  In 2+1 dimensions at finite temperature, spontaneous symmetry breaking of
global symmetries is precluded by large thermal fluctuations of the order
parameter. The holographic correspondence implies that analogous effects must
also occur in 3+1 dimensional theories with gauged symmetries in certain curved
spacetimes with horizon. By performing a one loop computation in the background
of a holographic superconductor, we show that bulk quantum fluctuations wash
out the classical order parameter at sufficiently large distance scales. The
low temperature phase is seen to exhibit algebraic long range order. Beyond the
specific example we study, holography suggests that IR singular quantum
fluctuations of the fields and geometry will play an interesting role for many
3+1 dimensional asymptotically AdS spacetimes with planar horizon.
