Chapter 5, Part 2: Coarse Graining and Correlations in Landau Theory

5.4Inhomogeneous systems and coarse graining

Part 1 ended by noting that \(\eta(\vec r)\) can vary in space, for example in response to an inhomogeneous field \(H(\vec r)\). As \(T\to T_c\) the correlation length \(\xi(T)\) grows; it is the linear dimension of patches of spins in which the magnetization is approximately constant. Near \(T_c\), \(\xi\gg a\), the lattice spacing, so if we choose \(\Lambda^{-1}\lesssim\xi(T)\),

\[ a\ll\Lambda^{-1}\lesssim\xi(T), \]

and a block of side \(\Lambda^{-1}\) contains a large number, \(O(\xi^d/a^d)\), of spins, especially as \(T\to T_c\), with approximately uniform magnetization inside. Dividing the system into such blocks is coarse graining. Within each block it is sensible to define a local magnetization, centered at \(\vec r\),

\[ \begin{gathered} M_\Lambda(\vec r) = \frac{1}{N_\Lambda(\vec r)}\sum_{i\in\vec r}\langle S_i\rangle,\\ N_\Lambda(\vec r) = \Lambda^{-d}/a^d, \end{gathered} \]

the coarse-grained magnetization, and to assign a Landau free energy to each block. \(M_\Lambda\) is not uniquely defined (one must specify \(\Lambda\)), but it does not fluctuate wildly on the scale \(a\): it varies smoothly, with no Fourier components at wavenumbers greater than \(\Lambda\). It will be convenient to work in Fourier space, where wavenumbers are restricted to \(k\lt\Lambda\). Usually \(\Lambda\) is not mentioned explicitly, on the assumption that results do not depend on its choice.

The Landau free energy cannot simply be \(\sum_{\vec r}\mathcal L(M_\Lambda(\vec r))\): minimizing that would just give the equilibrium value of \(M_\Lambda\) in each block independently, yet large differences between adjacent blocks are obviously unfavorable, because domain walls are costly. The simplest analytic term penalizing such differences is

\[ \sum_{\vec r}\sum_{\vec\delta}\frac{\gamma'}{2}\left(\frac{M_\Lambda(\vec r) - M_\Lambda(\vec r+\vec\delta)}{\Lambda^{-1}}\right)^2, \]

where \(\vec\delta\) is a vector of magnitude \(\Lambda^{-1}\) pointing to a neighboring block and \(\gamma'\) is a positive constant (temperature dependent in principle). Since \(M_\Lambda\) varies slowly compared to \(a\), in the continuum this becomes

\[ \begin{gathered} L = \int_\Omega d^dr\,\Big[\mathcal L\big(M_\Lambda(\vec r)\big) + \tfrac12\gamma\big(\nabla M_\Lambda(\vec r)\big)^2\Big],\\ \mathcal L(\eta) = at\eta^2 + \tfrac12b\eta^4 - H\eta, \end{gathered} \]

with \(\gamma\) another positive constant whose temperature dependence is negligible near the critical point. \(L\) is now a functional of \(M_\Lambda(\vec r)\): computing it requires the entire function. It is the coarse-grained free energy, or effective Hamiltonian. What about a term \(M\nabla^2M\), which is also isotropic and invariant under \(M\to-M\)? Since \(\nabla\cdot(M\nabla M) = M\nabla^2M + (\nabla M)^2\), the divergence theorem gives

\[ \int_\Omega M\nabla^2M\,d^dr = \oint_{\partial\Omega}M\nabla M\cdot d\vec S - \int_\Omega(\nabla M)^2\,d^dr, \]

and the surface term is negligible, so the \((\nabla M)^2\) term already includes it.

