Ising to φ⁴: Hubbard-Stratonovich Mapping

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Hello! Welcome to the ninth module of our course.

In our previous lesson, we concluded our exploration of real-space renormalization group methods by examining their inherent limitations. We saw that for dimensions greater than one, the RG flow generates an unmanageable proliferation of new couplings, forcing us into uncontrolled, lattice-dependent approximations. This critical assessment motivated the need for a more powerful and systematic framework: a field-theoretic approach operating in momentum space.

Today, we take the first and most crucial step on that path. Our goal is to derive the continuous field theory that forms the starting point for momentum-space RG. We will see how a discrete lattice model, like the Ising model, can be formally mapped onto a continuous field theory.

The learning outcome for this lesson is to outline the mapping of the Ising model to a $ϕ^{4}$ field theory via the Hubbard-Stratonovich transformation. This powerful mathematical technique will allow us to trade the discrete spin variables for a continuous auxiliary field, which we can then identify as the order parameter field of the Landau-Ginzburg-Wilson theory.

1. The Mathematical Toolkit: Gaussian Integrals and the HS Trick

The entire procedure hinges on a clever application of Gaussian integrals, often called the Hubbard-Stratonovich (HS) or Gaussian transformation. Before we apply it to the complex Ising model, let's understand the core idea.

The simplest Gaussian integral is:

\int_{- \infty}^{\infty} e^{- a y^{2}} d y = \frac{π}{a}

A more useful form for our purposes involves a linear term in the exponent, which can be solved by completing the square:

\int_{- \infty}^{\infty} e^{- a y^{2} + b y} d y = \frac{π}{a} e^{b^{2} /4 a}

We can rearrange this identity to express a Gaussian function of $b$ as an integral:

e^{b^{2} /4 a} = \frac{a}{π} \int_{- \infty}^{\infty} e^{- a y^{2} + b y} d y

This is the essence of the HS transformation: a term that is quadratic in some variable ( $b$ ) in an exponent can be replaced by an integral over a new auxiliary field ( $y$ ), where the exponent is now linear in the original variable ( $b$ ).

Your background in physics and mathematics, particularly your coursework in QFT, means you are familiar with the extension of these ideas to many variables (vectors and matrices) and even to continuous functions (functional integrals). To formalize this and set the stage for our derivation, the following reading provides an excellent and concise review of the necessary mathematical machinery.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

This section from A. C. H. Cheung's lecture notes, 'Ginzburg-Landau Phenomenology', provides a formal summary of Gaussian integrals for single variables, multiple variables (using matrix notation), and their generalization to functional integrals over continuous fields. This will be the foundation for the transformation we are about to perform.

Please read section 2.3, 'Gaussian and Functional Integration'. Pay close attention to the generalization from the N-variable integral (Eq. 2.3) to the functional integral over a field φ(x). This bridge from discrete variables to continuous functions is precisely the conceptual leap we will be making.

2. Deriving the Field Theory from the Ising Model

With the mathematical tools in hand, we can now tackle the Ising model. Our goal is to transform the partition function, which is a sum over discrete spin configurations, into a functional integral over a continuous field. We will follow a clear four-step process.

The starting point is the partition function for the Ising model with nearest-neighbor interactions $J$ on a lattice, in the presence of an external magnetic field $h^{'}$ (we use $h^{'}$ to distinguish from the field in the final theory). For pedagogical convenience, we write the interaction term using a matrix $J_{ij}$ , which is $J$ if $i$ and $j$ are nearest neighbors and 0 otherwise.

Z = {s_{i} = \pm 1} \sum e^{- β H} = {s_{i} = \pm 1} \sum exp (\frac{β}{2} i, j \sum s_{i} J_{ij} s_{j} + β h^{'} i \sum s_{i})

Here, the quadratic term $\sum s_{i} J_{ij} s_{j}$ couples all the spins together, making the sum intractable (except in 1D and 2D). The HS transformation will allow us to decouple the spins.

The following reading provides a compact, step-by-step outline of the entire derivation. We will then break down each step in more detail.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

This section from Cheung's notes provides a bird's-eye view of the derivation we are about to undertake. It clearly lays out the four main stages of the transformation.

Read section 2.4, 'Derivation of GL Hamiltonian for the Ising model'. Focus on understanding the purpose of each of the four steps: introducing the field, integrating out spins, re-exponentiating, and expanding.

Let's now work through those steps with more detail, drawing on the logic from the reading.

