  Checkerboard patterns have been proposed in order to explain STM experiments
on the cuprates BSCCO and Na-CCOC. However the presence of these patterns has
not been confirmed by a bulk probe such as neutron scattering. In particular,
simple checkerboard patterns are inconsistent with neutron scattering data, in
that they have low energy incommsensurate (IC) spin peaks rotated 45 degrees
from the direction of the charge IC peaks. However, it is unclear whether other
checkerboard patterns can solve the problem. In this paper, we have studied
more complicated checkerboard patterns ("modulated checkerboards") by using
spin wave theory and analyzed noncollinear checkerboards as well. We find that
the high energy response of the modulated checkerboards is inconsistent with
neutron scattering results, since they fail to exhibit a resonance peak at
(pi,pi), which has recently been shown to be a universal feature of cuprate
superconductors. We further argue that the newly proposed noncollinear
checkerboard also lacks a resonance peak. We thus conclude that to date no
checkerboard pattern has been proposed which satisfies both the low energy
constraints and the high energy constraints imposed by the current body of
experimental data in cuprate superconductors.
