  The nuclear matrix elements for the momentum quadrupole operator are
important for the interpretation of precision atomic physics experiments that
search for violations of local Lorentz and CPT symmetry and for new
spin-dependent forces. We use the configuration-interaction nuclear shell model
and self-consistent mean field theory to calculate these matrix elements in
$^{21}$Ne, $^{23}$Na, $^{131}$Xe, $^{173}$Yb and $^{201}$Hg. These are the
first microscopic calculations of the nuclear matrix elements for the momentum
quadrupole tensor that go beyond the single-particle estimate. We show that
they are strongly suppressed by the many-body correlations, in contrast to the
well known enhancement of the spatial quadrupole nuclear matrix elements.
