Gaussian Fluctuations and the Ornstein-Zernike Correlation Function

Welcome back to our course on Advanced Statistical Mechanics.

In the previous lesson, we established a crucial bridge between the microscopic Ising model and the continuum description of phase transitions. By applying the Hubbard-Stratonovich transformation, we formulated the partition function as a functional integral over an order parameter field $Ψ (r)$ , governed by the Landau-Ginzburg (L-G) free energy functional. We then showed that the simplest approximation to this integral—the saddle-point approximation—is exactly equivalent to the phenomenological Landau mean-field theory, correctly reproducing its critical exponents.

While powerful, the saddle-point approximation only considers the single most probable field configuration, completely neglecting the thermal fluctuations that are central to statistical mechanics. Today, we will take our first and most important step beyond mean-field theory. We will analyze the impact of small, spatially varying fluctuations around the mean-field solution. By expanding the L-G functional to second order in these fluctuations—a technique known as the Gaussian approximation—we will calculate the two-point spatial correlation function. This will lead us directly to the celebrated Ornstein-Zernike form for the correlation function, or "propagator," which is a cornerstone of modern statistical field theory.

Expanding Around the Saddle Point: The Gaussian Approximation

Let's begin with the Landau-Ginzburg functional we derived:

β H [Ψ] = \int d^{d} r [\frac{K}{2} (\nablaΨ (r))^{2} + \frac{t}{2} Ψ^{2} (r) + u Ψ^{4} (r)]

Here, we've set the external field $h = 0$ . The saddle-point approximation corresponded to finding the uniform field $Ψ (r) = m_{0}$ that minimizes this functional.

To include fluctuations, we now write the field $Ψ (r)$ as the sum of the uniform mean-field value $m_{0}$ and a space-dependent fluctuation field $ϕ (r)$ :

Ψ (r) = m_{0} + ϕ (r)

Our goal is to find the effective Hamiltonian for the fluctuation field $ϕ (r)$ . To do this, we substitute this expression back into $β H [Ψ]$ and expand in powers of $ϕ$ . For simplicity, let's work in the high-temperature phase ( $T > T_{c}$ ), where $t > 0$ and the mean-field solution is simply $m_{0} = 0$ . In this case, $Ψ (r) = ϕ (r)$ , and the expansion is trivial:

β H [ϕ] = \int d^{d} r [\frac{K}{2} (\nabla ϕ)^{2} + \frac{t}{2} ϕ^{2} + u ϕ^{4}]

The full functional integral $Z = \int D ϕ e^{- β H [ϕ]}$ remains intractable due to the $ϕ^{4}$ term, which represents interactions between the fluctuation modes. The Gaussian approximation consists of neglecting this interaction term. This is physically justified when the fluctuations are small, which we expect to be true far from the critical point.

Under this approximation, the Hamiltonian becomes purely quadratic:

β H^{(2)} [ϕ] = \int d^{d} r [\frac{K}{2} (\nabla ϕ (r))^{2} + \frac{t}{2} ϕ^{2} (r)]

The partition function is now a Gaussian functional integral, which is exactly solvable.

The following video segment provides an excellent walkthrough of this expansion. While the lecturer considers the more general case below $T_{c}$ (distinguishing longitudinal and transverse fluctuations), the core logic of expanding around the mean-field solution and keeping only quadratic terms is the same.

4. The Landau-Ginzburg Approach Part 3

In this segment, Prof. Kardar from MIT OpenCourseWare details the expansion of the L-G functional around the mean-field configuration to obtain a quadratic (Gaussian) theory for the fluctuations.

Please watch from this section. Focus on the strategy: write the total field as a mean value plus a fluctuation, substitute into the L-G functional, and expand to second order. Note how the linear term vanishes because the expansion is around a minimum of the energy. For our purposes, you can focus on the logic, which simplifies significantly for the T>T_c case where m_0=0.

