Ornstein-Zernike Equation from Landau-Ginzburg Theory

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Hello! In our previous lesson, we constructed the Landau-Ginzburg free energy functional, which provides a powerful field-theoretic framework for describing phase transitions. We promoted the order parameter to a spatially-varying field $m (r)$ and, by including a gradient-squared term, we endowed this field with "stiffness," penalizing inhomogeneous configurations. Our functional, for a single-component order parameter, is:

F [m (r)] = \int d^{d} r [\frac{1}{2} a_{0} m^{2} + \frac{1}{4} b_{0} m^{4} + \frac{1}{2} κ (\nabla m)^{2}]

where $a_{0} = a (T - T_{c})$ .

Today, we will put this functional to use. Our goal is to derive the explicit mathematical form of the order parameter correlation function, $G (r) = ⟨ δ m (r) δ m (0)⟩$ , where $δ m$ represents fluctuations around the average value. To do this, we must evaluate averages over all possible field configurations, weighted by the Boltzmann factor $e^{- βF [m (r)]}$ . This involves a functional integral that is intractable due to the $m^{4}$ term.

We will therefore make a crucial simplification: the Gaussian approximation. This involves analyzing only the small, quadratic fluctuations around the mean-field ground state. This approximation not only makes the problem solvable but also serves as the first-order correction to mean-field theory, giving us our first quantitative look at the structure of correlations near criticality.

1. The Gaussian Approximation

The core idea of the Gaussian approximation is to expand the order parameter field $m (r)$ around its spatially uniform, mean-field value $\overset{m}{ˉ}$ , and keep only terms up to second order in the fluctuations $δ m (r)$ .

m (r) = \overset{m}{ˉ} + δ m (r)

By truncating the Landau-Ginzburg functional at the quadratic level, the functional integral becomes a Gaussian integral, which is exactly solvable. Given your familiarity with QFT, you'll recognize this as analogous to studying a free field theory of fluctuations around a classical background.

Let's apply this approximation. The following reading provides a clear and detailed derivation of the effective action for these fluctuations.

Mean-Field Theory and the Gaussian Approximation

Please read the following section from the text 'Mean-Field Theory and the Gaussian Approximation'. It systematically derives the effective quadratic (Gaussian) action for the fluctuation field, considering both the high-temperature and low-temperature phases.

Read section 2.3.1, 'Gaussian Effective Action' (pages 39-40). Start from Eq. (2.71) and follow the derivation to Eq. (2.75) for the high-temperature phase and Eq. (2.78) for the low-temperature phase. Pay close attention to how the coefficients of the quadratic terms change depending on the phase.

Let's summarize the key results from this reading.

Case 1: High-Temperature Phase ( $T > T_{c}$ )

In the disordered phase, the mean-field order parameter is zero, $\overset{m}{ˉ} = 0$ . Thus, $m (r) = δ m (r)$ . The Gaussian approximation corresponds to simply neglecting the quartic term $b_{0} m^{4}$ in the Landau-Ginzburg functional, as it is of higher order in the small fluctuations. The resulting Gaussian functional is:

F_{G} [δ m] \approx \int d^{d} r [\frac{1}{2} a_{0} (δ m)^{2} + \frac{1}{2} κ (\nabla δ m)^{2}] (T > T_{c})

Here $a_{0} = a (T - T_{c}) > 0$ .

Case 2: Low-Temperature Phase ( $T < T_{c}$ )

In the ordered phase, we have a non-zero mean-field value, $\overset{m}{ˉ}^{2} = - a_{0} / b_{0}$ . We substitute $m (r) = \overset{m}{ˉ} + δ m (r)$ into the full functional and expand. As shown in the text (and as you can verify by hand), the term linear in $δ m$ vanishes precisely because $\overset{m}{ˉ}$ satisfies the mean-field equation of state. The resulting quadratic functional for the fluctuations is:

F_{G} [δ m] \approx const. + \int d^{d} r [\frac{1}{2} (- 2 a_{0}) (δ m)^{2} + \frac{1}{2} κ (\nabla δ m)^{2}] (T < T_{c})

Here $a_{0} < 0$ . Notice that the coefficient of the $(δ m)^{2}$ term is now $- 2 a_{0}$ , which is positive as required for stability. This change arises from the expansion of the $m^{4}$ term. For simplicity, we will focus our derivation on the high-temperature phase, but the procedure is analogous for the ordered phase.

