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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.

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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.

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  • $\begingroup$ Thanks. Not sure if I misunderstand. It sounds like the RG starting point for this case is changed to the Luttinger liquid instead of the original free field theory. I want to see how it is worked out in more detail. Do you happen to know where to look? $\endgroup$
    – xiaohuamao
    Commented Jul 17, 2023 at 0:13
  • $\begingroup$ @xiaohuamao Free field theory is a special case of the Luttinger liquid. If you are right at the free point, of course the interaction is irrelevant. However, as you crank up the interaction, it changes the Luttinger parameter, so you should keep track of both in the RG flow. Many references do this carefully, one that comes to my mind now is Giamarchi's book "quantum physics in one dimension". $\endgroup$
    – Meng Cheng
    Commented Jul 17, 2023 at 3:02

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