Fluctuation-Dissipation Theorem and Static Susceptibility

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Hello! Welcome back to our exploration of Landau-Ginzburg theory.

In our previous lesson, we established the crucial concepts of the correlation function $G_{c} (r)$ and the correlation length $ξ$ . We saw that these quantities are essential for describing spatial fluctuations of the order parameter. The key takeaway was that as a system approaches a critical point, $ξ$ diverges, and correlations become long-ranged, decaying as a slow power law instead of a rapid exponential.

Today, we will build directly on that foundation to establish one of the most profound and elegant results in statistical physics. Our goal is to relate the static susceptibility to the integral of the correlation function via the fluctuation-dissipation theorem. This will forge a direct, quantitative link between the microscopic fluctuations we discussed last time and a macroscopic, measurable thermodynamic quantity. By the end of this lesson, it will be physically and mathematically clear why a diverging correlation length necessitates a diverging susceptibility.

1. The Fluctuation-Dissipation Theorem: A General Statement

The relationship we seek to establish is a cornerstone of statistical mechanics, known as the fluctuation-dissipation theorem. In its static form, it states that the linear response of a system to a uniform external field is proportional to the integrated sum of all its equilibrium correlations.

For a magnetic system, the susceptibility $χ$ measures the response of the magnetization to an applied magnetic field $H$ . The correlation function $G_{c} (r)$ measures the spatial correlations of magnetization fluctuations. The theorem connects them as follows:

χ = \frac{1}{k _{B} T} \int d^{d} r G_{c} (r)

(Note: Constant prefactors may vary depending on the precise definition of $χ$ and the order parameter).

Let's unpack the physical intuition behind this before diving into derivations. The susceptibility $χ$ tells us how easily the system as a whole can be polarized by an external field. The integral $\int G_{c} (r) d^{d} r$ measures the total, system-wide "amount" of correlation. It sums up the influence of a fluctuation at the origin on every other point in the system. The theorem tells us these two quantities are one and the same: a system is easy to polarize ( $χ$ is large) precisely because its internal fluctuations are correlated over large distances, allowing a small local perturbation to propagate and align the entire system.

We can make the link even more direct in Fourier space. Recall from the last lesson that the structure factor $S (q)$ is the Fourier transform of the correlation function:

S (q) = \int d^{d} r e^{- i q \cdot r} G_{c} (r)

The susceptibility is the response to a spatially uniform ( $q = 0$ ) field. It is therefore natural that it should be related to the $q = 0$ component of the structure factor:

χ \propto S (q = 0)

Evaluating the structure factor at $q = 0$ gives:

S (q = 0) = \int d^{d} r e^{- i 0 \cdot r} G_{c} (r) = \int d^{d} r G_{c} (r)

This provides a beautifully simple path to our main result. Now, let's see how this emerges from more rigorous starting points.

2. A Concrete Derivation: The Ising Model

To see that this is not just a phenomenological argument, let's derive the result from a microscopic Hamiltonian. The Ising model is the perfect setting for this.

The following text provides a clear, step-by-step derivation. Given your background, the "slick trick" of using a derivative with respect to the field $H$ to pull down a factor of the magnetization $M$ inside the thermal average should be a familiar and satisfying technique.

9.4: Correlation Functions in the Ising Model - Physics LibreTexts

Let's ground this relationship in a concrete microscopic model. The following text provides a rigorous derivation for the Ising model, showing precisely how the magnetic susceptibility emerges from summing up the spin-spin correlations across the entire lattice.

Please read section '9.4.2 Susceptibility from the correlation function'. Follow the derivative trick used to relate the susceptibility to the sum of correlations. The key result is equation (9.45). Also, pay close attention to the discussion at the end that explains why the sum over all sites, which seems to scale as N, does not lead to a non-extensive result on the right-hand side.

To summarize the derivation's key result, for a lattice of $N$ spins where each has magnetic moment $m$ :

χ (T, H) = \frac{m ^{2}}{k _{B} T} i, j \sum (⟨ s_{i} s_{j} ⟩ - ⟨ s_{i} ⟩ ⟨ s_{j} ⟩) = \frac{N m ^{2}}{k _{B} T} j \sum G_{j} (T, H)

Here, the correlation function $G_{j} = ⟨ s_{0} s_{j} ⟩ - ⟨ s_{0} ⟩^{2}$ is defined relative to an arbitrary origin site 0, and the sum runs over all sites $j$ in the lattice. This is the discrete lattice version of the integral relation we wrote down earlier.

3. The Continuum Field Theory Perspective

Now let's return to the Landau-Ginzburg continuum theory. This perspective is powerful because it's universal—it doesn't depend on the microscopic details of a specific model like the Ising Hamiltonian.

In the previous lesson, we saw that the structure factor, within the Gaussian approximation of Landau-Ginzburg theory, takes the Ornstein-Zernike form:

S (q) \propto G (q) \propto \frac{1}{q ^{2} + ξ ^{- 2}}

As argued above, the static susceptibility $χ$ is the system's response to a $q \to 0$ probe. Taking this limit, we find a direct relationship between susceptibility and correlation length:

χ \propto S (q \to 0) = q \to 0 lim \frac{1}{q ^{2} + ξ ^{- 2}} = \frac{1}{ξ ^{- 2}} = ξ^{2}

This simple but powerful result, $χ \propto ξ^{2}$ , is central to the Landau-Ginzburg description of critical phenomena.

