  In spite of their intrinsic one-dimensional nature matrix product states have
been systematically used to obtain remarkably accurate results for
two-dimensional systems. Motivated by basic entropic arguments favoring
projected entangled-pair states as the method of choice, we assess the relative
performance of infinite matrix product states and infinite projected
entangled-pair states on cylindrical geometries. By considering the Heisenberg
and half-filled Hubbard models on the square lattice as our benchmark cases, we
evaluate their variational energies as a function of both bond dimension as
well as cylinder width. In both examples we find crossovers at moderate
cylinder widths, i.e. for the largest bond dimensions considered we find an
improvement on the variational energies for the Heisenberg model by using
projected entangled-pair states at a width of about 11 sites, whereas for the
half-filled Hubbard model this crossover occurs at about 7 sites.
