Analytic continuation of numerical data obtained in imaginary time or frequency has become an essential part of many branches of quantum computational physics. It is, however, an ill-conditioned procedure and thus a hard numerical problem. The maximum-entropy approach, based on bayesian inference, is the most widely used method to tackle that problem. Although the approach is well established and among the most reliable and efficient ones, useful developments of the method and of its implementation are still possible. In addition, while a few free software implementations are available, a well-documented, optimized, general purpose and user-friendly software dedicated to that specific task is still lacking. Here we analyze all aspects of the implementation that are critical for accuracy and speed, and present a highly optimized approach to maximum-entropy. Original algorithmic and conceptual contributions include (1) numerical approximations that yield a computational complexity that is almost independent of temperature and spectrum shape (including sharp Drude peaks in broad background for example) while ensuring quantitative accuracy of the result whenever precision of the data is sufficient, (2) a robust method of choosing the entropy weight α that follows from a simple consistency condition of the approach and the observation that information-and noise-fitting regimes can be identified clearly from the behavior of χ 2 with respect to α, and (3) several diagnostics to assess the reliability of the result. Benchmarks with test spectral functions of different complexity and an example with an actual physical simulation are presented. Our implementation, which covers most typical cases for fermions, bosons and response functions, is available as an open source, user friendly software.

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.