Holography and the Coleman-Mermin-Wagner theorem

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 anti-de Sitter spacetimes with planar horizon.