Key Results for Special Limits of the O(n) Model

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the final lesson of our module on the Epsilon Expansion.

In our previous lesson, we generalized the scalar $ϕ^{4}$ theory to the O(n) vector model, a powerful framework for describing systems with n-component order parameters. We saw that the RG flow equations and the resulting critical exponents depend explicitly on this parameter $n$ , which formalizes the concept of universality classes being determined by dimension $d$ and symmetry $n$ .

A remarkable feature of the field-theoretic approach is that we can treat $n$ as a continuous parameter. This allows us to explore some fascinating and physically profound limits that are not immediately obvious. Today, we will state the key results for special limits of the O(n) model, specifically the cases where $n \to 0$ and $n \to \infty$ .

You will see how these abstract limits connect the sophisticated machinery of quantum field theory to two seemingly unrelated, classic problems in statistical physics:

The Self-Avoiding Walk (n→0): A fundamental model in polymer science and combinatorics.
The Spherical Model (n→∞): An exactly solvable variant of the Ising model that provides crucial insights into the role of dimensionality.

This exploration showcases the unifying power of the RG framework and provides a fitting conclusion to our study of the epsilon expansion.

1. The n → 0 Limit: Self-Avoiding Walks and Polymers

First, let's consider the perplexing limit where the number of order parameter components goes to zero. It turns out that this limit elegantly describes the statistical properties of self-avoiding walks (SAWs).

A SAW is simply a path on a lattice that never visits the same site twice. This problem is trivial to state but notoriously difficult to solve. It serves as a canonical model for a polymer chain in a good solvent, where the "excluded volume" effect prevents different segments of the chain from occupying the same space.

Representations of Self-Avoiding Walks and Polymer Coils — This image shows (a) a self-avoiding walk on a 2D lattice, (b) its continuous counterpart, (c) a representation of a polymer coil with its radius of gyration $R_c$, and (d) the interaction between two such coils. The SAW is the fundamental mathematical model for the configuration of a single polymer chain.

Two central questions in the study of SAWs are:

How many distinct SAWs of length $L$ exist, denoted $Z_{L}$ ?
What is the average size of such a walk, typically measured by the mean-square end-to-end distance, $⟨ R_{L}^{2} ⟩$ ?

For large $L$ , these quantities are believed to follow universal scaling laws:

Z_{L} \sim μ^{L} L^{γ - 1} and ⟨ R_{L}^{2} ⟩ \sim L^{2 ν}

Here, $μ$ is a non-universal lattice-dependent "connectivity constant," while $γ$ and $ν$ are universal critical exponents that depend only on the spatial dimension $d$ .

The Connection to the O(n) Model

The connection, first established by P.G. de Gennes, is a beautiful piece of theoretical physics reasoning. It involves representing the partition function of the O(n) model as a high-temperature expansion, which is a sum over diagrams (graphs) on the lattice.

One starts with the O(n) lattice Hamiltonian, $H = - J \sum_{⟨ ij ⟩} S_{i} \cdot S_{j}$ , where $S_{i}$ are n-component spins.
The partition function $Z = Tr [e^{- β H}]$ is expanded in powers of $J$ . This generates a sum over all possible paths and closed loops on the lattice.
A key step is to perform the trace over the spin components. Each closed loop in a diagram contributes a factor of $n$ (from $Tr (I_{n}) = n$ , where $I_{n}$ is the n-dimensional identity matrix).
If we now formally set $n = 0$ , any diagram containing a closed loop gets a multiplicative factor of zero and vanishes from the sum!
The only diagrams that survive are those without any closed loops—these are precisely the self-avoiding walks.

This stunning result means that the partition function of the O(n) model, in the limit $n \to 0$ , becomes the generating function for self-avoiding walks. Consequently, the critical exponents $γ$ and $ν$ for the SAW problem are precisely the critical exponents of the O(n) model evaluated at $n = 0$ .

We can now use our results from the epsilon expansion. Recall the one-loop expressions for the exponents:

ν = \frac{1}{2} + \frac{n + 2}{4 ( n + 8 )} ϵ + O (ϵ^{2}) and γ = 1 + \frac{n + 2}{2 ( n + 8 )} ϵ + O (ϵ^{2})

Setting $n = 0$ , we find the exponents for SAWs:

ν_{SAW} \approx \frac{1}{2} + \frac{2}{32} ϵ = \frac{1}{2} + \frac{ϵ}{16}

γ_{SAW} \approx 1 + \frac{2}{16} ϵ = 1 + \frac{ϵ}{8}

For a polymer in three dimensions, $d = 3$ , so $ϵ = 4 - 3 = 1$ . The RG prediction is $ν_{SAW} \approx 1/2 + 1/16 = 0.5625$ . This is already a significant improvement over the simple random walk result ( $ν = 1/2$ ) and remarkably close to the best numerical estimates ( $ν \approx 0.588$ ) and the famous Flory exponent ( $ν = 3/5 = 0.6$ ).

2. The n → ∞ Limit: The Spherical Model

Now let's turn to the opposite extreme: an infinite number of order parameter components. This limit connects the O(n) model to another celebrated solvable model in statistical mechanics: the spherical model, introduced by Berlin and Kac in 1952.

The spherical model is a variation of the Ising model. Instead of spins being restricted to $σ_{i} = \pm 1$ , they are continuous real variables. To prevent the partition function from diverging, a constraint is needed. Rather than constraining each spin individually, the spherical model imposes a single, global constraint on the sum of their squares:

i = 1 \sum N σ_{i}^{2} = N

where $N$ is the total number of spins.

