# What is the anomalous dimension for two-dimensional multi-component spin systems?

My question is: What is the (predicted) anomalous dimension $\eta$ for the two-dimensional $n$-vector model (or $O(n)$ model)?

Note: I am not looking for a derivation of $\eta$. A simple reference to its predicted value will do. Tables of critical exponents are easy to find for the case $n = 1$ (the Ising model), for which $\eta = 1/4$. However, I have looked through a number of sources (including Cardy, Goldenfeld, Henkel, Huang, Kardar, Zinn-Justin) and failed to find the $n > 1$ case in $d = 2$.

More precisely: Let $\Lambda_N \subset \mathbb{Z}^d$ be an increasing sequence of hypercubes and define the finite-volume Hamiltonian $H_{n,N}(\sigma) = -\sum_{x, y \in \Lambda_N} \sigma_x \cdot \sigma_y$ for $\sigma : \Lambda_N \to \mathbb{R}^n$ (and $\cdot$ the dot product). The two-point function for the $n$-vector model is defined by $$\langle \sigma_0 \cdot \sigma_x \rangle_{\beta,n} = \lim_{N\to\infty} \langle \sigma_0 \cdot \sigma_x \rangle_{\beta,n,N},$$ where $\langle \cdot \rangle_{\beta,n,N}$ denotes the expectation with respect to the finite-volume Gibbs measure $\mu_{\beta,n,N}(d\sigma) = Z_{\beta,n,N}^{-1} e^{-\beta H_{n,N}(\sigma)}$ (with $Z_{\beta,n,N}$ the partition function).

Then at the critical temperature $\beta_c = \beta_c(n)$, the two-point function should scale according to a power law of the form $\langle \sigma_0 \cdot \sigma_x \rangle_{\beta_c,n} \sim C_n |x|^{-(d-2+\eta)}$. The exponent $\eta$ is known as the anomalous dimension and should depend only on $d$ and $n$. When $d = 2$, I expect $\eta$ to be rational.

• What do you mean by $\beta_c(n)$ when $n>1$? When $n=2$, are you interested in the BKT transition? Note that in this case the system is critical at all $\beta>\beta_c(2)$? When $n>2$, there is no positive-temperature phase transition. Dec 29, 2016 at 9:10

For those who are still interested, one may have a look at the QFT literature of the $$\phi^4$$ model, which is the continuous limit of the model you defined above. In this framework, one can try to reach perturbatively the value of the critical exponents using Feynman diagram expansions. In fact, the problem is that the critical dimension for this model is $$d=4$$, and $$d=2$$ is far from 4, therefore perturbative expansions are converging quite slow in this case. See e.g. https://arxiv.org/abs/1705.06483 where they computed $$\eta$$ in arbitrary $$n$$ and $$d$$ with very high precision.