We apply field theoretical renormalization group (RG) methods to describe the Tomonaga–Luttinger model as an important test ground to deal with spin–charge separation effects in higher spatial dimensions. We calculate the RG equations for the renormalized forward couplings g 2 R and g 4 R up to two-loop order and demonstrate that they do not flow in the vicinities of the Fermi points (FPs). We solve the Callan–Symanzik equation for G ( a ) R in the vicinities of the FPs. We calculate the related spectral function and the momentum distribution function at p = k F , p 0 = ω. We compute the renormalized one-particle irreducible function Γ (2) (+) R ( p , p 0 = 0; Λ) and show that it carries important spin–charge separation effects in agreement with well-known results. Finally, we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.
We revisit the renormalization group (RG) analysis of the Kohn-Luttinger (KL) mechanism for superconductivity. The KL mechanism leads to superconductivity in a system with a repulsive bare interaction. The key ingredient is the screening effect that renders the induced interaction attractive in channels with nonzero angular momentum $\ensuremath{\ell}\ensuremath{\ne}0$, thereby triggering the Bardeen-Cooper-Schrieffer (BCS) instability. According to the original argument, the resulting gap is exponentially small, with its exponent scaling as $\ensuremath{-}{\ensuremath{\ell}}^{4}$. However, the KL mechanism was originally formulated within perturbation theory, where the series is known to converge poorly in certain cases---most notably, for the
p-wave paring gap induced by a repulsive
s-wave contact interaction. This poor convergence may be attributed to a divergent integrand in a specific class of diagrams containing both the BCS logarithm and the Kohn anomaly, suggesting that one must resum the Kohn anomaly contributions separately from the BCS logarithm. In this work, we incorporate the Kohn anomaly contribution into the $\ensuremath{\beta}$ function of the RG equation governing the BCS instability near the Fermi surface. Our solution shows that the KL gap exponent is then proportional to $\ensuremath{-}\ensuremath{\ell}$, indicating a significant enhancement of the KL mechanism beyond the previously known result. To illustrate this, we study the spin-triplet
p-wave pairing gap arising from a repulsive
s-wave contact interaction and compare our RG-based results with those obtained from the Bethe-Salpeter equation in perturbation theory.