2. Deriving the Correlation Function in Fourier Space

The gradient term $(\nabla δ m)^{2}$ couples neighboring points, making the real-space functional difficult to work with directly. The key to solving this is to transform to Fourier space, which diagonalizes the quadratic functional.

Let's define the Fourier transform of the fluctuation field:

δ m (r) = \int \frac{d ^{d} q}{( 2 π ) ^{d}} e^{i q \cdot r} δ m (q)

Using the properties of Fourier transforms, the gradient term becomes $\int d^{d} r (\nabla δ m)^{2} = \int \frac{d ^{d} q}{( 2 π ) ^{d}} q^{2} ∣ δ m (q) ∣^{2}$ .

The Gaussian functional for the high-temperature phase can now be rewritten as an integral over independent Fourier modes:

F_{G} [δ m (q)] = \frac{1}{2} \int \frac{d ^{d} q}{( 2 π ) ^{d}} (a_{0} + κ q^{2}) ∣ δ m (q) ∣^{2}

The probability of a particular fluctuation configuration ${δ m (q)}$ is proportional to $e^{- β F_{G}}$ . Since the free energy is a sum over $q$ -modes, the probability distribution factorizes into a product of independent Gaussian distributions for each mode $δ m (q)$ .

For a single real Gaussian variable $x$ with probability $P (x) \propto e^{- A x^{2} /2}$ , the variance is $⟨ x^{2} ⟩ = 1/ A$ . For a complex variable $z = x + i y$ with probability $P (z) \propto e^{- A ∣ z ∣^{2}}$ , the variance is $⟨ ∣ z ∣^{2} ⟩ = 1/ A$ . Our fluctuation modes $δ m (q)$ are complex variables. The "A" in our case is $β (a_{0} + κ q^{2})$ .

The following lecture segment masterfully walks through this calculation.

4. The Landau-Ginzburg Approach Part 3

Watch this portion of Prof. Kardar's lecture. He transforms the Landau-Ginzburg functional to Fourier space and uses the properties of Gaussian integrals to derive the two-point correlation function in q-space.

Watch from 13:53 to 34:09. The first part (until 25:33) covers the expansion to quadratic order. The key segment for our current step starts at 25:33, where he moves to Fourier space and derives the correlator (\langle \phi(q) \phi(q') angle). Pay attention to his definition of the longitudinal and transverse correlation lengths.

From this reasoning, we arrive at the central result for the two-point correlation function in Fourier space:

G (q) \equiv ⟨ ∣ δ m (q) ∣^{2} ⟩ = \frac{k _{B} T}{a _{0} + κ q ^{2}}

(Note: Often $β = 1/ k_{B} T$ is absorbed into the definition of the functional, in which case the numerator is 1. We will keep it for clarity.)

This expression is precisely the Ornstein-Zernike form for the correlation function. We can rewrite it to make its structure more apparent:

G (q) = \frac{k _{B} T / κ}{q ^{2} + a _{0} / κ} = \frac{G _{0}}{q ^{2} + ξ ^{- 2}}

where we have defined the correlation length $ξ$ :

ξ = \frac{κ}{a _{0}} = \frac{κ}{a ( T - T _{c} )}

This beautifully confirms the physical picture: as we approach the critical temperature $T_{c}$ , $a_{0} \to 0$ , and the correlation length $ξ$ diverges. The divergence follows a power law $ξ \propto ∣ T - T_{c} ∣^{- 1/2}$ , which tells us that the mean-field correlation length exponent is $ν = 1/2$ .

Test your understanding!

Using the result for the Gaussian action in the low-temperature phase ( $a_{0} \to - 2 a_{0}$ ), what is the expression for the correlation length $ξ$ for $T < T_{c}$ ? How does the amplitude of the divergence compare to the high-temperature side?

Show answer

For $T < T_{c}$ , the coefficient $a_{0}$ in the Gaussian action is replaced by $- 2 a_{0}$ . Since $a_{0} = a (T - T_{c})$ is negative in this phase, $- 2 a_{0}$ is positive.
The correlation length becomes:

ξ_{T < T_{c}} = \frac{κ}{- 2 a _{0}} = \frac{κ}{- 2 a ( T - T _{c} )} = \frac{κ}{2 a ( T _{c} - T )}

Comparing the amplitudes, we have $ξ_{T > T_{c}} \propto \frac{1}{a}$ and $ξ_{T < T_{c}} \propto \frac{1}{2 a}$ . The ratio of the amplitudes is $\frac{A _{+}}{A _{-}} = 2$ . This is a universal amplitude ratio predicted by mean-field theory.

