  Materials with strong electronic correlations may exhibit a superconducting
(SC) phase when tuning some parameters, but they almost always also have
multiple other phases, typically insulating ones, that are in close competition
with SC. It is highly challenging to resolve this competition with quantitative
numerics for the group of quasi-two-dimensional materials such as the cuprates.
This is the case even for the simplified minimal models of these materials, the
doped 2D Hubbard model with repulsive interactions, where clusters of
sufficient size to determine the phase in the thermodynamic limit can be
hard-to-impossible to treat in practice. The present work shows how
quasi-one-dimensional systems, 2D and 3D arrays of weakly coupled 1D correlated
electrons, are much more amenable to resolve the competition between SC and
insulating orders on an equal footing using matrix-product states (MPS). Using
the recently established MPS plus mean field (MPS+MF) approach for fermions, we
demonstrate that large systems are readily reachable in these systems, and thus
the thermodynamic regime by extrapolation. Focusing on basic model systems, 3D
arrays of negative-U Hubbard chains with additional nearest-neighbor
interaction V, we show that despite the MF component of the MPS+MF technique we
can reproduce the expected coexistence of SC and charge-density wave at V=0 for
density n=1. We then show how we can tune away from coexistence by both tuning
V and doping the system. This work paves the way to deploy two-channel MPS+MF
theory on some highly demanding high-$T_c$ SC systems, such as 3D arrays of
repulsive-U doped Hubbard ladders, where we have recently characterized the
properties of such arrays in single-channel MPS+MF calculations. The present
approach could thus conclusively show that this SC order would actually be
obtained, by explicitly comparing SC against its insulating competitors.
