Saddle-Point Approximation and Landau Theory

In our previous lesson, we successfully transformed the partition function of the discrete Ising model into a functional integral over a continuous order parameter field, $Ψ (r)$ . This powerful formalism frees us from the lattice and allows us to describe the system's long-wavelength behavior. We arrived at:

Z \propto \int D Ψ (r) exp (- β H [Ψ (r)])

where $β H [Ψ]$ is the Landau-Ginzburg (L-G) free energy functional. While exact, this expression involves an infinite-dimensional integral over all possible field configurations, which is generally intractable.

Today, we will take the first and most crucial step in approximating this integral. We will employ the saddle-point approximation, also known in this context as the mean-field approximation. You will see how this method, which seeks the most probable configuration of the field, precisely reproduces the results of the phenomenological Landau theory you've encountered before, including its predictions for critical exponents. This will firmly establish the connection between the microscopic statistical field theory and the more familiar mean-field picture of phase transitions.

The Saddle-Point Approximation (Laplace's Method)

Before applying it to our functional integral, let's first understand the saddle-point approximation for ordinary integrals. The core idea is that for an integral of the form $I_{N} = \int e^{- N f (x)} d x$ , where $N$ is a large parameter, the value of the integral is overwhelmingly dominated by the contribution from the region around the global minimum of $f (x)$ . The "large parameter" in our physical system is related to its size, as the free energy is an extensive quantity.

The following reading from an MIT OpenCourseWare resource provides a clear and rigorous derivation of this technique, known as Laplace's method. Given your mathematical background, you should find the step-by-step derivation both accessible and satisfying.

Laplace's Method and the Mean Field Ising Model

This section develops the mathematical machinery for the saddle-point approximation. It shows how to approximate an exponential integral by a Gaussian integral centered at the minimum of the exponent.

Please read section 3 Laplace’s Method and Evaluating Exponential Integrals. Follow the argument that begins by noting that the integrand e^{-Nf(x)} is highly peaked. Pay close attention to the Taylor expansion of f(x) around its minimum, x_1, and see how truncating it at second order leads to the final result in Equation (21). Also, note the discussion of the error term, which shows that the approximation becomes exact in the limit of large N.

The key result of Laplace's method is that the integral is approximated by the value of the integrand at the minimum of $f (x)$ , multiplied by a factor that comes from the Gaussian integral over the quadratic fluctuations around that minimum:

I_{N} = \int_{- \infty}^{\infty} d x e^{- N f (x)} ≃ \frac{2 π}{N f ^{''} ( x _{1} )} e^{- N f (x_{1})}

where $f^{'} (x_{1}) = 0$ and $f^{''} (x_{1}) > 0$ . The most important contribution to the free energy, which is proportional to $ln Z$ , comes from the exponential term, $e^{- N f (x_{1})}$ . Therefore, in the thermodynamic limit (large $N$ ), the free energy is simply given by the minimum value of the exponent.

Applying the Saddle-Point Method to the L-G Functional

We now apply this logic to our functional integral. The task is to find the "saddle point" of the integrand, which means finding the field configuration $Ψ_{0} (r)$ that minimizes the L-G functional $β H [Ψ (r)]$ .

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

This is a problem in the calculus of variations. However, we can simplify it enormously with a physical argument. The term $\frac{K}{2} (\nablaΨ)^{2}$ penalizes spatial variations in the order parameter. Since the coefficient $K$ is positive for a ferromagnet (as neighbors prefer to align), the absolute minimum of the functional must correspond to a spatially uniform field, $Ψ (r) = m_{0}$ .

The following video segment from an MIT OpenCourseWare lecture by Prof. Mehran Kardar lucidly explains this step.

3. The Landau-Ginzburg Approach Part 2

This video segment introduces the saddle-point approximation in the context of the L-G functional integral and justifies the simplification to a uniform order parameter.

Watch from the introduction of the saddle-point method. The key takeaway is that minimizing the functional under the assumption of a uniform order parameter reduces a complex functional minimization problem to a simple minimization of a regular function of one variable, \psi(m).

With $Ψ (r) = m_{0}$ , the gradient term vanishes, and the integral over space simply yields the system volume $V$ . The L-G functional becomes a simple function of the constant $m_{0}$ , and the free energy density $ψ_{0} = β H / V$ is:

ψ_{0} (m_{0}) = \frac{t}{2} m_{0}^{2} + u m_{0}^{4} - h m_{0}

The saddle-point approximation thus states that the equilibrium state of the system is found by simply minimizing this function with respect to $m_{0}$ . This potential should look familiar—it is precisely the phenomenological free energy posited in Landau theory.