Step 1: Introducing the Auxiliary Field

We apply the multi-variable Gaussian integral identity to the term $exp (\frac{β}{2} \sum_{i, j} s_{i} J_{ij} s_{j})$ . The general identity is:

exp (\frac{1}{2} x^{T} Ax) = \frac{1}{( 2 π ) ^{N} det ( A ^{- 1} )} \int i \prod d ϕ_{i} exp (- \frac{1}{2} ϕ^{T} A^{- 1} ϕ + ϕ^{T} x)

By identifying $A \to β J$ and $x \to s$ , we can rewrite the Boltzmann factor for the interaction term. Plugging this into the partition function gives:

Z \propto {s_{i}} \sum \int i \prod d ϕ_{i} exp (- \frac{1}{2 β} i, j \sum ϕ_{i} (J^{- 1})_{ij} ϕ_{j} + i \sum ϕ_{i} s_{i} + β h^{'} i \sum s_{i})

We have introduced a set of continuous auxiliary variables $ϕ_{i}$ , one for each lattice site. The crucial achievement is that the exponent is now linear in the spin variables $s_{i}$ .

Step 2: Integrating Out the Original Spins

Because the spins are now decoupled, we can swap the order of summation and integration and perform the sum over each $s_{i} = \pm 1$ independently:

Z \propto \int i \prod d ϕ_{i} exp (- \frac{1}{2 β} i, j \sum ϕ_{i} (J^{- 1})_{ij} ϕ_{j}) i \prod (s_{i} = \pm 1 \sum e^{(ϕ_{i} + β h^{'}) s_{i}})

The sum inside the product is straightforward:

s_{i} = \pm 1 \sum e^{(ϕ_{i} + β h^{'}) s_{i}} = e^{ϕ_{i} + β h^{'}} + e^{- (ϕ_{i} + β h^{'})} = 2 cosh (ϕ_{i} + β h^{'})

The partition function is now expressed solely as an integral over the auxiliary fields $ϕ_{i}$ :

Z \propto \int i \prod d ϕ_{i} exp (- \frac{1}{2 β} i, j \sum ϕ_{i} (J^{- 1})_{ij} ϕ_{j} + i \sum ln [2 cosh (ϕ_{i} + β h^{'})])

Step 3: From Lattice Variables to a Continuous Field

We have an effective theory for the fields $ϕ_{i}$ , but it's still defined on a lattice. To get to the Landau-Ginzburg-Wilson Hamiltonian, we make two approximations valid near the critical point:

Continuum Limit: Near criticality, the correlation length diverges, so the order parameter varies slowly on the scale of the lattice spacing $a$ . We can therefore replace the discrete variables $ϕ_{i}$ with a continuous field $ϕ (x)$ , where $x$ is the position of site $i$ . Sums over sites become integrals: $\sum_{i} \to \int \frac{d ^{d} x}{a ^{d}}$ .
Small Field Expansion: Near a second-order phase transition, the order parameter is small. We can therefore Taylor expand the $ln (cosh (\cdot))$ term. For simplicity, let's set the external field $h^{'}$ to zero for now. $ln [2 cosh (ϕ)] = ln (2) + \frac{1}{2} ϕ^{2} - \frac{1}{12} ϕ^{4} + O (ϕ^{6})$

The tricky part is the quadratic term $- \frac{1}{2 β} \sum_{i, j} ϕ_{i} (J^{- 1})_{ij} ϕ_{j}$ . It's much easier to analyze this in Fourier space. For nearest-neighbor interactions on a hypercubic lattice, the Fourier transform of the interaction matrix is $J (q) = 2 J \sum_{α = 1}^{d} cos (q_{α} a)$ . For long wavelengths ( $q a ≪ 1$ ), which dominate near criticality, we can expand the cosine:

J (q) \approx 2 J α = 1 \sum d (1 - \frac{( q _{α} a ) ^{2}}{2}) = 2 Jd - J a^{2} q^{2}

The quadratic term in the exponent involves $J (q)^{- 1}$ . Using the approximation $(1 - x)^{- 1} \approx 1 + x$ :

\frac{1}{J ( q )} \approx \frac{1}{2 Jd} (1 - \frac{a ^{2} q ^{2}}{2 d})^{- 1} \approx \frac{1}{2 Jd} (1 + \frac{a ^{2} q ^{2}}{2 d})

So the quadratic part of the effective Hamiltonian in Fourier space is $\propto \sum_{q} \frac{1}{2 β J ( q )} ∣ ϕ (q) ∣^{2}$ .

