Physics 212: Statistical mechanics II Lecture XI

Physics 212: Statistical mechanics II Lecture XI The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is fixed up, m(k) C 1 βj(1 R 2 k 2 ). (1) Its Fourier transform is m(r), which we argued above to be equal to G(r), the correlation function between two spins separated by r. First rewrite it as G(k) C KR 2 k 2 + 1 K = CR 2 k 2 + tr 2 = CR 2 k 2 + ξ 2 (2) with ξ = Rt 1/2, and t = (T T c )/T. The result ξ = Rt 1/2 defines the critical exponent ν = 1 2 for the growth of the correlation length near criticality. The Fourier transform described above, which in general dimensionality can be written in terms of Bessel functions, gives that the correlation function on long length scales falls off as (now forgetting about units; additional prefactors of C and the lattice spacing R must be added to get the units right) G(r) e r/ξ /r (d 1)/2 (3) starting from G(r) CR 2 d d k exp(ikx) k 2 + ξ 2 = CR 2 ξ 2 d Actually G(r) has two different asymptotic limits in MFT: for large r G(r) { e r/ξ d d k exp(ikx/ξ) k 2. (4) + 1 if r ξ r (d 1)/2 1 if r ξ. (5) r d 2 (Huang describes this Fourier transform in one problem but has a wrong power in the first limit.) Note that the first limit in d = 1 matches our result for the correlation function of the onedimensional Ising chain, in the limit of large separation. With this information about correlations, we are able to justify our earlier claim about the validity of mean-field theory above four dimensions. The approach will be to estimate the neglected part of the Hamiltonian and compare it to the part we found in the mean-field calculation. This procedure is carried out for lattices in the book of Cardy; instead of working with the lattice mean field theory, we now will introduce a phenomenological approach to phase transitions in the continuum known as Landau theory. The continuum approach to phase transitions is a rare example where phenomenology can give exactly correct answers even for certain numerical quantities, such as critical exponents. For now, think of this phenomenology as motivated by a desire to describe continuum systems such as the liquid-gas transition in water, which was claimed in Lecture I to share some properties, including critical exponents, with the (lattice) Ising model. 1

We start off by following Landau and conjecturing that the free energy density near a phase transition can be expanded in powers of the order parameter describing the transition. For the Ising transition, w choose the magnetization to be the order parameter: the requirement is that the symmetry that is broken at the transition should transform the order parameter from one ordered state to another. In terms of the dimensionless magnetization, we can write this expansion as F (m) = m H kt + 1 2 r 0m 2 + s 0 m 3 + u 0 m 4 +.... (6) Remark: We will later on want to consider spatially varying configurations and also fluctuations: let me quickly introduce a different way of looking at the Landau free energy. We assume that near the critical point, the coarse-grained spin m(r) varies only slowly for the lowest-energy configurations. Hence we guess that the Gibbs free energy in zero magnetic field takes the phenomenological form, ψ(m(r), H(r)) = 1 2 m(r) 2 + 1 2 r 0m(r) 2 + u 0 m(r) 4 +.... (7) Here the coefficients are unknown functions of T, and m is rescaled so that the first term has coefficient 1/2 (see next paragraph). The purpose of this is that the partition function can be written in terms of an integral over functions m(r): Z(T, H) C (Dm(r))e β d d r ψ(m,h). (8) This is similar to the improved derivation of MFT, where we first carried out the spin sum to obtain a free energy as a function of the uniform variational parameter m; the Landau free energy (1937) is essentially a phenomenological guess for a general form of the free energy. Choosing the coefficient of the gradient term to be 1 2 is just a convention that effectively defines the scale of m: if we started with a different coefficent, then we could rescale m to set it equal to 1 2. Note that here m is not even bounded between 1 and 1, although we will assume u 0 positive so that configurations with m large in magnitude are quite unlikely. This is known as a soft-spin free energy since there is no longer a hard constraint on the values of m. The Landau theory of phase transitions gives a powerful way of deriving such results as the Ginzburg criterion found below. It is currently believed that the phenomenological form, with a suitably defined functional integral, has exactly the same critical exponents and other universal properties as the original lattice Ising model. This is perhaps not too surprising if one remembers that the liquid-gas transition in the continuum has the same universal properties as well. End remark Looking at the terms in this free energy, we can interpret them quite easily, at least with zero magnetic field and symmetry between ±m, so s 0 = 0. Changing the sign of r 0 while keeping u 0 positive will change between having one or two minima of the energy in the uniform case. We will show now that this is analogous, under a simple assumption about the temperature dependence of r 0 to our solution of heuristic mean-field theory. (Landau mean-field theory is obtained by approximating the functional integral in the remark above by its saddle point, ignoring fluctuations; then we do not even need to worry about the functional integral.) The mean value m in the ordered state is then r0 m = ±. (9) 4u 0 2

