance, uniaxial magnets, for which the magnetic moment points preferentially along one axis, differ in their critical behavior (exponents and scaling functions) from Heisenberg-like magnets, in which spins can point in any direction. Nevertheless, it IS true (and very surprising) that uniaxial ferromagnets and fluid systems such as liquid-gas systems should behave in the same way! What is now known (and understood) is that all systems with the same underlying symmetry have the same critical behavior. A liquid-gas system, for instance, can be described by the presence/absence of a particle, just like an up/down spin in the Ising model for (uniaxial!) ferromagnetism. So, the critical exponents depend on the symmetry of the problem, but NOT on any details of how the components interact: what could be more different in detail, after all, than magnetic spins and particles colliding! This lack of dependence on molecular details is a very striking aspect of critical behavior that begs for some explanation. While we shall focus our attention on scaling theory as an approach, it is worth noting that some relationships between the various critical exponents were found by other means. For instance, based on very general thermodynamic principles, Rushbrooke showed that must be true. Similarly, Griffiths showed that α +2β +γ 2 (3.62) α +β(1+δ) 2. (3.63) Predictions of scaling theory do not conflict with these general conditions. But, both of these inequalities become equalities within scaling theory. Not surprisingly, many people attempted to derive exact expressions for various exponents directly from thermodynamics or rigorous statistical mechanics. Occasionally, people even reported derivations that were at odds with the above general inequalities. But, these all suffered the same fate as various reports of violations of the second law! So, we areleft with twoburning questions: (1) whydoes scalingtheorywork; and (2) why is the behavior universal, in that molecular details do not matter. Perhaps the most important conceptual breakthrough that actually lead to a simultaneous answer to both of these questions was due to Kadanoff in mid s. The beauty of this idea lies in its simplicity and physical intuition. It 3-18

19 does not require, for instance, complicated analysis and calculation. The downside of this approach is that is is essentially conceptual. It was not possible to turn it directly into a practical procedure to actually calculate an exponent. That had to wait for the work of Wilson in the 1970s, for which he (Wilson) was awardedthe Nobel prize in We shall NOT go through the latter approach. But, we can learn a lot from the insights of Kadanoff, which we can understand based on just one more property of critical phenomena that we have not focused on so far: fluctuations. 3.5 Critical Fluctuations One of the most striking features of a critical point is the large fluctuations that become apparent there. This is often refered to as critical opalescence, because of the opalescent or milky appearance of otherwise transparent systems at a critical point. For instance, in the case of the liquid-gas transition, (very!) near the critical temperature and pressure, the sample appears cloudy. This is because the system tends to fluctuate wildly between the two phases, since they are nearly indistinguishable near the critical point. These fluctuations between phases amount to fluctuations in density. These fluctuations are both large in magnitude (of density), as well as of large spatial extent. The density fluctuations result in fluctuations in the index of refraction. This, together with the large length scale of the fluctuations (i.e., comparable to the wavelength of light) means that the sample scatters light strongly, as do clouds. Not only are these fluctuations strong at the critical point, but in a sense they provide the key to understanding the key features of critical behavior that we have mentioned so far: (1) the scaling behavior, (2) the anomalous (nonmean-field) exponents, and even (3) universality and the fact that microscopic details of the system become irrelevant. Let s try to generalize Landau theory to allow for fluctuations. The result is usually known as Ginsburg-Landau theory. We consider a spatially varying magnetization m(x), and generalize our free energy to include a penalty for these spatial variations. (If there were no penalty, why would we have any macroscopically homogeneous phases at all?) We write the free energy as [ 1 F[m(x)] d d x 2 tm2 (x)+ 1 4 um4 + 1 ] 2 ξ2 0 ( m) 2, (3.64) where we have used a slightly simpler normalization of the various terms in the original Landau theory. We have normalized by b and let u = c/b. We have also introduced a new parameter with dimensions of a length, ξ 0. This new free energy is actually a functional of the spatially varying m(x). Otherwise, this is still in the spirit of Landau theory, in that we are expanding the free energy in small quantities, now including gradients or derivatives of the field m. We are keeping only the lowest order derivatives, with the assumption that these will dominate at long length scales. 3-19

20 We are not so interested in a precise function m(x) so much as how the magnetization, or order parameter is correlated from one point to another. With this in mind, we begin with replacing m by its Fourier series/transform: m(x) = k m k e ikx. (3.65) Furthermore, we shall focus on the case of t > 0, where we can ignore the fourth-order term, at least if m remains small enough. We can then write the free energy F d d x 1 ( t+ξ (ik)(ik ) ) e i(k+k )x m k m k = V k 1 2 k,k ( t+ξ 2 0 k 2) m k 2. (3.66) Here, we have used the fact that d d x e i(k+k )x = Vδ k,k, (3.67) and the fact that m k is the complex conjugate of m k. By using the equipartition theorem, we then find that m k 2 kt t+ξ0 2 (3.68) k2. We can measure the correlations of the fluctuating m(x) field by the correlation function Γ(x) = m(x+x )m(x ). (3.69) 3-20

