  Motivated by dimensional crossover in layered organic ${\kappa}$ salts, we
determine the phase diagram of a system of four periodically coupled Hubbard
chains with frustration at half filling as a function of the interchain hopping
${t_{\perp}/t}$ and interaction strength ${U/t}$ at a fixed ratio of
frustration and interchain hopping ${t'/t_{\perp}=-0.5}$. We cover the range
from the one-dimensional limit of uncoupled chains (${t_{\perp}/t=0.0}$) to the
isotropic model (${t_{\perp}/t=1.0}$). For strong ${U/t}$, we find an
antiferromagnetic insulator; in the weak-to-moderate-interaction regime, the
phase diagram features quasi-one-dimensional antiferromagnetic behavior, an
incommensurate spin-density wave, and a metallic phase as ${t_{\perp}/t}$ is
increased. We characterize the phases through their magnetic ordering,
dielectric response, and dominant static correlations. Our analysis is based
primarily on a variant of the density-matrix renormalization-group algorithm
based on an efficient hybrid-real-momentum-space formulation, in which we can
treat relatively large lattices albeit of a limited width. This is complemented
by a variational cluster approximation study with a cluster geometry
corresponding to the cylindrical lattice allowing us to directly compare the
two methods for this geometry. As an outlook, we make contact with work
studying dimensional crossover in the full two-dimensional system.
