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.