21 This correlation function must vanish for large separations x, since m = 0. Furthermore, it must also be independent of x. Thus, m(x+x )m(x ) = k,k m k e ik(x+x ) m k e ikx = k m k 2 e ikx. (3.70) Thus, we can find Γ(x) by Fourier transform. You might recognize the corresponding problem with t = 0, since the Fourier transform of the electric potential about a point-charge is proportional to 1/k 2. For t > 0, the result is Γ(x) e x /ξ. (3.71) x d 2 While writing most of the expressions above for one dimension (d = 1), this final expression exhibits the expected inverse dependence on separation x that is characteristic of the the electrostatic problem mentioned above. For finite t > 0, the correlation function also exhibits an exponential decay with distance x. The decay or screening length is ξ = ξ 0 t 1/2. This correlation length represents the typical size of a fluctuating domain (with correlated value of the order parameter, such as a magnetic domain in magnets). This correlation length diverges near the critical point with an exponent 1/2. Of course, this is what one would expect within Landau(-Ginsburg) theory. Real systems exhibit a somewhat stronger dependence ξ = ξ 0 t ν, (3.72) where the new (no pun intended) exponent ν 0.6. In fact, it is also observed that the exponent in the denominator of Eq is not the predicted one from Landau-Ginsburg theory: Γ(x) e x /ξ x d 2+η, (3.73) where η is observed to be non-zero, but small. So, what do these fluctuations tell us about the above puzzles of critical behavior? Well, the most important thing to note is that the correlation length not only can be large, but strictly diverges at the critical point. 3-21

22 3.6 Real-space Renormalization Since the correlation length ξ diverges near the critical point, if we look at the Ising model near the critical point, we expect to see neighboring spins to be highly correlated. This suggests that we can consider larger blocks of spins as effectively acting in concert. We introduce the concept of block spins, where groupsof spins are thought of as acting like a single spin. Consider, for instance, a2-dlatticeofspinsindicatedbys i, whichtakeonvalues±1. Ifwenowconsider squares of L spins on a side, which we define to be a block, then we can treat the resulting lattice of block spins as a magnet. (In the figure, L = 2.) We define these block spins S I = ( i s i)/l 2, which takes on values between -1 and 1. Provided that L is small enough so that the block spins are still smaller than ξ, we expect that the system is not fundamentally altered, i.e., that it still looks likean Isingmodel. Thisis becausethe spinsin eachblockarehighly correlated. This procedure amounts to a transformation of the original Hamiltonian from H = h i s i J i,j s i s j (3.74) to H = h I S I J I,J S I S J. (3.75) Here, we might expect that h L 2 h, since there are L 2 spins in a single block. Note that on scaling up by a factor of L, this amounts to moving AWAY from the criticalpoint. This alsomakessensefrom the point ofviewofthe correlation length ξ, since under this rescaling of the system, ξ decreases by a factor of L: ξ ξ = ξ/l. Given that ξ = ξ 0 t ν, this also corresponds to moving away from the critical point, as characterized by t t = L 1/ν t. All of this can be generalized to d dimensions, by noting that upon rescaling the system by a factor of L, we expect that the system behaves as though both t and h increase according to t t = L x t, (3.76) 3-22

23 and h h = L y h. (3.77) Here, as noted above, we expect that x = 1/ν 2 and y d, the number of dimensions. This is an expression of the scale invariance we discussed above. We really have not derived anything yet. We are simply arguing for a simple rescaling of the parameters t and h by factors that we might expect to be powers of L. We argued for this by appealing to the fact that near the critical point, since the range of correlations is large, we expect to be able to lump large groups of spins together to form blocks, resulting in a simple geometric increase in the effective field strength h. Likewise, the rescaling results in a simple decrease of the correlation length, as measured in the new lattice units. This is equivalent to an increase in t by a factor of L 1/ν. These arguments really only apply near the critical point, where the correlation length ξ is sufficiently large that we still have correlated block spins after scaling the system up: ξ is still larger than a lattice spacing. We can also see that the scale factors relating t to t and h to h must be simplepowersofl. Thisisreallyaconsequenceofscaleinvariance. Ifoursystem really is scale invariant, meaning that it still looks like an Ising ferromagnet near the critical point as we rescale our system by a factor of L, then we expect to have the same form for the Hamiltonian H and free energy F after rescaling. Consider, for instance, a rescaling of the system by a factor of L 1 followed by another rescaling of the system by a factor of L 2. Assuming that the effective temperature simply rescales by some factor φ that depends on L 1 : t t = φ(l 1 )t. (3.78) If we then follow this with another rescaling of the system by a factor of L 2, then t t = φ(l 2 )t = φ(l 2 )φ(l 1 )t. (3.79) But, this should be the same as if we simply rescale the original system by a factor of L 1 L 2 : t = φ(l 2 L 1 )t. (3.80) This is satisfied by any simple power φ(l) = L x. Likewise, we also expect the scaling of h to involve a (possibly different) power L y. This composition property of these rescaling/renormalization transformations suggests a grouplike structure, where following one transformation with another corresponds to a single transformation by a product of the scale factors. In fact, these operations do NOT form a group in the mathematical sense, since these transformation cannot, in general, be inverted: we only lose information in a renormalization transformation. 3-23