Similarly, in this saddle-point approximation, it is even easier than before to calculate mean-field critical exponents like γ = 1 and δ = 3, where the parameter driving the transition is r 0 = a 0 t for some constant a 0 and t the reduced temperature ( ) T Tc t =. (10) There are several assumptions that go into the Landau (continuum) version of mean-field theory: there is an order parameter that transforms under the broken symmetry; there is a Taylor series expansion of the free energy in terms of the order parameter, with coefficients that are analytic functions of temperature; and the saddle point of the integral is a good approximation to its behavior. The Ginzburg criterion for validity of mean-field theory is most easily derived on the ordered side of the transition. We want to compare the neglected fluctuations near the critical point with the mean m. A detailed calculation shows that the correlation length exponent ν just on the disordered side, which we derived earlier this lecture, is exactly the same as the correlation length exponent on the ordered side (recall that the correlation function on the ordered side is defined as the correlation of the fluctuation part). We want to estimate the relative magnitude of the double-fluctuation term, δm(r)δm(r ) m 2 = G(r r ) m 2. (11) At a separation r r of order ξ, this becomes (using the Coulomb law derived previously) ξ 2 d ( r 0 /4u 0 ) = T c ξ 2 d ( r 0 /4u 0 ) = ξ 2 d ( a 0 t/4u 0 ). (12) The mean-field prediction is that (since ν = 1/2) the correlation length grows as ξ t 1/2, (13) so the fluctuations scale with t as t (d 2)/2 /t = t d 4/2. This blows up near the transition for d < 4, but goes to zero as d > 4, suggesting that close to the transition mean field theory becomes a worse approximation in low dimensions, but a better approximation in high dimensions. The model obtained from neglecting the cubic, quartic, and higher terms in the Landau free energy is known as the Gaussian model. We will have more to say about this model later. One main result of the previous two lecures was a certain form for the correlation functions near a critical point. In fact such scaling forms were conjectured historically by Widom, Kadanoff, and others before the modern RG approach to phase transitions. The point of this lecture is to show that a great deal of information can be derived from simple scaling assumptions. An example of such a scaling assumption is that correlations near the transition are controlled by the diverging length scale ξ, as we showed for mean-field theory last time. This part will have fewer equations than the previous sections on phase transitions: its goal is to put the specific calculations done before in the broader context of the scaling theory of phase transitions. One motivation for the scaling laws is that they seem to hold exactly for almost all of the transitions encountered in physical systems, even though many of the laws have still not been proven rigorously. (To be precise, most of the laws have been rigorously proven as inequalities, 3

although they are believed to hold as equalities.) Assuming the scaling laws hold, only two of the critical exponents at a classical phase transition need to be known in order to deduce all the others. Let us make a list of critical exponents and in the process review the critical exponents discussed so far for mean-field theory. First we give a list of six critical exponents and their definitions for the Ising model, and then for a liquid-gas transition near the critical point. For brevity we define the reduced temperature ( ) T Tc t (14) and reduced magnetic field T c h H kt c. (15) 1. The critical exponent α describes the singularity in the specific heat at a second-order transition. Recall that the specific heat is proportional to the second derivative of the free energy with respect to temperature: S = F T, S NC = T, (16) and that at a generic second-order point the first derivative is continuous and the second derivative is singular. Here N is the number of spins in the system in order to make the specific heat an intensive quantity. We might expect different singularities on the ordered and disordered sides of the transition, C { A t α if t > 0 A t α if t < 0. (17) It turns out that α = α (this is predicted by the RG, and confirmed by exact results) so we will just call this exponent α. The amplitudes A and A are nonuniversal (but it happens that their ratio A/A is universal; such universal amplitude ratios are an example of universal critical quantities other than critical exponents!) We didn t calculate the mean-field theory specific heat because it is actually somewhat pathological; it is discontinuous but not otherwise singular, so α = 0. 2. The critical exponent β describes the spontaneous magnetization (that is, H = 0) on the ordered side of the transition: M ( t) β, t < 0. (18) For mean-field theory we calculated the value β = 1 2. 3. The critical exponent γ was not discussed so far: it gives the divergence of the susceptibility near the transition. The susceptibility is related to the second derivative of the free energy with respect to magnetic field (the first derivative is the magnetization), so should in general have a singularity for the same reason that the specific heat does. This singularity is described as χ = ( M H ) H=0 t γ. (19) As for the specific heat, we might have expected different values for γ on the ordered and disordered sides of the transition, but theory and exact results suggest γ + = γ so we will just write γ. Meanfield theory predicts γ = 1. 4. The critical exponent δ describes the magnetization in a field H 0 at the critical temperature (T = T c ): M h 1/δ. (20) 4

