Landau-Ginzburg from Ising: Hubbard-Stratonovich Approach

Welcome to your first lesson in Advanced Statistical Mechanics. In your foundational studies, you explored phase transitions using tools like the mean-field approximation for the Ising model. While powerful, you know that mean-field theory is an approximation that neglects fluctuations, which become dominant near a critical point. Our goal in this course is to develop a more robust framework to handle these fluctuations systematically.

This lesson marks our first step in that direction. We will derive the celebrated Landau-Ginzburg free energy functional, not from phenomenological arguments alone, but by starting from the microscopic Ising model. This will transition us from a discrete spin system to a continuous field theory. The key mathematical tool we will master is the Hubbard-Stratonovich transformation, which allows us to trade interactions between spins for an integral over an auxiliary field. This process will culminate in expressing the partition function as a functional integral, a concept analogous to the path integrals you may have encountered in quantum mechanics, and the central object of study for much of this course.

From Spins to Fields: The Coarse-Graining Picture

Before diving into the mathematics, it's crucial to build a physical intuition for what we are trying to achieve. Near a critical point, the correlation length diverges, meaning that fluctuations are correlated over vast distances compared to the lattice spacing. The microscopic details of the spin interactions become less important than the collective, long-wavelength behavior of the system.

This motivates the idea of "coarse-graining": we average the microscopic spins over a region that is large compared to the lattice spacing but small compared to the correlation length. This procedure gives us a smooth, continuous field variable—the order parameter field $m (x)$ —which captures the essential physics of the transition.

The following reading from A. C. H. Cheung's notes on phase transitions provides an excellent introduction to this phenomenological viewpoint.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

This text introduces the foundational ideas of Ginzburg-Landau theory, focusing on the concept of coarse-graining and constructing a general Hamiltonian based on symmetries.

Please read the first section, "2.1 Ginzburg-Landau Theory". Pay close attention to the justification for replacing discrete spins with a continuous magnetization field m(\mathbf{x}) and how symmetry principles (locality, rotational invariance) constrain the form of the resulting Hamiltonian, shown in Equation (2.1). This phenomenological form is precisely what we are about to derive from a microscopic model.

The Hubbard-Stratonovich Transformation: A Mathematical Bridge

Now, we will develop the primary mathematical tool for this lesson. The Hubbard-Stratonovich (HS) transformation (also known as a Gaussian decoupling) is a technique that replaces an interaction term that is quadratic in some variables (like $s_{i} s_{j}$ ) with an integral over a new auxiliary field that couples linearly to those variables. The core of this "trick" rests on a simple Gaussian integral identity.

Instead of just stating the result, you will derive it yourself. The following problem set provides a step-by-step guide. Your background in linear algebra will be useful here, as the proof for the multidimensional case involves changing to the eigenbasis of the interaction matrix.

[PDF] RG Methods in Statistical Field Theory: Problem Set 1

This problem set guides you through the proof of the Gaussian integral identity that underpins the Hubbard-Stratonovich transformation and its initial application to the Ising Hamiltonian.

First, focus on part (a) of the problem set. Follow the hint provided to prove the identity for the integral over N variables \phi_i. The key is to diagonalize the matrix A and perform a change of variables to its eigenbasis. Next, work through parts (b) and (c). These steps show you how to use the identity from part (a) to rewrite the Boltzmann factor e^{-\beta H} for the Ising model as an integral over auxiliary fields. This is the HS transformation in action.

The general identity you proved allows us to replace an exponential of a quadratic term, $e^{\frac{1}{2} s^{T} Js}$ , with an integral over a field $m$ , like so:

e^{\frac{1}{2} s^{T} Js} \propto \int D m exp (- \frac{1}{2} m^{T} J^{- 1} m + s^{T} m)

Note that we have effectively "decoupled" the spins. The complicated $s_{i} s_{j}$ interaction is gone, and each spin $s_{i}$ now couples linearly to a corresponding field component $m_{i}$ . This makes the next step—summing over the spin configurations—tractable.

