RESISTIVITY
OF
NON-GALILEAN-INVARIANT FERMI- AND NON-FERMI LIQUIDS

H.K. Pal^{a}, V.I. Yudson^{b}, and D.L. Maslov^{a}

^{a}Department of
Physics, University of Florida, Gainesville, FL 32611-8440, USA

E-mail: maslov@phys.ufl.edu

^{b}Institute for
Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow
Region, 142190, Russia

Received 13 April 2012; accepted 7 June 2012

H.K. Pal

E-mail: maslov@phys.ufl.edu

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 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/E_{F} (E_{F} 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 Bi_{2}Te_{3} 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.

Keywords: normal-state
electron transport, Fermi-liquid theory, quantum phase transitionsPACS: 71.10.Ay, 71.10.Hf, 73.20.-r

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