Phase diagram and critical properties of a two-dimensional lattice-gas model of oxygen ordering in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{\mathrm{z}}$

Using transfer-matrix finite-size scaling, we investigate a square lattice-gas model for the oxygen ordering in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{\mathrm{z}}$, which reproduces the experimentally observed orthorhombic and double-cell ordered phases. Contrary to the results of cluster-variational methods of calculation, the phase boundary between the (disordered) tetragonal and double-cell phases is continuous, as is the boundary between the orthorhombic and double-cell phases. Critical properties of the model can be understood by drawing an analogy with the Ashkin-Teller model; numerical results are consistent with this picture. At low temperature and low oxygen concentration, there is complicated behavior, possibly indicative of a second orthorhombic phase.