Is the lattice spacing $a$ a dangerously irrelevant parameter? Near a renormalization group fixed point, we can perform a scale transformation of length $L' = b^{-1} L$. In this case the relative lattice spacing should transform as $a' = b^{-1} a$. After $n$ renormalization group steps the lattice spacing is $a' = b^{-n}a$ which vanishes as $n \to \infty$. From this power counting argument the lattice spacing is irrelevant, however, it is known that the lattice spacing is important in defining the anomalous dimension exponent of the correlation function. From dimensional analysis, we know that $$G(\mathbf{r}) \propto |\mathbf{r}|^{2-d} \left( \frac{a}{|\mathbf{r}|}\right)^\eta .$$
Does this imply that the lattice spacing is dangerously irrelevant? If we arbitrarily set it to zero the anomalous dimension will vanish.
 A: Relevant, irrelevant, or marginal apply to the eigenvalues of the linearized RG transformation (i.e., a transformation of coupling constants) more than to physical observables like the lattice constant.
A single renormalization step in the usual implementation of RG to lattice systems corresponds to two conceptual operations:

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*renormalization of the Hamiltonian, i.e., a coarse-graining procedure aimed to reduce the number of degrees of freedom. In a lattice system, it is implemented by blocking technique, introducing a new lattice of larger lattice constant, $a'= b~a$.

*rescaling all the lengths by a factor $b^{-1}$ to return the new lattice constant to the original value $a$.

Then, it is clear that the lattice constant of the renormalized lattice always remains the same. It is the original lattice constant that is progressively reduced. Its presence at each renormalization step introduces a second length scale, in addition to the correlation length, eventually responsible for the anomalous dimension resulting from the analysis of the scaling of the correlation functions.
