Several numerical methods used in the study of tensor network renormalization are introduced, including the power, Lanczos, conjugate gradient, Arnoldi methods, and quantum Monte Carlo simulation.

Chapter 7 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. Newton-Cotes integration methods are first studied ad hoc, via Taylor expansions and, second, building on the interpolation machinery of the previous chapter. Standard techniques like the trapezoid rule and Simpson’s rule are introduced, including the Euler-Maclaurin summation formula. The error behavior is employed to produce an adaptive-integration routine and also, separately, to introduce the topic of Romberg integration. The theme of integration from interpolation continues, when Gauss-Legendre quadrature is explicitly derived, including the integration abscissas, weights, and error behavior. Emphasis is placed on analytic manipulations that can help the numerical evaluation of integrals. The chapter then turns to Monte Carlo, namely stochastic integration: this is painstakingly introduced for one-dimensional problems, and then generalized to the real-world problem of multidimensional integration. The chapter is rounded out by a physics project, on variational Monte Carlo for many-particle quantum mechanics, and a problem set.

A burgeoning number of research studies are emerging where scientific questions are being successfully addressed because of the combination of information revealed from atom probe microscopy and density functional theory (DFT). Situations where high-quality experimental data alone would not wholly answer the question at hand and, equally, situations where atomistic simulations would have no obvious starting place were it not for the atom probe. Atomic-scale analytical tomography holds great potential to expand the realm of mediation between experimentation and computer simulation of materials properties. Any model framework is applicable, but we have delved into detail for the case of DFT because it is a self-consistent theory that has arguably the most immediate and exciting intersection with ASAT data.

In this chapter, we discuss how to build effective many-body models starting from first principles electronic structure calculations and apply this general approach to graphene. We present quantitative results for the Fermi velocity renormalization, which were preliminary announced in Chapter 8. After that, we discuss many-body effects in graphene electron spectrum, static screening, and optical conductivity based on the results of lattice quantum Monte Carlo simulations. At the end, we consider many-body renormalization of minimal conductivity in graphene within the concept of environment-induced suppression of quantum tunneling.