Optical and dc conductivity of the two-dimensional Hubbard model in the pseudogap regime and across the antiferromagnetic quantum critical point including vertex corrections

Abstract

The conductivity of the two-dimensional Hubbard model is particularly relevant for hightemperature superconductors. Vertex corrections are expected to be important because of strongly momentum dependent self-energies. To attack this problem, one must also take into account the Mermin-Wagner theorem, the Pauli principle and crucial sum rules in order to reach nonperturbative regimes. Here, we use the Two-Particle Self-Consistent approach that satisfies these constraints. This approach is reliable from weak to intermediate coupling. A functional derivative approach ensures that vertex corrections are included in a way that satisfies the f sum-rule. The two types of vertex corrections that we find are the antiferromagnetic analogs of the Maki-Thompson and Aslamasov-Larkin contributions of superconducting fluctuations to the conductivity but, contrary to the latter, they include non-perturbative effects. The resulting analytical expressions must be evaluated numerically. The calculations are impossible unless a number of advanced numerical algorithms are used. These algorithms make extensive use of fast Fourier transforms, cubic splines and asymptotic forms. A maximum entropy approach is specially developed for analytical continuation of our results. These algorithms are explained in detail in appendices. The numerical results are for nearest neighbor hoppings. In the pseudogap regime induced by two-dimensional antiferromagnetic fluctuations, the effect of vertex corrections is dramatic. Without vertex corrections the resistivity increases as we enter the pseudogap regime. Adding vertex corrections leads to a drop in resistivity, as observed in some high temperature superconductors. At high temperature, the resistivity saturates at the Ioffe-Regel limit. At the quantum critical point and beyond, the resistivity displays both linear and quadratic temperature dependence and there is a correlation between the linear term and the superconducting transition temperature. A hump is observed in the mid-infrared range of the optical conductivity in the presence of antiferromagnetic fluctuations.

Absence of a quantum limit to charge diffusion in bad metals

“…For a doped Hubbard model it was found in a 4 site DCA calculation that the vertex corrections to the optical conductivity were not significant, except very close to the Mott insulator 43 . A study of the doped two-dimensional Hubbard model using a two-particle self-consistent approach found that vertex corrections altered the calculated resistivity by less than a factor of two 44 . As has been pointed out 40 the interacting many electron system in strongly correlated materials so far has no gravity description and hence the strongly coupled gauge theory has no dual description.…”

Shear viscosity of strongly interacting fermionic quantum fluids

“…For a doped Hubbard model, it was found in a dynamical cluster approximation calculation based on a four-site cluster that the vertex corrections to the optical conductivity were not significant, except very close to the Mott insulator [47]. A study of the same model using a two-particle self-consistent approach found that vertex corrections changed the calculated resistivity by less than a factor of 2 [48]. In light of the above, it seems unlikely that vertex corrections would increase the viscosity by more than an order of magnitude compared to the DMFT results.…”

Damping effects in doped graphene: The relaxation-time approximation

“…(iii) The Aslamazov-Larkin contributions (diagrams 4C and 4D) [12,43] lead to the strong suppression of the normal backward scattering processes, and, therefore, causes a further reduction in M LL α (k, ω). In simple weakly interacting systems the result is the imaginary part of M LL α (k, ω) which is dominated by the Umklapp backward scattering processes, in agreement with the common Fermi liquid theory.…”