24 It is very interesting to think about how this loss of information happens. Let us say, for instance, that in our original system/lattice we have some free energy that depends on a collection of parameters: t, h 1, h 2,... Then, the arguments above suggest that near the critical point, these all transform as h j h j = upon rescaling. If the exponent here y Lyj j happens to be less than zero, then the corresponding parameter becomes irrelevant near the critical point. In fact, it becomes increasingly irrelevant near the transition, since the large correlation length means that we can rescale by a large factor, making h j small. This is one of the reasons why most of the microscopic details of the system become increasingly irrelevant near the transition. We actually only care about those parameters for which y > 0. This also begins to explain why we have universality, and why it is valid only close to the critical point. So, what does this tell us about scaling? Consider the free energy f(t,h). Under a renormalization transformation, we have argued that t L x t and h L y h. Thus, we expect that f(t,h) f(l x t,l y h). But, we have to be careful here. Since the free energy is an extensive quantity, depending on the size of our sample, we must account for the fact that our sample effectively shrinks under this renormalization: there are fewer block spins than original spins. Thus, it is more appropriate to describe the free energy per spin. Since we have L d original spins in a block, the free energy per spin in the new lattice must be f(l x t,l y h) = L d f(t,h). (3.81) This is just like Eq. (3.48) above, with Λ = L d. Thus, a = x/d and b = y/d. Using Eqs. (3.55), (3.58), (3.59) and (3.61), we also obtain the various critical exponents in terms of x and y. We also have, from above, that which also means that ν = 1/x, (3.82) α = 2 dν. (3.83) This illustrates once again that there are only two independent critical exponents. This is even true if we consider the exponent η above, since it can be shown that γ = (2 η)ν. As it stands, we now have a concrete theoretical procedure/approach to quantitatively understand critical behavior. If we can learn about how Hamiltonians and free energies transform under the rescaling/renormalization transformations outlined above, then we can, for instance, extract x and y. This is 3-24

25 more or less what is done. But, this procedure is complicated to do in practice (one needs to consider transformations in the space of Hamiltonians), and the so-called real-space procedure of Kadanoff is not actually a practical one. Nevertheless, we have at least conceptually identified the origin of scaling, and closely associated aspects of critical phenomena such as universality. It remains now to explain just why/when mean-field theory fails, and when/if it can ever be valid. Before doing so, however, we an already learn something about this from what we have just derived. In particular, there is something very odd about the scaling relation in Eq.(3.83). It depends on dimensionality. All of the mean-field exponents are independent of dimension. On the one hand, this would appear to tell us that mean-field theory CANNOT be right. On the other hand, we also learn that there is a special dimension, d = 4, for which Eq. (3.83) is satisfied for mean-field theory. This is an important point that we shall return to: four dimensions is a special case. In fact, for any dimension d > 4, mean-field theory works! We are either incredibly unfortunate, or lucky that we live in a world with funny critical exponents. I would argue that it is LUCK. The developments of phase transitions and critical phenomena would have concluded with Landau theory, and we would not have learned anything about scaling, renormalization, etc The Breakdown of Mean-field Theory We have seen that fluctuations play an important role in the behavior of systems near a critical point. These fluctuations are also the cause of the breakdown of mean-field theory. We can see this by making an estimate of the size of, e.g., magnetization fluctuations in a magnet. If these fluctuations are larger than or comparable to the average value of the magnetization, then clearly the whole idea behind mean-field theory is questionable at best. In order to estimate the magnitude of the fluctuations, we consider the case of t < 0, where we expect to find a non-zero average of the order parameter m. From the Ginsburg-Landau theory in Eq. (3.64), for t < 0 we expect to find m 2 = m 2 0 = t/u = t /u. About this value, we expect to find fluctuations δm(x). The free-energy can then be expressed as F[m(x)] F 0 + d d x [ t δm(x) ] ξ20 ( m)2, (3.84) where we have expanded the integrand about the minimum at m = ±m 0. We have also indicated by F 0 the mean-field value of the free energy, i.e., the free energy for a uniform m(x) = ±m 0. Note that d 2 dm 2 [ 1 2 tm um4 ] m=±m 0 = 2 t. (3.85) We can estimate the remaining integral by taking very seriously (of course, a little too seriously in detail) the idea of the correlationlength ξ, and the figure 3-25