For mean-field theory we found δ = 3. 5. The divergence of the correlation length near the transition is given by the critical exponent ν. The correlation function goes as and G(r) = (σ 0 m)(σ r m) e r/ξ (polynomial factors), (21) ξ t ν. (22) Once again, we might worry that ν would take different values on the different sides of the transition, but that does not appear to be the case. Note that ν and the following exponent η both are defined in terms of the correlation functions, while the first four exponents were defined just in terms of the free energy and other translation-invariant quantities. For mean-field theory we found ν = 1 2. 6. At criticality (H = 0, T = T c ), the correlation falls off as a power-law rather than as an exponential, because the correlation length has gone to infinity. The coefficient of this power-law is defined using the critical exponent η: For mean-field theory η = 0. G(r) 1 r d 2+η at criticality. (23) (7. For completeness we mention a critical exponent that comes up in anisotropic critical phenomena, which we will probably not deal with in this course. You can imagine a problem where, for example, the diverging correlation length scale in the x direction might scale differently from the diverging correlation length scale in the y direction; then one defines the anisotropy critical exponent z through ξ x (ξ y ) z. (24) This is chiefly important for dynamical problems where the time scale near criticality is a power of the length scale, and time can be thought of as an extra dimension. For isotropic critical phenomena z = 1.) Now we quickly give the definitions of the first four exponents in the fluid case, which are natural analogies to those in the Ising magnet. The specific heat exponent α in the spin case corresponds to the specific heat at constant volume: C V t α at ρ = ρ c. The exponent β gives the density difference near the critical point, which also determines the shape of the coexistence curve in a system of fixed volume: ρ L ρ G ( t) β. The critical exponent γ is related to the isothermal compressibility χ T t γ. Finally, the critical exponent δ gives the density difference in response to a change in pressure at the critical temperature: ρ L ρ G = (p p c ) 1/δ. (25) The six main critical exponents turn out to be governed by just two independent quantities. This will turn out to be a very simple prediction of the renormalization-group theory of the Ising model. Before this modern understanding, however, it is important to note that by intuitive/physical arguments, people already understood scaling relations such as α = 2 dν, (26) 5

γ = ν(2 η), (27) α + 2β + γ = 2, (28) α + β(1 + δ) = 2. (29) Now we introduce the scaling forms for free energy and correlation functions in order to understand where the above relations come from. Let us assume that the free energy per site near a phase transition can be separated into a smooth part f n and a part f s whose derivatives contain the interesting singularities: f(t, h,...) = f n (t, h,...) + f s (t, h,...). (30) Now focus on the singular part f s and assume that the only variables controlling it are t and h. At h = 0, we know that the singular part should give rise in its second derivative to the specific heat singularity, or f s (t, 0) t 2 α. (31) It will be convenient to introduce two scales t 0 and h 0 that stay finite at the transition in order to make quantities dimensionless: then this becomes f s (t, 0) = t/t 0 2 α. (32) Now the main assumption of the scaling theory is that tuning away from the critical point with some magnetic field h is related to tuning away from the critical point with temperature, so that rather than having f s (t, h) be an arbitrary function of t and h, its dependence is restricted to f s (t, h) = t/t 0 2 α Φ( h/h 0 t/t 0 κ ) (33) for some single-variable function Φ (known as the scaling function) and some exponent κ. By our definitions Φ(0) = 1. Our current understanding is that different problems in the same universality class differ only in the scales h 0 and t 0 : the scaling function Φ, not just the individual critical exponents that it predicts, is universal. We can relate κ to the other critical exponents as follows: the spontaneous magnetization at h = 0 is, assuming Φ (0) to be a finite number, f h ( t)2 α κ β = 2 α κ. (34) The susceptibility is given by the second derivative of f s with respect to h, or 2 f h 2 ( t)2 α 2κ γ = α + 2κ 2. (35) Just from the above simple assumption about the form of f s, we can now derive one of the identities for critical exponents: combining 2β + γ in order to eliminate the unknown quantity κ, we find 2β + γ = 2 α α + 2β + γ = 2, (36) one of the scaling relations mentioned above. A slightly more involved trick can be used to express δ in terms of α and κ to obtain another exponent identity. We will return to these identities once the RG theory has been developed to explain the scaling form used above. 6

Lastly, consider a scaling form for the correlation function. A first guess, which turns out not to work except for simple cases like mean-field theory, is that near criticality the only important length scale is the correlation length ξ, so that G(r) = r λ f(r/ξ), (37) where λ is some exponent and f some unknown scaling function. For somewhat subtle reasons, this formula is not sufficiently general: it essentially describes a case where all the fluctuations are on the length scale ξ, where what we want in general is a description of fluctuations on all length scales between lattice spacing a and the correlation length ξ. Hence consider the more general form G(r) = r λ f(r/ξ, a/ξ). (38) The value λ = (d 2) is suggested by our mean-field calculation. This prefactor will be justified in more detail later when we discuss dimensional analysis; for now focus on the function f. The assumption corresponding to the scaling assumption in the free energy is that f(r/ξ, a/ξ) (a/ξ) η g(r/ξ) for ξ a. (39) To see why this is a powerful assumption, recall (or convince yourself) that the susceptibility is related to the correlation function through χ G(r)d d r. (40) Then the above scaling form predicts χ = ξ 2 η or γ = ν(2 η), another identity for critical exponents. For simple cases with η = 0, it is acceptable to use the simpler form of f with only one argument, but for most transitions (like the 3D Ising model) this does not give the correct picture of correlations. The next lectures will explain why η 0 is an example of an anomalous dimension, i.e., a critical exponent not calculable simply from dimensional analysis, and how this insight led to the modern theory of critical phenomena. 7