Ornstein-Zernike Correlation from Landau-Ginzburg

Introduction

In our last lesson, we constructed the Landau-Ginzburg free energy functional, $F [ψ]$ , by generalizing Landau's theory to accommodate spatially varying order parameters. We established that this functional acts as a coarse-grained effective Hamiltonian for the order parameter field, $ψ (r)$ . The mean-field approximation, which has been our primary tool so far, corresponds to finding the single field configuration that minimizes this functional, while ignoring all other possibilities.

Today, we move beyond this approximation to study the effect of thermal fluctuations around the mean-field state. This is a pivotal step, as these fluctuations are the very heart of critical phenomena. Our primary goal is to derive the two-point correlation function, which quantifies how a fluctuation at one point in space is correlated with a fluctuation at another. This will lead us to the celebrated Ornstein-Zernike form of the correlator and allow us to precisely define and calculate the correlation length, $ξ$ , the characteristic scale of critical fluctuations. This analysis will provide a deeper, microscopic understanding of the cooperative behavior near a phase transition.

Analyzing Fluctuations around the Mean-Field State

Let's consider the Landau-Ginzburg functional for an $n$ -component vector order parameter $m (r)$ , which we'll write in a slightly more general form, incorporating a wavevector-dependent susceptibility through the parameter $K$ :

F [m] = \int d^{d} r [\frac{t}{2} m^{2} + \frac{u}{4} m^{4} + \frac{K}{2} (\nabla m)^{2} - h \cdot m]

Here, $t \propto (T - T_{c})$ , and $u, K > 0$ .

The mean-field state, $m ˉ$ , is the uniform field that minimizes this functional. Above $T_{c}$ ( $t > 0$ ), this is simply $m ˉ = 0$ . Below $T_{c}$ ( $t < 0$ ), spontaneous symmetry breaking occurs, and the system chooses a ground state with uniform magnetization, say along the $e^_{1}$ direction, with magnitude $m ˉ = - t / u$ .

To study fluctuations, we expand the order parameter field around this mean-field value:

m (r) = m ˉ + δ m (r)

Below $T_{c}$ , it is crucial to decompose the fluctuation vector $δ m (r)$ into a component parallel (longitudinal) to the ordering direction and components perpendicular (transverse) to it.

m (r) = (m ˉ + ϕ_{l} (r)) e^_{1} + α = 2 \sum n ϕ_{t, α} (r) e^_{α}

$ϕ_{l}$ represents fluctuations in the magnitude of the order parameter, while the $n - 1$ components of $ϕ_{t}$ represent fluctuations in its direction.

We now substitute this expansion back into the functional $F [m]$ and keep terms only up to quadratic order in the fluctuation fields $ϕ_{l}$ and $ϕ_{t}$ . The terms linear in $ϕ$ must vanish, as we are expanding around a minimum of the energy functional. After some algebra, the functional separates into independent parts for the longitudinal and transverse modes.

The most convenient way to analyze this quadratic functional is by moving to Fourier space. The gradient terms $\nabla^{2}$ become simple $q^{2}$ factors, and the integral over space decouples into a sum over independent fluctuation modes for each wavevector $q$ . The probability of observing a fluctuation mode $ϕ (q)$ becomes Gaussian, $P (q) \propto exp (- β F [ϕ (q)])$ .

The following video provides an exceptionally clear step-by-step derivation of this procedure.

4. The Landau-Ginzburg Approach Part 3

This lecture from MIT OpenCourseWare by Prof. Mehran Kardar provides a complete derivation of the correlation function by analyzing quadratic fluctuations.

First, watch the section where he expands the Landau-Ginzburg energy to quadratic order in fluctuations around the mean-field value, separating them into longitudinal (\phi_l) and transverse (\phi_t) components. This part is from the expansion. Pay close attention to how the linear terms vanish. Next, see how this quadratic energy is diagonalized by transforming to Fourier space, leading to a Gaussian probability distribution for each independent q-mode, from Fourier transform. He derives the functional form for the two-point correlator \langle |\phi(\mathbf{q})|^2 \rangle, which is known as the Ornstein-Zernike form.

The Correlation Function and Correlation Length

As demonstrated by Prof. Kardar, the analysis of Gaussian fluctuations yields the two-point correlation function in Fourier space. This function, often called the static structure factor $S (q)$ , is what is directly measured in scattering experiments. Its form is a Lorentzian:

G_{α} (q) = ⟨ ∣ ϕ_{α} (q) ∣^{2} ⟩ \propto \frac{1}{K ( q ^{2} + ξ _{α}^{- 2} )}

where $α$ can be longitudinal ( $l$ ) or transverse ( $t$ ). This is the celebrated Ornstein-Zernike correlation function. The parameter $ξ_{α}$ is the correlation length for the $α$ -component of the fluctuations.