The Propagator in Fourier Space

The quadratic Hamiltonian $β H^{(2)} [ϕ]$ becomes much simpler if we work in Fourier space. The gradient term $(\nabla ϕ)^{2}$ , which couples neighboring points in real space, becomes a simple algebraic term in momentum space. Let's define the Fourier transform of the fluctuation field:

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

Substituting this into $β H^{(2)}$ and performing the integration over $r$ , the Hamiltonian diagonalizes into a sum over independent contributions from each momentum mode $q$ :

β H^{(2)} [ϕ ~] = \frac{1}{2} \int \frac{d ^{d} q}{( 2 π ) ^{d}} (K q^{2} + t) ∣ ϕ ~ (q) ∣^{2}

The partition function $Z$ is now a product of independent Gaussian integrals, one for each mode $ϕ ~ (q)$ . We can now calculate expectation values. The object of interest is the two-point correlation function, defined as $G (r - r^{'}) = ⟨ ϕ (r) ϕ (r^{'}) ⟩_{c}$ , where the subscript $c$ denotes the connected correlator. For $T > T_{c}$ , the average magnetization $⟨ ϕ (r)⟩$ is zero, so this is just $⟨ ϕ (r) ϕ (r^{'})⟩$ .

In Fourier space, this corresponds to calculating $G (q) = ⟨ ∣ ϕ ~ (q) ∣^{2} ⟩$ . The statistical weight for each mode is $e^{- \frac{1}{2} (K q^{2} + t) ∣ ϕ ~ (q) ∣^{2}}$ . For a simple Gaussian random variable $x$ with probability $P (x) \propto e^{- a x^{2} /2}$ , the variance is $⟨ x^{2} ⟩ = 1/ a$ . Applying this principle to each Fourier mode, we find:

G (q) = ⟨ ∣ ϕ ~ (q) ∣^{2} ⟩ = \frac{k _{B} T}{K q ^{2} + t}

Ignoring the overall factor of $k_{B} T$ (or absorbing it into the field definition), we can rewrite this in a more suggestive form. We define the correlation length $ξ$ as:

ξ^{2} = \frac{K}{t}

With this definition, the correlation function in momentum space takes the famous Ornstein-Zernike form:

G (q) = \frac{1}{K ( q ^{2} + ξ ^{- 2} )}

This expression is fundamental. It is the propagator of the Gaussian field theory, describing how a fluctuation at a certain momentum propagates through the system.

For a very clear and concise written derivation using a variational approach, which you might appreciate given your background, please refer to the following notes. The method is slightly different, but it arrives at the identical Ornstein-Zernike form.

[PDF] Lecture 3: φ4 theory and Ornstein-Zernike form

These lecture notes from Prof. Naoki Kawashima at the University of Tokyo use the Gibbs-Bogoliubov-Feynman variational principle with a Gaussian trial Hamiltonian to find the optimal quadratic description of the system. This elegant method provides a rigorous path to the same result.

Please start reading at section [3-2] Variational approximation to φ4 model (on slide 10). Follow the derivation through slides 12-14, where a trial Gaussian Hamiltonian is introduced and the variational free energy is minimized. The key result appears on slide 15, where the minimization condition directly yields G_k \propto 1/(k^2 + \kappa^2), with \kappa = 1/\xi. This is precisely the Ornstein-Zernike form.

Physical Interpretation: Real-Space Correlations

The momentum-space propagator $G (q)$ is what is directly measured in scattering experiments (e.g., neutron scattering for magnets, light scattering for fluids). The intensity of scattering at a momentum transfer $q$ is proportional to $G (q)$ . The Lorentzian shape $1/ (q^{2} + ξ^{- 2})$ is a classic signature seen in countless experiments near critical points.

To gain a more intuitive physical picture, we must transform back to real space by computing the inverse Fourier transform:

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

This integral is the Green's function for the Helmholtz operator $(\nabla^{2} - ξ^{- 2})$ . You have likely encountered similar calculations for the Yukawa potential in nuclear physics or the screened Coulomb potential in plasma physics.

Prof. Kardar's lecture provides an outstanding and detailed derivation of the real-space form of this correlation function by solving the differential equation.

4. The Landau-Ginzburg Approach Part 3

This segment shows how to obtain the real-space correlation function G(r) by inverse Fourier transforming the Ornstein-Zernike form. The method involves solving the associated differential equation, which beautifully illustrates the physical meaning of the result.

