There is quite a lot of discussion on SE about correlation functions in lattice models. So I would say that it is well known that the two-spin (two-point) correlation function has the following asymptotics at large distances $$ \langle \sigma_{\vec{R}}\sigma_{\vec{R}+\vec{r}}\rangle \sim \frac{f(\vec{n})}{r^a}e^{-r/\xi(\vec{n})},\quad r=|\vec{r}|,\vec{n} = \vec{r}/r. $$ Here the vector $\vec{n}$ defines a direction on the lattice, $\xi$ is a direction-dependent correlation length, $f$ is a direction-dependent and distance-independent amplitude, and the parameter $a$ I will call the pre-exponential factor index. I am not sure that the last term is accepted. In his answer to another question, Yvan Velenik wrote that for $d$-dimensional ferromagnetic systems, the typical value of the index $a$ is $(d-1)/2$. I became interested in whether it is possible that the index $a$ depends on direction in some non-ferromagnetic models. Are there any known examples of models where $a$ changes continuously or discontinuously with the variation of the direction $\vec{n}$? One somewhat strange motivation for asking this question is that among lattice models, some are known for having critical indices that vary along the critical line. I am far from claiming that there is an analogy here, but I cannot help feeling that the objects called indices in these different situations may share the property of not being constant.
Update. After comments by Syrocco and Yvan Velenik, I wanted to clarify that my question concerns correlations in translationally invariant systems with short-range interactions and far from boundaries, i.e. in the bulk. The system may also be close to criticality, but this is not necessarily implied.
