Resistivity of non-Galilean-invariant Fermi- and non-Fermi liquids

While it is well-known that the electron-electron (ee) interaction cannot affect the resistivity of a Galilean-invariant Fermi liquid (FL), the reverse statement is not necessarily true: the resistivity of a non-Galilean-invariant FL does not necessarily follow a T 2 behavior. The T 2 behavior is guaranteed only if Umklapp processes are allowed; however, if the Fermi surface (FS) is small or the electron-electron interaction is of a very long range, Umklapps are suppressed. In this case, a T 2 term can result only from a combined – but distinct from quantum-interference corrections – effect of the electron-impurity and ee interactions. Whether the T 2 term is present depends on (i) dimensionality [two dimensions (2D) vs three dimensions (3D)], (ii) topology (simply- vs multiply-connected), and (iii) shape (convex vs concave) of the FS. In particular, the T 2 term is absent for any quadratic (but not necessarily isotropic) spectrum both in 2D and 3D. The T 2 term is also absent for a convex and simply-connected but otherwise arbitrarily anisotropic FS in 2D. The origin of this nullification is approximate integrability of the electron motion on a 2D FS, where the energy and momentum conservation laws do not allow for current relaxation to leading – second – order in T /EF (EF is the Fermi energy). If the T 2 term is nullified by the conservation law, the first non-zero term behaves as T 4 . The same applies to a quantum-critical metal in the vicinity of a Pomeranchuk instability, with a proviso that the leading (first non-zero) term in the resistivity scales as T^(D+2)/3 (T^(D+8)/3). We discuss a number of situations when integrability is weakly broken, e. g., by inter-plane hopping in a quasi-2D metal or by warping of the FS as in the surface states of topological insulators of the Bi2 Te3 family. The paper is intended to be self-contained and pedagogical; review of the existing results is included along with the original ones wherever deemed necessary for completeness.