Interpreting the Landau free energy

First, \(L\) is not the Gibbs free energy. That would be

\[ \Gamma_\Omega(M) = F_\Omega\big(H(M)\big) + NM\,H(M), \]

where \(F_\Omega\) is the Helmholtz free energy, \(e^{-\beta F_\Omega(H)} = \operatorname{Tr}\,e^{-\beta\mathcal H_\Omega}\), and \(H(M)\) is defined implicitly by \(N(\Omega)M(H) = -\partial F_\Omega/\partial H\): the external field required for the magnetization to have the value \(M\). Dropping the subscript \(\Omega\), the equation of state is

\[ \begin{gathered} \frac{\partial\Gamma}{\partial M} = \frac{\partial F}{\partial H}\frac{\partial H}{\partial M} + NH + NM\frac{\partial H}{\partial M} = NH,\\ \text{i.e.}\quad \frac{\partial\Gamma}{\partial(NM)} = H, \end{gathered} \]

since \(\partial F/\partial H = -NM\). At zero field, \(\partial\Gamma/\partial(NM) = 0\): minimizing \(\Gamma\) with respect to \(M\) specifies the state. So \(\Gamma\) resembles \(L\), at least for \(H=0\). In nonzero field a candidate is \(L'(M,\hat H)\equiv\Gamma(M) - M\hat HN = F(H(M)) + NM\big(H(M)-\hat H\big)\), with \(\hat H\) a parameter fixed by \(\partial L'/\partial M = \partial\Gamma/\partial M - \hat HN = 0\); for \(\hat H = 0\), \(L' = \Gamma\). But there are two important differences:

\(\Gamma(M)\) is convex (Exercise 2-1(c)), whereas \(L(M)\) is not. In the thermodynamic limit, for \(-M_s\lt M\lt M_s\), \(H(M) = 0\) (the vertical part of the \(M(H)\) curve of Figure 2.10), so \(\partial\Gamma/\partial M = 0\) and \(\Gamma\) is flat there. For \(|M|>M_s\), \(H(M)\ne0\) with \(\partial H/\partial M>0\), so \(\partial\Gamma/\partial M = NH\) is positive for \(M>M_s\) and negative for \(M\lt -M_s\). Compare the double-well Landau function: \(\Gamma\) is indeed convex, while \(\mathcal L\) is not (book's Figure 5.3).
\(\Gamma(M)\) is a thermodynamic function. A spatially varying \(M(\vec r)\) has no meaning for it, because all spatial information has been integrated out in the trace: \(\Gamma\) depends on \(\operatorname{Tr}e^{-\beta\mathcal H}\) and so cannot depend on particular microscopic configurations.

The meaning of coarse graining. \(L\{M_\Lambda(\vec r)\}\) is the free energy of the system constrained to be in the configuration specified by \(M_\Lambda(\vec r)\), and it depends on \(\Lambda\) implicitly. Many spin configurations \(\{S_i\}\) are consistent with a given \(M_\Lambda(\vec r)\). Formally,

\[ \begin{aligned} e^{-L\{M_\Lambda(\vec r)\}/k_BT} &= \operatorname{Tr}\Big\{e^{-\beta\mathcal H_\Omega\{S_i\}}\\ &\qquad\times\prod_{\vec r}\delta^{(K)}\Big[\sum_{i\in\vec r}S_i - M_\Lambda(\vec r)N_\Lambda(\vec r)\Big]\Big\}\\ &\equiv \operatorname{Tr}'\,e^{-\beta\mathcal H\{S_i\}}, \end{aligned} \]

where the Kronecker delta is 1 if the block sum matches and 0 otherwise, and \(\operatorname{Tr}'\) is the partial trace over only those microscopic configurations with the coarse-grained magnetization \(M_\Lambda(\vec r)\). This differs from \(\operatorname{Tr}\,e^{-\beta\mathcal H_\Omega} = Z\), in which all configurations are included. Split the configurations into \(\{\bar S\}\), consistent with a particular profile \(\bar M_\Lambda(\vec r)\), and \(\{S^*\}\), labeling the profiles themselves. Then

\[ Z = e^{-\beta F} = \sum_{\{S\}}e^{-\beta\mathcal H_\Omega\{S\}} = \sum_{\{S^*\}}\sum_{\{\bar S\}}e^{-\beta\mathcal H_\Omega\{S\}} = \sum_{\{S^*\}}e^{-\beta L\{\bar M_\Lambda(\vec r)\}}, \]

\[ Z = \int\mathcal DM_\Lambda\;e^{-\beta L\{M_\Lambda(\vec r)\}}, \]

a functional integral: a sum over all smoothly varying functions \(M_\Lambda(\vec r)\), with \(e^{-\beta L}\) proportional to the probability of finding the configuration \(M_\Lambda(\vec r)\). Once we realize that \(L\) is a coarse-grained free energy, not the thermodynamic one, there is no theorem requiring it to be convex. Nevertheless the true thermodynamic Gibbs free energy is convex: integrating over the semi-macroscopic degrees of freedom restores convexity. This interpretation also explains why only analytic contributions to \(\mathcal L\) appear: the partial trace involves a finite number of states in a finite block, so \(\mathcal L\) should be analytic. (For a system defined on a continuum, a suitable regularization must first be given.)