26 above. We ll assume that at any given time, the sample can be decomposed into regions of size ξ that fluctuate independently. Because the field δm is strongly correlated over distances smaller than ξ, we will treat the field as uniform within a single region. Furthermore, these regions effectively behave independently of each other since the the field δm is weakly correlated over distances larger than ξ. Thus, we can calculate the contribution to the free energy above of one of these regions as F ξ d t δm 2. (3.86) We have neglected the gradient term in this free energy, but you can convince yourself that this is of the same order as the above, since m 2 δm 2 /ξ 2. This means that the probability distribution of δm is given by the exponential of F/kT. In fact, we have actually normalized the free energy by kt. Thus, δm 2 t 1 ξ d t dν 1 ξ d 0. (3.87) But, for mean-field theory to make sense, we have to have that δm 2 /m Thus, we must have t dν 2 ξ d 0/u, (3.88) since m 2 0 = t /u. Of course, ν = 1/2 within mean-field theory, so that this condition becomes t d 4 ξ0 2d /u 2. (3.89) This can always be satisfied close to the critical point for d > 4, but cannot be satisfied close to the critical point for d < 4. In three dimensions, this can only be satisfied for t > t 0, there t 0 (u/ξ d 0) 2. (3.90) This shows us that mean-field theory cannot be valid arbitrarily close to the the critical point. As t 0, the fluctuations will always dominate the average, 3-26

27 or mean field. But, depending on the system in question, it may be possible to satisfy the inequality above for even very small t. This can happen especially if the bare correlation length ξ 0 is large enough. This is the case, for instance, for the transition to superconductivity in many metals. This is because the natural small distance, and therefore ξ 0, in this problem is the size of a Cooper pair, which can be much larger than the atomic/molecular scale. 3.8 Finite-size Scaling We have found that physical quantities such as the heat capacity and susceptibility diverge, seemingly to infinity at the critical point. It is natural to ask, in what sense can these things be infinite. In fact, in any finite system, the heat capacity and susceptibility remain finite and continuous through the transition. The divergence is, strictly speaking, only in the thermodynamic limit that N, where N is the number of degrees of freedom. But, Avagadro s number N A is BIG. For all practical purposes in most experiments, these quantities appear to get ever larger near the critical point. If one were to perform experiments on much smaller samples, say with N 10 6 or less, one would see that these divergences are rounded off. One place where one definitely deals with small systems is numerical simulation. Here, one must deal with systems that have many fewer degrees of freedom than N A. In fact, simulation systems are usually so limited in size it is even hard to recognize what would be critical phenomena in much larger systems. This raises a challenge for numerical simulations to see critical phenomena. This is a particular problem with second-order phase transitions, since the correlation length ξ diverges. When ξ becomes comparable to the size of the system, the behavior begins to deviate strongly from the thermodynamic limit. One can actually use these finite-size limitations to ones advantage. Since the correlation length diverges as ξ ξ 0 t ν, for a finite system of linear di- 3-27

28 mension L, once ξ becomes comparable to L, the system cannot get any more critical than it already is. Again, we identify the divergence of ξ as the defining characteristic of criticality, and once ξ reaches the full system size, we have correlated fluctuations that span the whole system: ξ is as big as we can measure! We expect this to happen when t t L = (L/ξ 0 ) 1/ν. In practice, this will mean that the divergence of χ is rounded off and rendered finite in a region of order t L on either side of the transition. Thus, we expect that for χ χ 0 t γ in an infinite system, one should observe a maximum value χ max L γ/ν. This is something that can easily be checked in numerical simulations on different size samples. What is more, if we consider the normalized susceptibility χ/l γ/ν, it should approach a constant, independent both of t and the size of the system, below t of order t L. Only for t > t L should we see the critical behavior, and here χ should be independent of L. Thus, χ(t,l) should satisfy ( ) L γ/ν χ = φ L 1/ν t, (3.91) where φ(x) is a function that approaches a constant for small x and must vary as x γ for large x. Thus, by plotting L γ/ν χ versus L 1/ν t, one should see a collapse of data obtained for different values of L onto a single scaling curve φ (actually, two curves φ ± for t > 0 and t < 0), provided that γ and ν are chosen appropriately. A very similar approach is also possible with the order parameter m. If L β/ν m is plotted versus L 1/ν t (for t < 0), one obtains data collapse onto a universal function that approaches a constant for small argument and increases as a power law with exponent β for large arguments. 3-28