  We simulate two dimensional QED with two degenerate Wilson fermions and
plaquette gauge action. As a consequence of the Mermin-Wagner theorem, in the
continuum limit chiral symmetry is realized a la Wigner. This property affects
also the size of the cutoff effects. That can be understood in view of the fact
that the leading lattice artifacts are described, in the continuum Symanzik
effective theory, by chirality breaking terms. In particular, vacuum
expectation values of non-chirality-breaking operators are expected to be O(a)
improved in the chiral limit. We provide a numerical confirmation of this
expectation by performing a scaling test.