3. Real-Space Correlation Function

The Ornstein-Zernike form in Fourier space is powerful, but to get a more intuitive physical picture, we need to transform it back to real space.

G (r) = \int \frac{d ^{d} q}{( 2 π ) ^{d}} e^{i q \cdot r} G (q) = \frac{k _{B} T}{κ} \int \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{e ^{i q \cdot r}}{q ^{2} + ξ ^{- 2}}

The evaluation of this integral is a classic problem in mathematical physics. The exact form depends on the spatial dimension $d$ . Let's explore the most important asymptotic behaviors, which are derived in the following resource.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

This section of the notes from Cambridge provides the asymptotic results for the real-space correlation function obtained by Fourier transforming the Ornstein-Zernike form.

Focus on section 2.5, specifically Eq. (2.14) and the asymptotic results in Eq. (2.15). You don't need to work through the details of the Fourier transform (footnote 6) but understand the two limiting behaviors.

The key results are:

At the critical point ( $T = T_{c}$ ): Here, $ξ \to \infty$ , and the integral becomes $\int \frac{d ^{d} q}{q ^{2}} e^{i q \cdot r}$ . This is the Fourier transform of $1/ q^{2}$ , which is the Green's function for the Laplacian operator. The result is a power-law decay:
$G (r) \propto \frac{1}{r ^{d - 2}} (for r ≫ a, where a is the lattice spacing)$
Comparing this to the general definition $G (r) \propto 1/ r^{d - 2 + η}$ , we find that the mean-field prediction for the exponent $η$ is $η = 0$ .
Away from the critical point ( $T \neq = T_{c}$ ):
- For short distances ( $r ≪ ξ$ ), the function still looks like the critical power law, $G (r) \propto 1/ r^{d - 2}$ . The fluctuations don't "feel" the finite correlation length yet.
- For long distances ( $r ≫ ξ$ ), the $ξ^{- 2}$ term dominates the denominator for small $q$ , and the integral is dominated by an exponential decay: $G (r) \propto \frac{e ^{- r / ξ}}{r ^{(d - 1) /2}}$
This shows that $ξ$ is indeed the characteristic length scale beyond which correlations decay exponentially to zero. For the important case of $d = 3$ , this gives the Yukawa potential form: $G (r) \propto \frac{e ^{- r / ξ}}{r}$ .

oz-form-real-space

CAPTION: A sketch of the real-space correlation function G_0(r) as derived from the Gaussian approximation. At short distances (r \ll \xi), it follows a power-law decay characteristic of the critical point. At long distances (r \gg \xi), it is dominated by an exponential decay with a characteristic scale set by the correlation length \xi. Source: 'Mean-Field Theory and the Gaussian Approximation' (Resource LINK), Figure 2.4.

Conclusion

Key Takeaways:

By applying the Gaussian approximation (expanding the Landau-Ginzburg functional to second order in fluctuations), we can explicitly calculate the two-point correlation function.
In Fourier space, the correlation function takes the Ornstein-Zernike form: $G (q) \propto \frac{1}{q ^{2} + ξ ^{- 2}}$ .
This derivation provides a direct link between the parameters of the Landau-Ginzburg functional and the correlation length: $ξ = κ / a_{0}$ .
From the temperature dependence $a_{0} \propto (T - T_{c})$ , we extract the mean-field critical exponent for the correlation length: $ν = 1/2$ .
The real-space correlation function decays as a power law $1/ r^{d - 2}$ at $T_{c}$ (implying $η = 0$ ) and decays exponentially as $e^{- r / ξ}$ for distances $r ≫ ξ$ away from $T_{c}$ .

Preview of the Next Lesson:

We have now calculated the correlation function and seen its structure. However, the Gaussian approximation neglects the crucial $m^{4}$ interaction term between fluctuation modes. When are we justified in doing this? In our next lesson, we will investigate the validity of mean-field theory by comparing the size of the fluctuations to the mean value of the order parameter. This will lead us to derive the Ginzburg criterion, a powerful tool for determining when mean-field theory breaks down.