RESISTIVITY OF NON-GALILEAN-INVARIANT FERMI- AND NON-FERMI LIQUIDS
H.K. Pal^a, V.I. Yudson^b, and D.L. Maslov^a
aDepartment of Physics, University of Florida, Gainesville, FL 32611-8440, USA
E-mail: maslov@phys.ufl.edu
^bInstitute for Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, 142190, Russia

Received 13 April 2012; accepted 7 June 2012

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 T² behavior. The T² 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 T² term can result only from a combined – but distinct from quantum-interference corrections – effect of the electron-impurity and ee interactions. Whether the T² 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 T² term is absent for any quadratic (but not necessarily isotropic) spectrum both in 2D and 3D. The T² 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/E_F (E_F is the Fermi energy). If the T² term is nullified by the conservation law, the first non-zero term behaves as T⁴. 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 Bi₂Te₃ 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.