The Connection to the O(n) Model

The equivalence between the $n \to \infty$ limit of the O(n) model and the spherical model is based on the law of large numbers. In the O(n) model, each spin $S_{i}$ is an n-component vector, typically with a fixed length constraint $∣ S_{i} ∣^{2} = n$ . In the $n \to \infty$ limit, the fluctuations of the length of an individual spin vector become statistically insignificant compared to the average. The many local constraints effectively merge into a single global constraint on the total length of all spin vectors, $\sum_{i} ∣ S_{i} ∣^{2} = N n$ , which is precisely the form of the spherical model constraint.

Solving the Spherical Model and its Critical Exponents

The great utility of the spherical model is that it is exactly solvable in any dimension. The global constraint is handled elegantly by introducing a Lagrange multiplier, which reduces the calculation of the partition function to a Gaussian integral. The final step involves a saddle-point integration over the Lagrange multiplier, which becomes exact in the thermodynamic limit.

The following resource provides a detailed derivation of the solution and the critical exponents. Given your mathematical background, you should find the derivation accessible and rewarding.

[PDF] 1. The spherical model

These lecture notes by Gordon Semenoff provide a self-contained, step-by-step solution of the spherical model. The analysis reveals how its critical behavior depends on dimensionality.

Please read the entire document (pages 1-8). Pay close attention to: How the spherical constraint is implemented with a Lagrange multiplier (Eq. 2). The derivation of the saddle-point equation for the parameter κ² (Eq. 9). The analysis of how κ² scales with temperature for different dimensions (Eq. 16), which is the key to finding the exponents. The final summary table of critical exponents and how they compare to mean-field theory (Eq. 30).

The key results from the spherical model are profound:

It exhibits a phase transition only for dimensions $d > 2$ .
For $2 < d < 4$ , the critical exponents are non-mean-field and depend continuously on the dimension $d$ .
For $d > 4$ (the upper critical dimension for this model), the exponents saturate to their classical mean-field values.

The table of exponents from the resource is particularly important:

Exponent	Mean-Field Theory	Spherical Model (2 < d < 4)	Spherical Model (d > 4)
$α$	0	$\frac{d - 4}{d - 2}$	0
$β$	1/2	1/2	1/2
$γ$	1	$\frac{2}{d - 2}$	1
$δ$	3	$\frac{d + 2}{d - 2}$	3

The spherical model was the first solvable model to demonstrate non-classical exponents that vary with dimension, providing crucial early validation for the ideas that would later be fully developed by the renormalization group.

Test your understanding!

Let's verify the consistency between the $\epsilon$-expansion and the spherical model for the exponent $\gamma$. The spherical model gives $\gamma = \frac{2}{d-2}$ for $2 < d < 4$. The O(n) model gives $\gamma \approx 1 + \frac{n+2}{2(n+8)}\epsilon$ to first order in $\epsilon = 4-d$.

Show that in the limit $n \to \infty$ , the $ϵ$ -expansion for $γ$ matches the Taylor expansion of the spherical model result around $d = 4$ .

Show answer

First, let's take the $n \to \infty$ limit of the $\epsilon$-expansion formula: $$ \lim_{n\to\infty} \gamma(n, \epsilon) = \lim_{n\to\infty} \left( 1 + \frac{n+2}{2(n+8)}\epsilon \right) = 1 + \frac{1}{2}\epsilon $$ Now, let's expand the exact spherical model result around $d=4$, or $\epsilon=0$. Let $d = 4-\epsilon$: $$ \gamma_{\text{spherical}} = \frac{2}{d-2} = \frac{2}{(4-\epsilon)-2} = \frac{2}{2-\epsilon} = (1 - \epsilon/2)^{-1} $$ Using the geometric series expansion $(1-x)^{-1} = 1 + x + O(x^2)$, we get: $$ \gamma_{\text{spherical}} = 1 + \frac{\epsilon}{2} + O(\epsilon^2) $$ The two results match to first order in $\epsilon$, confirming the consistency of the O(n) $\to \infty$ limit.

Conclusion

This lesson concludes our exploration of the epsilon expansion by examining the rich physical content hidden in the formal analytic continuation of the order parameter dimension $n$ .

Key Takeaways:

The parameter $n$ in the O(n) model can be treated as a continuous variable in the RG framework, leading to powerful and non-obvious connections.
The $n \to 0$ limit of the O(n) model describes the universal scaling properties of self-avoiding walks, which are fundamental to polymer physics. The critical exponents for SAWs can be calculated via the epsilon expansion.
The $n \to \infty$ limit of the O(n) model is equivalent to the exactly solvable spherical model.
The spherical model is a crucial theoretical laboratory: it correctly predicts a lower critical dimension of $d = 2$ , an upper critical dimension of $d = 4$ , and demonstrates non-classical, dimension-dependent exponents in between.

Preview of the Next Lesson:

We have now completed our deep dive into the Landau-Ginzburg-Wilson paradigm and the momentum-space RG. This theoretical machinery provides a nearly complete picture of continuous phase transitions. However, there are important phenomena that lie beyond this framework. In the next module, we will explore some of these "Advanced Topics in Critical Phenomena." We will begin by formally proving the Mermin-Wagner theorem, which rigorously explains why continuous symmetries ( $n \geq 2$ ) cannot be spontaneously broken in low dimensions ( $d \leq 2$ ), a result we've seen hinted at by both the spherical model and the behavior of Goldstone modes.