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

Abstract

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+83 ). 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.

Departure from the Wiedemann–Franz law in WP2 driven by mismatch in T-square resistivity prefactors

“…A second source of q-selectivity concerns momentum relaxing electron-electron scattering (See Fig.2.b). The quadratic temperature dependence of resistivity in a Fermi liquid is a manifestation of such scattering [22]. This is because the phase space for collision between two fermionic quasi-particles scales with the square of temperature.…”

Collision-dominated conductance in clean two-dimensional metals

“…This contrasts with the much milder power law growth of the relaxation length in 2D [see Eq. (55)]. Hence, we infer that as spatial dimensions are increased from 1D to 2D, the current relaxation is enhanced, which results in a stronger T dependence of the correction to the conductance.…”

Lorentz ratio of a compensated metal

“…As a result, the resistivity increases with temperature from its residual value at the lowest temperatures towards another impurity-controlled limiting value at the highest temperatures. 17,18 If the band masses differ substantially, so do the low-and high-temperature limits of the resistivity, and there is a well-defined intermediate region in which ρ scales just as T 2 even in 2D, without an extra logarithmic factor. Also, if a 2D metal is compensated but has an unequal number of electron and hole pockets (as it is the case, e.g., for the parent state of iron-based superconductors 22 ), the Fermi momenta of electrons and holes are different and, as result, the resistivity also scales just as T 2 , without an extra logarithmic factor.…”