Step 4: Assembling the $ϕ^{4}$ Hamiltonian

Putting everything together, the exponent (the effective Hamiltonian) in the continuum limit becomes an integral over $x$ .
The terms are:

From $\frac{1}{2 β J ( q )}$ at $q = 0$ : A term $\propto ϕ (x)^{2}$ .
From the $q^{2}$ part of $\frac{1}{2 β J ( q )}$ : A term $\propto (\nabla ϕ (x))^{2}$ , since $q^{2}$ in Fourier space corresponds to $- \nabla^{2}$ in real space.
From the $ln (cosh)$ expansion: Terms $\propto ϕ (x)^{2}$ and $\propto ϕ (x)^{4}$ .

Combining the $ϕ^{2}$ terms and relabeling the coefficients, we arrive at the celebrated Landau-Ginzburg-Wilson Hamiltonian for a field $ϕ (x)$ (often denoted $m (x)$ to evoke magnetization):

β H_{e ff} [ϕ] = \int d^{d} x [\frac{r}{2} ϕ (x)^{2} + u ϕ (x)^{4} + \frac{c}{2} (\nabla ϕ (x))^{2} - h ϕ (x)]

The partition function is now a functional or path integral over all possible field configurations:

Z = \int D ϕ (x) exp (- β H_{e ff} [ϕ])

We have successfully mapped the discrete Ising model onto a continuous $ϕ^{4}$ field theory. The following problem set walks through this derivation in a guided, step-by-step fashion, which is a fantastic way to solidify your understanding.

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

This problem set from a course on RG Methods provides an explicit, hands-on derivation of the φ⁴ theory from the Ising model. It forces you to work through the algebraic steps we have just outlined, including deriving the final coefficients.

I recommend you work through parts (a) to (f) of this problem set with pen and paper. This will cement your understanding of the entire process. Part (a) is a proof of the Gaussian integral identity. Parts (b)-(d) apply the identity and integrate out the spins, leading to the effective lattice theory for the auxiliary field. Parts (e)-(f) take the continuum limit, perform the Taylor expansions, and ask you to find the final expressions for the coefficients r, u, and c in the φ⁴ Hamiltonian in terms of the microscopic Ising parameters (J, T, d).

Test your understanding!

In our derivation, the continuous field $ϕ (x)$ emerged as a mathematical tool (an auxiliary field) to simplify the calculation. What is the physical interpretation of this field? And why is it justifiable to expand the effective Hamiltonian in powers of $ϕ$ , keeping only terms up to $ϕ^{4}$ ?

Show answer

The field $ϕ (x)$ represents a coarse-grained, local average of the microscopic spins around the position $x$ . It is essentially the local magnetization density, which is the order parameter for the ferromagnetic phase transition.

The expansion in powers of $ϕ$ is justified because we are interested in the physics near a second-order critical point. At the critical point, the average magnetization is zero, and just below it, the spontaneous magnetization is very small. We assume the fluctuations around the ordered state are also small. Therefore, we can describe the system's free energy with a Taylor series expansion in the small order parameter $ϕ$ . Truncating at the $ϕ^{4}$ term is the minimal model that can describe a second-order phase transition, as it's the lowest-order even power that can stabilize the free energy (the $ϕ^{2}$ term's coefficient changes sign at the transition).

Conclusion

In this lesson, we have forged the essential link between the microscopic world of discrete spins and the macroscopic, continuous world of field theory. By using the Hubbard-Stratonovich transformation, we have rigorously shown how the statistical mechanics of the Ising model can be mapped onto the Landau-Ginzburg-Wilson $ϕ^{4}$ model.

Key Takeaways:

The Hubbard-Stratonovich transformation is a mathematical tool that uses a Gaussian integral to replace a term quadratic in an exponent (like $s^{T} Js$ ) with an integral over an auxiliary field that is coupled linearly to the original variables.
Applying this to the Ising partition function allows us to integrate out the discrete spin variables, leaving an effective theory for the continuous auxiliary field $ϕ_{i}$ .
By taking the continuum limit and assuming the field $ϕ (x)$ is small and slowly varying (valid near a critical point), we can expand the effective action.
This procedure yields the $ϕ^{4}$ Landau-Ginzburg-Wilson Hamiltonian, the universal starting point for studying critical phenomena in the Ising universality class. The coefficients of this field theory ( $r, u, c$ ) are directly related to the microscopic parameters ( $J, β, d$ ) of the original Ising model.

Preview of the Next Lesson:

We have successfully derived our target field theory. Now, we are equipped to apply the powerful machinery of the renormalization group to it. In the next lesson, we will begin our study of momentum-space RG by setting up the momentum-shell RG procedure for the Landau-Ginzburg-Wilson Hamiltonian. This will involve integrating out high-momentum "fast" modes to see how the effective theory for the remaining low-momentum "slow" modes is changed.