Thermal melting of density waves on the square lattice

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We present the theory of the effect of thermal fluctuations on commensurate "p x p" density wave ordering on the square lattice (p >= 3, integer). For the case in which this order is lost by a second order transition, we argue that the adjacent state is generically an incommensurate striped state, with commensurate p-periodic long range order along one direction, and incommensurate quasi-long-range order along the orthogonal direction. We also present the routes by which the fully disordered high temperature state can be reached. For p=4, and at special commensurate densities, the "4 x 4" commensurate state can melt directly into the disordered state via a self-dual critical point with non-universal exponents.