A non-convex Landau function and a convex Gibbs function, computed exactly

\(k_BT/J_0\)

Constrained free energy \(\mathcal L(m)\) per spin

Gibbs function \(\Gamma_N(m)/N\) for increasing \(N\)

The Curie–Weiss magnet makes both objects exact. Restricting the trace to configurations with magnetization \(m\) (the partial trace) gives, per spin as \(N\to\infty\), \(\mathcal L(m) = -\tfrac12J_0m^2 - k_BT\,s(m)\), where \(s(m)\) is the entropy of mixing. Below \(T_c\) this is a non-convex double well. The Gibbs function \(\Gamma_N(m) = F_N(H) + NmH\), built from the full trace by a Legendre transform, is convex for every \(N\) and approaches the convex hull of \(\mathcal L\), flat between \(\pm m_s\) (magenta dashes), as \(N\) grows. Both are shifted to vanish at \(m=0\).

Phenomenological and microscopic Landau theory

In practice \(L\{\eta_\Lambda(\vec r)\}\) is never really derived by direct integration over microscopic degrees of freedom. The real-space renormalization group, described later, comes close but is hard to implement systematically. Landau theory can be motivated microscopically in other ways, such as the Hubbard–Stratonovich transformation (Exercise 3-3), but this is not a systematic derivation: the approximations become uncontrolled as \(T\to T_c\), perturbation theory diverges, and renormalization and the renormalization group are required to make sense of the procedure. In some cases this "microscopic Landau theory" can be used without renormalization, as long as \(T\) is not too close to \(T_c\). Superconductivity is the classic example: the asymptotic critical behavior occurs so close to \(T_c\) as to be inaccessible, the experimentally accessible regime is one where fluctuations about mean-field theory are small, and Gor'kov derived the coarse-grained Ginzburg–Landau free energy from the BCS theory.

In most cases, though, one follows the phenomenological prescription: write down the terms required by symmetry with more or less arbitrary coefficients. We will see later that those coefficients do not matter near \(T_c\), since they acquire corrections dependent on \(T-T_c\); they become renormalized, and close to \(T_c\) they tend toward special values characteristic of the universality class.

Mathematical preliminaries

For the Ising universality class,

\[ L = \int d^dr\,\Big[\frac\gamma2(\nabla\eta)^2 + at\eta^2 + \tfrac12b\eta^4 - H(\vec r)\eta(\vec r)\Big]. \]

The continuum limit. \(Z = \operatorname{Tr}_{\eta(\vec r)}e^{-\beta L\{\eta(\vec r)\}}\) is similar to \(Z = \operatorname{Tr}_{\{S\}}e^{-\beta\mathcal H_\Omega}\), but the variables are labeled by a continuous \(\vec r\in\Omega\) rather than a discrete \(i = 1,\dots,N(\Omega)\). In the continuum limit \(a\to0\) and \(N(\Omega)\to\infty\) with \(V = N(\Omega)a^d\) fixed. For a finite set of variables \(-\infty\lt\eta_i\lt\infty\), \(\operatorname{Tr}\equiv\int\prod_{i=1}^{N(\Omega)}d\eta_i\), an infinite product of integrals in the limit. Ignoring fluctuations more abrupt than \(\Lambda^{-1}\), the length scale of correlated spins, we perform the trace in Fourier space, \(\operatorname{Tr}\equiv\int\prod_{|\vec k|\lt\Lambda}d(\operatorname{Re}\eta_{\vec k})\,d(\operatorname{Im}\eta_{\vec k})\), with

\[ \begin{gathered} \eta(\vec r) = \frac1V\sum_{\vec k}\eta_{\vec k}e^{i\vec k\cdot\vec r},\\ \eta_{\vec k} = \int_Vd^dr\,\eta(\vec r)\,e^{-i\vec k\cdot\vec r}. \end{gathered} \]

