  The identification of the different phases of a two-dimensional (2d) system,
which might be in solid, hexatic, or liquid, requires the accurate
determination of the correlation function of the translational and of the
bond-orientational order parameters. According to the
Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory, in the solid phase
the translational correlation function decays algebraically, as a consequence
of the Mermin-Wagner long-wavelength fluctuations. Recent results have however
reported an exponential-like decay. By revisiting different definitions of the
translational correlation function commonly used in the literature, here we
clarify that the observed exponential-like decay in the solid phase results
from an inaccurate determination of the symmetry axis of the solid; the
expected power-law behaviour is recovered when the symmetry axis is properly
identified. We show that, contrary to the common assumption, the symmetry axis
of a 2d solid is not fixed by the direction of its global bond-orientational
parameter, and introduce an approach allowing to determine the symmetry axis
from a real space analysis of the sample.
