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.

We present a theory of quantum oscillations in insulators that are particle-hole symmetric and non-topological but with arbitrary band dispersion, at both zero and non-zero temperature. At temperatures T less than or comparable to the gap, the dependence of oscillations on T is markedly different from that in metals and depends crucially on the position of the chemical potential µ in the gap. If µ is in the middle of the gap, oscillations do not change with T ; however, if µ is asymmetrically positioned in the gap, surprisingly, oscillations go to zero at a critical value of the inverse field determined by T and µ and then change their phase by π and grow again. Additionally, the temperature dependence is different for quantities derived from the grand canonical potential, such as magnetization and susceptibility, and those derived from the density of states, such as resistivity. However, the non-trivial features arising from asymmetric µ are present in both.Quantum oscillations provide one of the most commonly used experimental tools to study metallic band structures, in both weakly and strongly correlated systems [1-6]. They arise from Landau levels (LLs) crossing the Fermi level periodically as a function of the magnetic field. Such oscillations, therefore, are expected only in metallic systems with a Fermi surface.Recently, this canonical understanding has been challenged by the observation [7] of quantum oscillations in SmB 6 which is believed to be a topological Kondo insulator [8][9][10][11][12]. While the exact origin of the oscillations is still being debated [13,14], it raises the questions: can quantum oscillations arise in insulators? If yes, how are they different from oscillations in metals? Two recent works have addressed these questions for specific models. Ref.[15] considered a model-inspired by the experiment [7]-of a flat band hybridized with a dispersive band leading to a gap, and found oscillations in magnetization with a temperature dependence that is non-monotonic. However, it is not clear to what extent such findings depend on the enhanced density of states due to the flat band and the resulting strong particle-hole asymmetry. In contrast, Ref.[16] considered a model of a topological insulator and found multiple phase changes in oscillations in density of states (DOS) accompanied by a non-monotonic temperature dependence. These features, however, are entirely a consequence of the topological properties of the model and are not expected in an ordinary insulator. A theoryand general understanding-of oscillations in insulators is missing.In this Letter, we present a theory of quantum oscillations in insulators, at both zero and non-zero temperature. We construct our theory for a class of systems that are particle-hole symmetric and non-topological, but with arbitrary band dispersion. The motivation for adopting such a model is not to simply contrast our results with those of Refs.[15] and [16]: realistic systems with narrow gap and inverted bands where oscillations could be observed (reason f...