Since \(\eta(\vec r)\) is usually real, \(\eta^*(\vec r) = \eta(\vec r)\) implies \(\operatorname{Re}\eta_{\vec k} = \operatorname{Re}\eta_{-\vec k}\) and \(\operatorname{Im}\eta_{\vec k} = -\operatorname{Im}\eta_{-\vec k}\). Integrating over all \(\vec k\) would count each Fourier component twice, so the trace is restricted to half of \(\vec k\)-space (e.g. \(k_z>0\)): \(\operatorname{Tr}\equiv\int\prod'_{\vec k}d\eta_{\vec k}\). Substituting one transform into the other,

\[ \begin{gathered} \eta(\vec r) = \frac1V\sum_{\vec k}\int_Vd^dr'\,\eta(\vec r')\,e^{i\vec k\cdot(\vec r-\vec r')}\\ \Longrightarrow\quad \frac1V\sum_{\vec k}e^{i\vec k\cdot(\vec r-\vec r')} = \delta(\vec r-\vec r'), \end{gathered} \]

the "resolution" of the delta function; in the continuum limit \(\frac1V\sum_{\vec k}\to\int\frac{d^dk}{(2\pi)^d}\). Conversely, \(\int_Vd^dr\,e^{i(\vec k-\vec k')\cdot\vec r} = V\delta_{\vec k\vec k'}\) (a Kronecker delta), and as \(V\to\infty\), \(V\delta_{\vec k\vec k'}\to(2\pi)^d\delta(\vec k-\vec k')\).

Functional differentiation. For finitely many degrees of freedom with a linear coupling to sources, \(Z\{H_i\} = \operatorname{Tr}\exp[-\beta(\mathcal H_\Omega - \sum_iH_i\eta_i)]\), correlation functions follow from \(\langle\eta_i\eta_j\rangle = \beta^{-2}Z^{-1}\,\partial^2Z/\partial H_i\partial H_j\), and the \(H_k\) may be set to zero afterward. In field theory the partition function becomes a generating functional, \(Z\{H(\vec r)\} = \operatorname{Tr}\exp[-\beta(\mathcal H_\Omega - \int d^dr\,H(\vec r)\eta(\vec r))]\), and partial derivatives are replaced by functional derivatives,

\[ \frac{\delta F}{\delta\eta(\vec r)}\equiv\lim_{\epsilon\to0}\frac{F\{\eta(\vec r') + \epsilon\,\delta(\vec r'-\vec r)\} - F\{\eta(\vec r')\}}{\epsilon}. \]

Directly from the definition,

\[ \begin{gathered} \frac{\delta}{\delta\eta(\vec r)}\int d^dr'\,\eta(\vec r') = 1,\\ \frac{\delta}{\delta\eta(\vec r)}\eta(\vec r') = \delta(\vec r-\vec r'),\\ \frac{\delta}{\delta\eta(\vec r)}\int d^dr'\,\eta^n(\vec r') = n\eta^{n-1}(\vec r). \end{gathered} \]

For the gradient term, \(\big(\nabla(\eta+\epsilon\delta)\big)^2 = (\nabla\eta)^2 + 2\epsilon\nabla\eta\cdot\nabla\delta + O(\epsilon^2)\), and integrating by parts with \(\nabla\cdot(\vec u\,\delta) = \delta\,\nabla\cdot\vec u + \vec u\cdot\nabla\delta\) (the surface term vanishes),

\[ \begin{aligned} \frac{\delta}{\delta\eta(\vec r)}\int d^dr'\,\tfrac12\big(\nabla\eta(\vec r')\big)^2 &= \int d^dr'\,\nabla\eta(\vec r')\cdot\nabla\delta(\vec r'-\vec r)\\ &= -\nabla^2\eta(\vec r). \end{aligned} \]

Response and correlation functions

For a system described by Landau theory, the free energy \(F\{H(\vec r)\}\) generates the expectation value of the order parameter and the generalized isothermal susceptibility,

\[ \begin{gathered} \langle\eta(\vec r)\rangle = -\frac{\delta F}{\delta H(\vec r)},\\ \chi_T(\vec r,\vec r') = \frac{\delta\langle\eta(\vec r)\rangle}{\delta H(\vec r')}. \end{gathered} \]