Derivation of the Landau-Ginzburg Functional

With the HS transformation in hand, we are ready to derive the Landau-Ginzburg functional for the Ising model. The process involves four key steps:

Introduce the Auxiliary Field: Apply the HS transformation to the Ising partition function to replace the spin-spin interaction term with an integral over a field.
Integrate out the Spins: With the spins decoupled, perform the sum over all spin configurations ( $s_{i} = \pm 1$ ).
Expand for Small Fields: Assume we are near the critical point where the order parameter field is small, and expand the resulting expression in a power series.
Take the Continuum Limit: Treat the field as a slowly varying function of space and expand its spatial dependence in terms of gradients.

The following resource presents a concise version of this full derivation. Having worked through the initial steps in the problem set, you should be well-equipped to follow this derivation.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

This section provides the full, four-step derivation of the Landau-Ginzburg Hamiltonian from the discrete Ising model.

Read section 2.4 Derivation of GL Hamiltonian for the Ising model. Follow the logic from Step 1 to the final result. Steps 1 & 2 formalize what you just saw in the problem set, culminating in the partition function expressed as an integral over the fields \Psi_i with the \ln \cosh term. Step 3 is the crucial expansion. The \ln \cosh(\Psi_i) is expanded as \frac{1}{2}\Psi_i^2 - \frac{1}{12}\Psi_i^4 + \dots. This is where the famous \Psi^2 and \Psi^4 terms of the Landau free energy originate. Step 4 handles the spatial dependence. The derivation switches to Fourier space to invert the interaction matrix G_{ij}. Notice how the expansion of \cos(q\alpha a) for small wave-vector q gives rise to the |q|^2 term. In real space, a q^2 term corresponds to a gradient-squared term, (\nabla\Psi)^2.

After these steps, we arrive at the final form of the partition function as a functional integral:

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

where $β H [Ψ (r)]$ is the Landau-Ginzburg free energy functional:

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

Here, the coefficients $t$ , $K$ , and $u$ are directly related to the parameters of the original Ising model ( $J$ and the lattice coordination number $d$ ). For instance, the reduced temperature $t \propto (\frac{1}{K _{B} T _{c}} - \frac{1}{K _{B} T})$ changes sign at the mean-field critical temperature.

The Landau free energy density, $\psi_0$, as a function of the order parameter, $m_0$, for different temperatures. Above the critical temperature ($t > 0$), the minimum is at $m_0 = 0$. Below the critical temperature ($t < 0$), a "mexican hat" potential develops, and the system spontaneously chooses a non-zero magnetization $\pm m_s$ to minimize its free energy.

Conclusion

In this lesson, we have built a crucial bridge from the microscopic world of discrete spins to the macroscopic world of continuous fields. You have seen that the Landau-Ginzburg functional, which is often introduced phenomenologically, can be rigorously derived from the Ising model.

The key takeaways are:

The Hubbard-Stratonovich transformation is a powerful mathematical technique that converts a model with pairwise interactions into a field theory.
This procedure expresses the partition function as a functional integral over all possible configurations of an order parameter field.
The terms in the resulting Landau-Ginzburg functional have clear physical interpretations: the coefficient of the $Ψ^{2}$ term, $t$ , measures the distance from the critical point; the $(\nablaΨ)^{2}$ term penalizes spatial inhomogeneities (like domain walls); and the $Ψ^{4}$ term ensures the stability of the system.

We now have a formidable tool at our disposal. But how do we use it? The functional integral is an infinite-dimensional integral that is generally impossible to solve exactly. In our next lesson, we will explore the simplest approximation for evaluating it: the saddle-point approximation. You will show that applying this method to the functional integral we derived today reproduces the familiar results of mean-field theory, confirming the consistency and power of our new formalism.

Recommended resources

From Spins to Fields: The Coarse-Graining Picture

The Hubbard-Stratonovich Transformation: A Mathematical Bridge

Derivation of the Landau-Ginzburg Functional

Conclusion