Emergent quasiparticles at Luttinger surfaces

In periodic systems of interacting electrons, Fermi and Luttinger surfaces refer to the locations within the Brillouin zone of poles and zeros, respectively, of the single-particle Green’s function at zero energy and temperature. Such difference in analytic properties underlies the emergence of well-defined quasiparticles close to a Fermi surface, in contrast to their supposed non-existence close to a Luttinger surface, where the single-particle density-of-states vanishes at zero energy. We here show that, contrary to such common belief, dispersive ‘quasiparticles’ with infinite lifetime do exist also close to a pseudo-gapped Luttinger surface. Thermodynamic and dynamic properties of such ‘quasiparticles’ are just those of conventional ones. For instance, they yield well-defined quantum oscillations in Luttinger surface and linear-in-temperature specific heat, which is striking given the vanishing density of states of physical electrons, but actually not uncommon in strongly correlated materials.

(S1.2) We emphasise that the equivalence between the two sums in Eq. (S1.2) is a trivial consequence of the summation over all Matsubara frequencies n = (2n and of the fact that the series decays faster than 1/ n , which implies that we can safely set η = 0 before performing the summation over n, and thus that no boundary term appears.
Luttinger's theorem is valid if, upon defining the following equivalence holds for ω = 2πT → 0, thus for T → 0, in which case we are allowed to drop the last term on the right hand side of Eq. (S1.1), and thus recover Luttinger's standard expression of the number of particles.
The equivalence (S1.4) might seem obvious, but in reality is not so. We first note that, since G(i , k) and Σ(i , k) have discontinuous imaginary parts at = 0, the summation I must be dealt with care in the T → 0 limit, since the functions that are summed may be on different sides of the imaginary axis. Therefore, we can write so that the two summations n ≥ 1 and n ≤ −2 only involve functions on the same side of the imaginary axis. At this stage, it is tempting to straight take the T → 0 limit and conclude that which is correct to leading order in T . On the contrary, the T → 0 limit of I L does not pose any problem, and reads (S1.7) Since I = 0, it follows that is just a boundary term on the imaginary frequency axis. We emphasise that Eq. (S1.8) is just the leading order in T , hence higher order terms are neglected.
If those terms can be indeed neglected, Eq. (S1.8) implies that is smooth around = 0 and where, by definition, (S1.10) Eq. (S1.9) does holds under the analytic assumptions we make in the article and that lead (S1.11) Therefore, those same assumptions seem to imply that I L = 0 at leading order in T , and thus the conventional Luttinger's theorem applies.
Under the above assumptions, (S1.13) in which case the Luttinger integral would be quantised in integers. We note that the above result is actually not incompatible with Eq. (S1.8) being zero. Indeed, since Im Ω(i ) ∝ i for → 0, then Im ln Ω(iπT ) = Im ln Re Ω(iπT ) + i Im Ω(iπT ) Im Ω(iπT ) Re Ω(iπT ) + O T 3 ∼ T , (S1.14) naïvely vanishes at leading order in T , despite the whole series may converge to ±π. Similarly, Eq. (S1.8) is only the first term of a series expansions in powers of T . That series may indeed converge to a value different from the leading term. Re G + ( , k) Im Σ + ( , k) = 0 , (S1.15) which is fulfilled under the analytic assumptions we make in the work, only guarantees that the Luttinger integral I L is quantised in integer values, namely that Luttinger's theorem generically leads to an estimate of the electron number wrong by an integer number.

S2. BEYOND CONVENTIONAL LUTTINGER'S THEOREM
Let us elaborate further on this point, still closely following Ref. [6]. Eq. (S1.1) at T = 0 and so In the perturbative regime, where Landau's adiabatic hypothesis is valid, the Luttinger integral which is the standard expression of Luttinger's theorem. The point at which perturbation theory breaks down is also that at which a Luttinger surface first emerges. Suppose that k lies on that surface. It follows that right before the breakdown (b.b.) Assuming that conventional Luttinger's theorem holds true even at the breakdown point, we must conclude that n * (k) = 1, since δ(0 + , k) = −π/2 when Re G(0, k) = 0, namely when a double zero of Re G(i , k) = Re G(−i , k) appears right at = 0. After the breakdown (a.b.), that double zero generically splits into two, one zero moving along the > 0 semi axis, and the other symmetrically along < 0. It follows that after the breakdown Re G a.b. (0, k) has the opposite sign of Re G b.b. (0, k) before the breakdown. Should the breakdown occur simultaneously at all momenta k, it would correspond to a Mott transition, beyond which n * (k) remains pinned at one, i.e., half-filled density. What does it happen when instead the breakdown just gives rise to a Luttinger surface with a single-particle pseudo gap rather than a hard one? According to Ref. [6] also there n * (k) remains pinned at one, which would imply that, in absence of additional Fermi pockets, the system should be incompressible despite the existence of 'quasiparticles'. Correspondingly, doping away from half-filling necessarily leads to the emergence of Fermi pockets, contributing to two electrons per k-point when they are electron-like, and zero when they are hole-like.
Therefore, one can write for a generic k and just after perturbation theory has broken down which is the formula that use in the work. We observe that n * (k) in the generalised expression (S2.9) differs from the conventional one 2θ − E(k) by an integer, either positive or negative, in agreement with the previous discussion.