From the quadratic expansion of the functional, we can read off the expressions for the inverse squared correlation lengths:

K ξ_{l}^{- 2} = t + 3 u m ˉ^{2} and K ξ_{t}^{- 2} = t + u m ˉ^{2}

Let's analyze their behavior near the critical point ( $t \to 0$ ):

Above $T_{c}$ ( $t > 0$ ):
Here, $m ˉ = 0$ . The distinction between longitudinal and transverse vanishes.
$K ξ^{- 2} = t ⟹ ξ = \frac{K}{t} = K / a_{0} (T - T_{c})^{- 1/2}$
The correlation length diverges as we approach the critical temperature from above. This divergence is characterized by the critical exponent $ν$ , defined by $ξ \propto ∣ T - T_{c} ∣^{- ν}$ . Our mean-field calculation gives $ν = 1/2$ .
Below $T_{c}$ ( $t < 0$ ):
Here, $m ˉ^{2} = - t / u$ .
- Longitudinal: $K ξ_{l}^{- 2} = t + 3 u (- t / u) = - 2 t$ . $ξ_{l} = \frac{K}{- 2 t} = K /2 a_{0} (T_{c} - T)^{- 1/2}$ The longitudinal correlation length also diverges with the same exponent $ν = 1/2$ . Note the universal amplitude ratio $ξ_{0}^{+} / ξ_{0}^{-} = 2$ .
- Transverse: $K ξ_{t}^{- 2} = t + u (- t / u) = 0$ . $ξ_{t} = \infty$ The transverse correlation length is infinite for all $T < T_{c}$ . This is a profound result: it signifies the presence of long-range correlations associated with the massless Goldstone modes. Any fluctuation that corresponds to a rotation of the order parameter in the degenerate ground-state manifold costs zero energy in the long-wavelength limit.

The following text provides a concise written summary of this derivation, which is a perfect complement to the video lecture.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

This chapter from a textbook by Ben Simons provides an excellent written account of the derivation of the correlation function and length.

Please focus on Section 2.5, "Fluctuations, Correlations & Susceptibilities". Start by reviewing the expansion of the Hamiltonian in terms of longitudinal and transverse fluctuations. Then, see how the coefficients of the quadratic terms define the correlation lengths \xi_l and \xi_t in Eq. (2.12). Finally, observe how the Fourier transform leads to the Ornstein-Zernike form in Eq. (2.13).

The Correlation Function in Real Space

The Ornstein-Zernike form in Fourier space tells us about the response to periodic perturbations of a given wavelength. To understand how a localized fluctuation at the origin affects its surroundings, we must transform the correlation function back to real space. This requires computing the inverse Fourier transform:

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

This integral is the Green's function for the Helmholtz operator, satisfying the differential equation:

(\nabla^{2} - ξ^{- 2}) G (r) \propto - δ^{d} (r)

Your experience with Green's functions in electromagnetism and QFT is directly applicable here. The solutions to this equation are known as Yukawa potentials or screened Coulomb potentials. The key physical insight comes from analyzing the asymptotic behavior of $G (r)$ .

Prof. Kardar's lecture continues with a detailed derivation of this real-space behavior.

4. The Landau-Ginzburg Approach Part 3

Let's now follow the derivation of the real-space correlation function and its physical interpretation.

Watch the segment from this point, where Prof. Kardar sets up and solves the differential equation for the real-space correlator G(\mathbf{r}). He analyzes the solution in two important limits: for separations r \ll \xi and for r \gg \xi. After finding the spatial form, he connects all the pieces together. In the final segment, from this part, he explicitly shows how \xi is the length scale that governs the exponential decay of correlations and relates its divergence (\nu=1/2) to the divergence of the susceptibility (\gamma=1) through the scaling relation \gamma=2\nu.

The analysis yields two distinct behaviors for the real-space correlation function $G (r)$ :

For short distances, $r ≪ ξ$ :
The $ξ^{- 2}$ term in the denominator is negligible. The correlation function is dominated by the $q^{- 2}$ term, which corresponds to the potential of a point charge in $d$ dimensions.
$G (r) \propto \frac{1}{r ^{d - 2}}$
The correlations exhibit a power-law decay.
For long distances, $r ≫ ξ$ :
The behavior of the integral is dominated by the exponential factor.
$G (r) \propto \frac{e ^{- r / ξ}}{r ^{(d - 1) /2}}$
The correlations exhibit exponential decay, with a characteristic decay length set precisely by $ξ$ .

This confirms the physical meaning of $ξ$ : it is the fundamental length scale that separates power-law decay from exponential decay. As $T \to T_{c}$ , $ξ \to \infty$ , and the power-law behavior extends to all length scales. At the critical point, the correlations become long-ranged.

Conclusion

In this lesson, we have moved beyond the simple mean-field picture by including quadratic fluctuations. This has allowed us to derive some of the most fundamental concepts in the theory of critical phenomena from the Landau-Ginzburg functional.

Key Takeaways:

By expanding the Landau-Ginzburg functional to second order around the mean-field solution, we can analyze the behavior of thermal fluctuations.
In Fourier space, the correlation function takes the Ornstein-Zernike form, $G (q) \propto (q^{2} + ξ^{- 2})^{- 1}$ , which is a Lorentzian directly observable in scattering experiments.
The parameter $ξ$ is the correlation length. Our calculation shows it diverges at the critical point as $ξ \propto ∣ T - T_{c} ∣^{- ν}$ with the mean-field exponent $ν = 1/2$ .
Below the critical temperature, transverse fluctuations associated with Goldstone modes have an infinite correlation length, corresponding to long-range, power-law correlations.
In real space, correlations decay as a power law ( $\sim r^{- (d - 2)}$ ) for distances $r ≪ ξ$ and decay exponentially ( $\sim e^{- r / ξ}$ ) for $r ≫ ξ$ .

We have successfully used the Landau-Ginzburg functional to describe the nature of correlations. However, our analysis was still an approximation, valid for small fluctuations. The question remains: when is this approximation valid? When do the fluctuations become so large that they overwhelm the mean-field behavior and invalidate our results? In our next lesson, we will answer this by deriving the Ginzburg criterion, which establishes the conditions for the validity of mean-field theory and points the way toward the more powerful framework of the Renormalization Group.

Recommended resources

Introduction

Analyzing Fluctuations around the Mean-Field State

The Correlation Function and Correlation Length

The Correlation Function in Real Space

Conclusion