So a small change \(\delta H(\vec r)\) of the field changes the free energy by \(\delta F = -\int d^dr'\,\langle\eta(\vec r')\rangle\,\delta H(\vec r')\) and the order parameter by \(\delta\langle\eta(\vec r)\rangle = \int d^dr'\,\chi_T(\vec r,\vec r')\,\delta H(\vec r')\). As in §3.7,

\[ \begin{aligned} \chi_T(\vec r,\vec r') &= -\frac{\delta^2F}{\delta H(\vec r)\delta H(\vec r')}\\ &= \frac{1}{k_BT}\Big\{\langle\eta(\vec r)\eta(\vec r')\rangle - \langle\eta(\vec r)\rangle\langle\eta(\vec r')\rangle\Big\} = \beta G(\vec r,\vec r'). \end{aligned} \]

For a translationally invariant system,

\[ \begin{gathered} G(\vec r-\vec r') = k_BT\,\chi_T(\vec r-\vec r'),\\ \hat\chi_T(\vec k) = \beta\hat G(\vec k),\\ \chi_T\equiv\lim_{k\to0}\hat\chi_T(\vec k) = \beta\int d^dr\,G(\vec r), \end{gathered} \]

an example of a principal result of linear response theory: response functions are proportional to correlation functions. The last equality, with carets denoting Fourier transforms and \(\hat\chi_T(\vec k)\) the wavevector-dependent susceptibility, is the static susceptibility sum rule.

The two-point correlation function

The order parameter minimizes \(L\), \(\delta L/\delta\eta(\vec r) = 0\). Using the functional derivatives above,

\[ -\gamma\nabla^2\eta(\vec r) + 2at\,\eta(\vec r) + 2b\,\eta^3(\vec r) - H(\vec r) = 0. \]

Taking \(\delta/\delta H(\vec r')\) of this equation,

\[ \begin{gathered} \big[-\gamma\nabla^2 + 2at + 6b\eta^2(\vec r)\big]\chi_T(\vec r,\vec r') = \delta(\vec r-\vec r'),\\ \beta\big(-\gamma\nabla^2 + 2at + 6b\eta^2\big)G(\vec r-\vec r') = \delta(\vec r-\vec r'): \end{gathered} \]

\(G\) is a Green function. For a translationally invariant system \(\eta\) is the uniform equilibrium value. For \(t>0\), \(\eta = 0\); for \(t\lt 0\), \(\eta = \pm(-at/b)^{1/2}\) and \(2at + 6b\eta^2 = -4at\). Either way

\[ \begin{gathered} \big(-\nabla^2 + \xi^{-2}\big)G(\vec r-\vec r') = \frac{k_BT}{\gamma}\,\delta(\vec r-\vec r'),\\ \xi_>(t) = \Big(\frac{\gamma}{2at}\Big)^{1/2},\quad \xi_\lt (t) = \Big(-\frac{\gamma}{4at}\Big)^{1/2}. \end{gathered} \]

Since \(\xi\sim|t|^{-1/2}\) on both sides, \(\nu = \nu' = \tfrac12\), with \(\xi_>/\xi_\lt = \sqrt2\) at equal \(|t|\).

Solving the Landau equation: a domain wall, and the response to a point field

\(t\)

Profile \(\eta(x)\) relaxing under \(\partial_\tau\eta = -\delta L/\delta\eta\)

\(|\eta(x)-\eta_\infty|\), log scale

The one-dimensional Landau equation \(-\gamma\eta'' + 2at\eta + 2b\eta^3 = H(x)\) with \(\gamma = a = b = 1\), solved by gradient descent on \(L\) (the relaxation is shown slowly). Domain wall (\(t\lt 0\), opposite bulk states at the ends): the solution is the kink \(\eta_s\tanh(x/2\xi_\lt)\) (magenta dashes), whose width is set by the correlation length. Point field (a small \(H\) concentrated at \(x=0\)): the response decays as \(e^{-|x|/\xi}\) with \(\xi_>\) above \(T_c\) and the shorter \(\xi_\lt\) below, the one-dimensional Green function of the operator above. For a wall with \(t>0\) there is no ordered state to support it, and the profile simply decays to zero.

