University of Basel

We performed the dimensional reduction of the interacting spin-degenerate D-dimensional electron gas with arbitrary interaction within the semiclassical limit $k_F r \gg 1$, $E_F \tau \gg 1$, where $k_F$ and $E_F$ are the Fermi momentum and the Fermi energy, respectively; $r$ and $\tau$ are spatial and temporal arguments of correlation functions. The dimensional reduction is performed exactly in all orders of perturbation theory. Our results agree with the multidimensional bosonization in assumption that the fermion loop cancellation theorem (FLCT) is valid. The FLCT states that all symmetrized fermion loops with more than two interaction vertices vanish. The FLCT is based on the linearity of the electron spectrum near the Fermi surface as well as on the assumption that the backscattering leading to the Friedel oscillations of dressed interaction is negligible. However, we find that the diagrams containing large number of fermion loops acquire large infrared-divergent factors in any D > 1. We estimate that this infrared divergence makes the spectral curvature near the Fermi surface a relevant perturbation in D > 2, the backscattering interaction is relevant in D > 1. As both the spectral curvature and the backscattering interaction explicitly violate the FLCT, we believe that the diagrams containing large number of fermion loops may be important for the infrared physics near the Fermi surface. The dimensional reduction allows us to classify the most infrared-divergent diagrams. In particular, we suggest new self-consistent approaches based on the resummation of the most infrared-divergent diagrams. The preprint is available on arXiv:2303.16732