  In one-dimension, quantum fluctuations prevent the appearance of long-range
order in a supersolid, and only quasi long-range order can survive. We derive
this quantum critical behavior and study its influence on the superfluid
response and properties of the solid. The analysis is based on an effective
low-energy description accounting for the two coupled Goldstone modes. We find
that the quantum phase transition from the superfluid to the supersolid is
shifted by quantum fluctuations from its mean-field prediction. However, for
current experimental parameters with dipolar atomic gases, this shift is not
observable and the transition appears to be mean-field like.