The following video provides an excellent summary of this derivation within the field theory context and then uses it to connect the critical exponents.

4. The Landau-Ginzburg Approach Part 3

Professor Mehran Kardar provides a clear explanation of these concepts from the perspective of the Landau-Ginzburg field theory. He first derives the Ornstein-Zernike form for the correlation function and then explicitly uses it to connect susceptibility to the correlation length.

Please watch the segment from 25:33 to 34:09, and then from 1:05:13 to 1:11:05. The first part recaps the derivation of the Ornstein-Zernike form from the quadratic (Gaussian) approximation of the free energy. The second part (starting at 1:05:13) directly connects the susceptibility to the integral of the correlation function using a scaling argument and derives the relationship between their critical exponents.

The same argument is presented concisely in the notes from Cambridge, which you used in the last lesson. You can refer to section 2.5 of resource LINK (specifically, the unnumbered equation leading to Eq. 2.16 on page 24) for a textual version of this derivation.

4. Consequence: The Fisher Scaling Relation

We can now use the fluctuation-dissipation theorem to derive our first scaling relation—an identity that constrains the values of the critical exponents.

Let's recall the definitions of the exponents $γ$ , $ν$ , and $η$ :

Susceptibility: $χ \propto ∣ t ∣^{- γ}$
Correlation length: $ξ \propto ∣ t ∣^{- ν}$
Correlation function at $T_{c}$ : $G_{c} (r, t = 0) \propto \frac{1}{∣ r ∣ ^{d - 2 + η}}$

Now, let's use our main result, $χ \propto \int G_{c} (r) d^{d} r$ . As we approach the critical point, the correlation function $G_{c} (r)$ decays like the power law $∣ r ∣^{- (d - 2 + η)}$ for distances $∣ r ∣ < ξ$ , and then decays exponentially for $∣ r ∣ > ξ$ . Therefore, the integral is dominated by the region within one correlation volume. We can approximate the integral with a simple scaling argument:

χ \sim \int_{0}^{ξ} r^{d - 1} d r G_{c} (r) \sim \int_{0}^{ξ} r^{d - 1} d r (\frac{1}{r ^{d - 2 + η}}) = \int_{0}^{ξ} r^{1 - η} d r

Evaluating the integral gives:

χ \propto ξ^{2 - η}

Now, we substitute the power-law definitions for $χ$ and $ξ$ :

∣ t ∣^{- γ} \propto (∣ t ∣^{- ν})^{2 - η} = ∣ t ∣^{- ν (2 - η)}

By comparing the exponents on both sides, we arrive at Fisher's scaling relation:

γ = ν (2 - η)

This is a remarkable result. It shows that the three exponents describing the divergence of susceptibility, the divergence of the correlation length, and the decay of correlations at criticality are not independent. This is our first glimpse into the highly structured and constrained nature of critical phenomena, a topic we will explore fully in the module on Scaling and Universality.

Test your understanding!

Within mean-field theory (like the Landau-Ginzburg theory in the Gaussian approximation we've been using), the anomalous dimension exponent $η$ is exactly zero.

What does Fisher's scaling relation simplify to in mean-field theory?
In previous modules, you will have seen that mean-field theory predicts $γ = 1$ for the susceptibility and $ν = 1/2$ for the correlation length. Verify that these mean-field exponents satisfy the simplified scaling relation.

Show answer

With $η = 0$ , the Fisher scaling relation $γ = ν (2 - η)$ simplifies to $γ = 2 ν$ . This is precisely the result $χ \propto ξ^{2}$ that we derived directly from the Ornstein-Zernike form.
Plugging the mean-field values into the relation:
- Left side: $γ = 1$
- Right side: $2 ν = 2 \times (1/2) = 1$
  The relation $1 = 1$ holds. This demonstrates that while mean-field exponents may not be quantitatively correct for real systems (e.g., in 3D), they form a self-consistent theoretical framework that correctly obeys the scaling laws.

Conclusion

Key Takeaways:

The fluctuation-dissipation theorem provides a fundamental link between microscopic fluctuations and macroscopic response.
For a static, uniform perturbation, it states that the susceptibility is proportional to the spatial integral of the two-point correlation function: $χ \propto \int G_{c} (r) d^{d} r$ .
This relationship provides the physical mechanism for critical phenomena: as $T \to T_{c}$ , the correlation length $ξ$ diverges, causing the integral of $G_{c}$ to diverge, which in turn causes the susceptibility $χ$ to diverge.
In the language of Landau-Ginzburg theory and the Ornstein-Zernike approximation, this takes the simple form $χ \propto ξ^{2}$ .
This theorem leads to scaling relations among critical exponents, such as Fisher's relation: $γ = ν (2 - η)$ .

Preview of the Next Lesson:

We have now established the full machinery for describing fluctuations in the Gaussian approximation (i.e., keeping only quadratic terms in the fluctuations $ϕ$ ). This approximation is powerful, but it has its limits. When are we justified in neglecting the higher-order interaction terms (like $ϕ^{4}$ ) in the Landau-Ginzburg Hamiltonian? In the next lesson, we will answer this by deriving the Ginzburg criterion, which quantifies the validity of mean-field theory by comparing the size of fluctuations to the mean value of the order parameter.