Please watch from this section on real-space correlations. The key steps are: Setting up the Fourier integral for G(\mathbf{r}). Showing this integral is the Green's function for the equation (\nabla^2 - \xi^{-2})G(\mathbf{r}) \propto \delta(\mathbf{r}). Solving this equation in the two important asymptotic limits: r \ll \xi and r \gg \xi. Pay close attention to the final result. For large distances (r \gg \xi), the correlation function decays exponentially: G(r) \sim \frac{e^{-r/\xi}}{r^{(d-1)/2}} This confirms that \xi is indeed the characteristic length scale over which fluctuations are correlated. For distances much shorter than \xi, the exponential factor is irrelevant, and the correlation function decays as a power law: G(r) \sim \frac{1}{r^{d-2}}

New Critical Exponents: $ν$ and $η$

This analysis has not only given us the form of the correlation function but has also allowed us to calculate two new critical exponents within the mean-field/Gaussian approximation.

The Correlation Length Exponent, $ν$ :
We defined the correlation length as $ξ = K / t$ . Since $t \propto (T - T_{c})$ , the correlation length diverges as the critical point is approached:
$ξ \propto (T - T_{c})^{- 1/2}$
The critical exponent $ν$ is defined by $ξ \propto ∣ T - T_{c} ∣^{- ν}$ . Thus, our Gaussian approximation yields the mean-field value $ν = 1/2$ .
The Anomalous Dimension Exponent, $η$ :
Precisely at the critical point ( $T = T_{c}$ ), we have $t = 0$ and the correlation length $ξ \to \infty$ . In this case, the short-distance power-law behavior holds for all $r$ . The correlation function is generally defined to decay as $G (r) \propto 1/ r^{d - 2 + η}$ at $T_{c}$ . Our result, $G (r) \propto 1/ r^{d - 2}$ , implies that the mean-field value for the anomalous dimension is $η = 0$ .

The following notes provide a very compact summary of this derivation, which is a good final check on your understanding.

[PDF] Phase Transitions and Collective Phenomena - TCM Group

This resource from the University of Cambridge provides a concise summary of the entire procedure, from expanding the Hamiltonian to finding the real-space correlation function and extracting the critical exponents.

Read section 2.6 Fluctuations, Correlations & Susceptibilities. It begins by expanding the field (here for the more general case T<T_c). Then, it moves to Fourier space and immediately writes down the Ornstein-Zernike correlator (Eq. 2.22). Finally, it discusses the real-space form and extracts the exponent \nu=1/2.

Conclusion

In this lesson, we have moved beyond the simple mean-field picture by incorporating the effect of Gaussian fluctuations. This is a significant step, as it allows us to investigate the spatial structure of the system near a phase transition.

Our key takeaways are:

By expanding the L-G functional to second order in fluctuations around the mean-field solution, we arrive at a solvable Gaussian theory.
In Fourier space, this theory predicts a two-point correlation function, or propagator, of the Ornstein-Zernike form: $G (q) \propto 1/ (q^{2} + ξ^{- 2})$ .
The parameter $ξ$ is the correlation length, which represents the characteristic scale of correlated fluctuations. It diverges at the critical point, and this formalism predicts the mean-field exponent $ν = 1/2$ .
In real space, correlations decay exponentially beyond the distance $ξ$ . At the critical point ( $ξ \to \infty$ ), they decay as a power law, $1/ r^{d - 2}$ , which implies the mean-field exponent $η = 0$ .

Although the critical exponents we found are still the mean-field ones, we have developed a powerful framework. We have neglected the $ϕ^{4}$ interaction term, and it is precisely this term that will be responsible for corrections to mean-field theory and the emergence of non-trivial critical exponents.

In the next lesson, we will use our new understanding of fluctuations to assess the validity of the mean-field approximation itself. We will derive the famous Ginzburg criterion, which compares the size of the fluctuations we just calculated to the mean-field order parameter, telling us when and why mean-field theory breaks down.

Recommended resources

Expanding Around the Saddle Point: The Gaussian Approximation

The Propagator in Fourier Space

Physical Interpretation: Real-Space Correlations

New Critical Exponents: ν and η

Conclusion

New Critical Exponents: $ν$ and $η$