Fourier transform. Transforming the equation for \(G\),

\[ \hat G(\vec k) = \frac{k_BT}{\gamma}\frac{1}{k^2+\xi^{-2}}. \]

At \(T=T_c\), \(\xi=\infty\) and \(\hat G(k)\sim k^{-2}\), so \(\eta = 0\) (the exponent of Chapter 3). Transforming back with a scaling argument, \(G(\vec r,T_c)\propto1/r^{d-2}\) for \(d>2\), as advertised in §3.7. For \(T\ne T_c\), the sum rule \(\hat G(0) = k_BT\chi_T\) gives \(\chi_T = \xi^2/\gamma\), which with \(\nu = \tfrac12\) gives \(\gamma = 1\) (the exponent, not the coefficient). The final form is the Ornstein–Zernike result of Chapter 4,

\[ \hat G(\vec k) = \frac{k_BT\chi_T(t)}{1+k^2\xi^2(t)}. \]

Real space. Fourier transforming in \(d\) dimensions is tedious because of the angular integrals, so return to the differential equation. Set \(\vec r'=0\); since \(\delta(\vec r)\) is spherically symmetric, write \(\nabla^2\) in radial coordinates. With \(\rho\equiv r/\xi\), \(\mathcal G(\rho)\equiv G(r/\xi)\), and \(\delta(\rho\xi) = \xi^{-d}\delta(\rho)\),

\[ \begin{gathered} \Big[-\frac{1}{\rho^{d-1}}\frac{\partial}{\partial\rho}\rho^{d-1}\frac{\partial}{\partial\rho} + 1\Big]\mathcal G(\rho) = g\,\delta(\rho),\\ g\equiv\frac{k_BT}{\gamma}\xi(t)^{2-d}, \end{gathered} \]

where \(g\) is a dimensionless measure of the strength of fluctuations. The solution involves the modified Bessel function of the third kind \(K_n(\rho)\):

\[ \frac1g\mathcal G(\rho) = (2\pi)^{-d/2}\rho^{-(d-2)/2}K_{(d-2)/2}(\rho), \]

which for \(d=1\) reduces to \(\tfrac12e^{-\rho}\). Using \(K_n(\rho)\sim(\pi/2\rho)^{1/2}e^{-\rho}\) as \(\rho\to\infty\), \(K_n(\rho)\sim\tfrac12\Gamma(n)(\rho/2)^{-n}\) as \(\rho\to0\) (\(n>0\)), and \(K_0(\rho)\sim-\log\rho\), with \(\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\):

\[ \begin{gathered} G(r) = \frac{\pi^{(1-d)/2}}{2^{(1+d)/2}}\,\frac{k_BT}{\gamma}\,\frac{e^{-r/\xi}}{r^{(d-1)/2}\,\xi^{(d-3)/2}}\quad(T\ne T_c,\ r\gg\xi),\\ G(r) = \frac{k_BT}{\gamma}\frac{\Gamma\big(\frac{d-2}{2}\big)}{4\pi^{d/2}}\frac{1}{r^{d-2}}\quad(T=T_c,\ d>2). \end{gathered} \]

At \(T_c\), \(\xi = \infty\) and the relevant limit is \(r\ll\xi\); for \(d>2\), \(\xi\) obligingly cancels out.

The Ornstein–Zernike correlation function in \(d\) dimensions

Dimension \(d\) Correlation length \(\xi\)

\(G(r)\) in units of \(k_BT/\gamma\), log–log

\(\hat G(k)\) in units of \(k_BT/\gamma\), log–log

The exact solution, with \(K_\nu\) evaluated numerically from \(K_\nu(x) = \int_0^\infty e^{-x\cosh u}\cosh\nu u\,du\), so non-integer \(d\) works too. Dashes show the two limits: the critical power law \(r^{-(d-2)}\) (logarithm for \(d=2\)) for \(r\ll\xi\), and \(e^{-r/\xi}/r^{(d-1)/2}\) for \(r\gg\xi\). For \(d\le2\) the critical form does not decay at all: the correlation function at \(r\ll\xi\) grows with \(\xi\), a first hint that fluctuations are especially important in low dimensions. In Fourier space every \(d\) gives the same Lorentzian, flat for \(k\xi\ll1\) and \(\propto k^{-2}\) for \(k\xi\gg1\).

