  Using SU(2) lattice QCD in two dimensions, we study diagonal and off-diagonal
gluon propagators in the maximally Abelian gauge (MAG) with U(1)$_3$ Landau
gauge fixing. These propagators are investigated both in momentum space and
coordinate space. The Monte Carlo simulation is performed at $\beta=7.99,
30.5,$ and $120$ on $62^2, 128^2,$ and $256^2$ at the quenched level. In the
momentum space, the transverse component of the diagonal gluon propagator shows
suppression with increasing $\beta$ in the infrared region and the dressing
function at $\beta=120$ has a maximum at $p^2 \simeq 4$GeV, while the
transverse component of the off-diagonal gluon propagator does not show the
$\beta$-dependence and the dressing function does not have a maximum. This
behavior indicates that the effect of the Gribov copies is found for the
diagonal gluon, consistent with the result obtained by the Gribov-Zwanziger
action in the MAG. In addition, this behavior supports that the Abelian
dominance is not found in two dimensions. In the coordinate space, the diagonal
gluon propagator decreases as $\beta$ increases at long distance. In
particular, at $\beta=120$ the diagonal propagator decreases more rapidly with
increasing distance than the off-diagonal propagator. These behaviors also
indicate the presence of Gribov copies and no Abelian dominance in two
dimensions. Furthermore, we also study these propagators at
zero-spatial-momentum. The result suggests that all of the spectral functions
of diagonal and off-diagonal gluons would have negative regions and thus they
show the violation of the Kallen-Lehmann representation.
