Renormalization-group approach to half-filling charge density wave

Question

In Shankar's noted review paper on the renormalization group (RG) approach to many-body physics, Sec. IV deals with RG in a 1D lattice nearest-neighbour (quartically) interacting model, which leads to the conclusion of marginal quartic interaction $u$ and hence a Luttinger liquid without gap opening.

It is only vaguely mentioned at the very end of that section (p85 in the linked arxiv version) that one exception is the half-filling case with a CDW gap at moderately large $u$. The argument is the following. This case allows the (RR$\leftrightarrow$LL) umklapp scattering of $u$, which eventually becomes relevant. I want to see in more detail how it can become relevant and why the operator dimensions would change from free-field values. The paper doesn't seem to have described any of such machinery up to that point.

Answer 1

The scaling dimension of the umklapp scattering term is determined by the Luttinger parameter, which in turn depends on the interaction strength. Therefore, when the interaction strength reaches a critical value, the umklapp perturbation becomes relevant and opens a gap.