The coefficient \(\gamma\)

The strength of correlations is controlled by \(\gamma\), which depends on the range of interactions in the microscopic physics. One way to see this is frankly phenomenological. From \(\xi = (\gamma/2at)^{1/2}\) for \(T>T_c\), write \(\xi(t) = \xi(1)|t|^{-1/2}\) with \(\xi(1) = (\gamma/2a)^{1/2}\): the correlation length extrapolated from mean-field theory to \(T = 2T_c\). So \(\xi(1)\) measures the range \(R\) of correlations well away from \(T_c\), and

\[ \gamma\propto R^2. \]

This is qualitatively correct, but "extrapolate" is important: away from the critical region the actual \(\xi(T)\) may well differ from the mean-field asymptotics, because when \(T\) is not close to \(T_c\) the coefficients \(a_n\) cannot be Taylor expanded to lowest order in \((T-T_c)/T_c\), which is no longer small.

The second way follows microscopic Landau theory. In mean-field theory the Gibbs free energy is \(\Gamma_{\rm MFT} = -\tfrac12\sum_{ij}J_{ij}m_im_j + \Phi\{m_i\}\), with \(\Phi\) the homogeneous contribution and \(m_i\equiv\langle S_i\rangle\). If \(J_{ij} = J\) for \(0\lt|\vec r_i-\vec r_j|\lt R\) and zero otherwise,

\[ \begin{aligned} \sum_{ij}J_{ij}m_im_j &= -\frac{Ja^2}{4}\sum_i\sum_{|\vec\delta|\lt R}\Big\{\Big(\frac{m_{i+\delta}-m_i}{a}\Big)^2\\ &\qquad\qquad - \Big(\frac{m_{i+\delta}+m_i}{a}\Big)^2\Big\}\\ &\approx -\frac{Ja^2}{4}\sum_i\sum_{|\vec\delta|\lt R}\Big(\frac{\vec\delta\cdot\nabla m(\vec r_i)}{a}\Big)^2 + O(m^2)\\ &\approx -\frac{zJR^2}{4}\int\frac{d^dr}{a^d}\big(\nabla m(\vec r)\big)^2 + O(m^2), \end{aligned} \]

where \(z\) is approximately the coordination number. This gives \(\gamma\approx R^2(Jz/4a^d) = R^2k_BT_c/a^d\), using \(k_BT_c\propto zJ\) (Exercise 5-1(a)), in agreement with \(\gamma\propto R^2\). Longer-ranged interactions make the gradient term stiffer, so fluctuations are suppressed and mean-field theory works better. This idea becomes quantitative in the Ginzburg criterion.

The semester in review

This chapter completes the classical theory of phase transitions. Looking back over the course:

Scaling and power laws (Ch. 1): critical points show power laws with non-trivial, universal exponents, which no smooth expansion explains.
The thermodynamic limit (Ch. 2, Part 2): non-analyticity requires \(N\to\infty\); spontaneous symmetry breaking and ergodicity breaking explain how a symmetric Hamiltonian produces an ordered state; the lattice gas makes fluids and magnets one problem.
Exact and approximate calculations (Ch. 3): transfer matrices, low-temperature expansions, and mean-field theory, with its exponents \(\beta = \tfrac12\), \(\gamma = 1\), \(\delta = 3\).
Fluids (Ch. 4): the van der Waals equation is a mean-field theory with the same exponents; density fluctuations and critical opalescence connect correlations to experiment.
Landau theory (Ch. 5 and this page): symmetry and analyticity alone produce every mean-field exponent, including \(\nu = \tfrac12\) and \(\eta = 0\) from the coarse-grained free energy.

Yet the experiments of Chapter 1 and the exact solutions of Chapters 2 and 3 disagree with these exponents: \(\beta = \tfrac18\) in the two-dimensional Ising model, about 0.33 in three dimensions. The missing ingredient is the very thing mean-field theory discards: the fluctuations integrated over in \(Z = \int\mathcal DM\,e^{-\beta L}\). Deciding when they matter (the Ginzburg criterion), and computing what they do when they do, is the task of the renormalization group, the subject that follows.

This concludes the course material for the semester.

Chapter 5: Landau theoryPart 2: coarse graining, correlations, and linear response