Tomonaga-Luttinger liquids and Landau-Fermi liquids

We review the two theories widely used in condensed matter physics to describe generic interacting quantum metallic phases, beyond the Fermi gas paradigm. The Landau-Fermi liquid theory focuses on the characterization of quantum metals in two and higher spatial dimensions. However, this paradigm fails in one dimension, where the understanding of metallic states is done by resorting to the Tomonaga-Luttinger liquid theory. We discuss the striking differences between interacting metals in one dimension and their higher dimensional counterparts. Finally, we exemplify the use of the Tomonaga-Luttinger liquid theory in the description of the ground state and the quantum dynamics of quantum metals in a lattice.