The Landau free energy density, $\psi_0(t, m_0)$, as a function of the uniform order parameter $m_0$ for different reduced temperatures $t$. For $t>0$, the minimum is at $m_0=0$. For $t<0$, the potential takes on its characteristic "Mexican hat" shape, and the system minimizes its free energy by acquiring a non-zero spontaneous magnetization, $\pm m_s$.

Recovering Mean-Field Critical Exponents

By minimizing $ψ_{0} (m_{0})$ , we can recover all the standard results of mean-field theory. The condition for the minimum is $\frac{\partial ψ _{0}}{\partial m _{0}} = 0$ , which gives the mean-field equation of state:

t m_{0} + 4 u m_{0}^{3} - h = 0

Let's analyze this equation to find the critical exponents.

Spontaneous Magnetization (Exponent $β$ ):
In zero field ( $h = 0$ ), the equation is $m_{0} (t + 4 u m_{0}^{2}) = 0$ .
- For $t > 0$ ( $T > T_{c}$ ), the only real solution is $m_{0} = 0$ .
- For $t < 0$ ( $T < T_{c}$ ), non-trivial solutions appear: $m_{0} = \pm - t / (4 u)$ .
  Since $t \propto (T - T_{c})$ , the spontaneous magnetization $m_{0}$ scales as $(T_{c} - T)^{1/2}$ . This gives the mean-field exponent $β = 1/2$ .
Critical Isotherm (Exponent $δ$ ):
At the critical point ( $t = 0$ ), the equation of state simplifies to $4 u m_{0}^{3} = h$ .
This implies that $m_{0} \propto h^{1/3}$ . The definition of the critical exponent $δ$ is $m_{0} \propto h^{1/ δ}$ at $T_{c}$ , so we find $δ = 3$ .
Magnetic Susceptibility (Exponent $γ$ ):
The zero-field susceptibility is $χ = (\frac{\partial m _{0}}{\partial h})_{h = 0}$ . It's easier to compute its inverse: $χ^{- 1} = (\frac{\partial h}{\partial m _{0}})_{h = 0}$ . From the equation of state, $\frac{\partial h}{\partial m _{0}} = t + 12 u m_{0}^{2}$ .
- For $t > 0$ , $m_{0} = 0$ , so $χ^{- 1} = t$ . Thus, $χ \propto t^{- 1} \propto (T - T_{c})^{- 1}$ , which gives $γ = 1$ .
- For $t < 0$ , $m_{0}^{2} = - t / (4 u)$ , so $χ^{- 1} = t + 12 u (- t /4 u) = - 2 t$ . Thus, $χ \propto (- t)^{- 1} \propto (T_{c} - T)^{- 1}$ , which gives $γ^{'} = 1$ .

The following segment of the video lecture meticulously works through these derivations.

3. The Landau-Ginzburg Approach Part 2

In this segment, Prof. Kardar explicitly derives the mean-field critical exponents by minimizing the Landau potential.

Please watch from this concluding part of the lecture. It directly connects the minimization of \psi(m) to the calculation of the exponents \beta, \delta, and \gamma. Note the interesting universal prediction for the ratio of susceptibility amplitudes above and below T_c.

For a concise written summary of this procedure, you can refer to the following text.

[PDF] Chapter 5 Broken Symmetry and Collective Phenomena

This short section provides a compact review of the minimization of the Ginzburg-Landau Hamiltonian at the mean-field level to find the average magnetization.

Read the final part of the chapter, starting from the paragraph "Following this philosophy...". Focus on how minimizing the Landau Free energy directly yields the expression for the average magnetization \bar{S} below the critical temperature.

Conclusion

In this lesson, you have solidified the connection between the functional integral representation of the partition function and conventional mean-field theory. We have shown that the leading-order approximation for evaluating the functional integral—the saddle-point approximation—is mathematically equivalent to the physical procedure of minimizing the Landau free energy.

Our key takeaways are:

The saddle-point approximation reduces the problem of evaluating the functional integral to finding the field configuration that minimizes the L-G functional.
For a translationally invariant system, this configuration is a uniform order parameter, $m_{0}$ .
Minimizing the resulting Landau free energy density, $ψ_{0} (m_{0})$ , precisely reproduces the mean-field equation of state and the classical mean-field critical exponents ( $β = 1/2, γ = 1, δ = 3$ ).

We have now confirmed that our powerful field-theoretic formalism contains the familiar Landau theory as its simplest approximation. But the real power of this formalism lies in its ability to go beyond mean-field theory. In the next lesson, we will take the next step: we will analyze the effect of small, Gaussian fluctuations around the saddle-point solution. This will allow us to calculate the spatial correlation function and derive the famous Ornstein-Zernike form for the propagator.

Recommended resources

The Saddle-Point Approximation (Laplace's Method)

Applying the Saddle-Point Method to the L-G Functional

Recovering Mean-Field Critical Exponents

Conclusion