  Simulations of quantum transport in coherent conductors have evolved into
mature techniques that are used in fields of physics ranging from electrical
engineering to quantum nanoelectronics and material science. The most efficient
general-purpose algorithms have a computational cost that scales as $L^{6 \dots
7}$ in 3D, which on the one hand is a substantial improvement over older
algorithms, but on the other hand still severely restricts the size of the
simulation domain, limiting the usefulness of simulations through strong
finite-size effects. Here, we present a novel class of algorithms that, for
certain systems, allows to directly access the thermodynamic limit. Our
approach, based on the Green's function formalism for discrete models, targets
systems which are mostly invariant by translation, i.e. invariant by
translation up to a finite number of orbitals and/or quasi-1D electrodes and/or
the presence of edges or surfaces. Our approach is based on an automatic
calculation of the poles and residues of series expansions of the Green's
function in momentum space. This expansion allows to integrate analytically in
one momentum variable. We illustrate our algorithms with several applications:
devices with graphene electrodes that consist of half an infinite sheet;
Friedel oscillation calculations of infinite 2D systems in presence of an
impurity; quantum spin Hall physics in presence of an edge; the surface of a
Weyl semi-metal in presence of impurities and electrodes connected to the
surface. In this last example, we study the conduction through the Fermi arcs
of the topological material and its resilience to the presence of disorder. Our
approach provides a practical route for simulating 3D bulk systems or surfaces
as well as other setups that have so far remained elusive.

Mesoscale and Nanoscale Physics