In two dimensions, crystals provide another loophole in a well-known result, known as the Mermin-Wagner theorem. Hohenberg, Mermin, and Wagner, in a series of papers, proved in the 1960's that two-dimensional systems with a continuous symmetry cannot have a broken symmetry at finite temperature. At least, that's the English phrase everyone quotes when they discuss the theorem: they actually prove it for several particular systems, including superfluids, superconductors, magnets, and translational order in crystals. Indeed, crystals in two dimensions do not break the translational symmetry: at finite temperatures, the atoms wiggle enough so that the atoms don't sit in lock-step over infinite distances (their translational correlations decay slowly with distance). But the crystals do have a broken orientational symmetry: the crystal axes point in the same directions throughout space. (Mermin discusses this point in his paper on crystals.) The residual translational correlations (the local alignment into rows and columns of atoms) introduce long-range forces which force the crystalline axes to align, breaking the continuous rotational symmetry. Mermin, Wagner, and Hohenberg's methods apply very generally, but are not general enough to apply to this case (for good reason!)

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).