Commensurate-Incommensurate Transitions in Two Dimensions: Discrete Models and Square Symmetric Substrates

The phase diagram and the critical behavior of a discrete two-dimensional model of the commensurate-incommensurate transition is determined with use of an approximate mapping onto the six-vertex model. For an uniaxial substrate the critical behavior is that of the continuum case. For a square substrate either a single first-order transition or a second-order transition followed by a first-order transition is found.