Numerical investigation of a model for oxygen ordering in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$

We report the results of large-scale Monte Carlo and transfer-matrix studies of a model lattice-gas Hamiltonian that has previously been introduced to stidy the oxygen ordering responsible for the orthorhombic-to-tetragonal transition in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$. We analyze the data using finite-size scaling to obtain the phase diagram and critical exponents. At high temperatures we find qualitative agreement with the cluster-variation method. However, at low temperatures we find only second-order transitions, in disagreement with the cluster-variation method. The critical exponents found are consistent with the universality classes of the d=2 Ising model and the XY model with cubic anisotropy. Consequently, any experimental evidence of a first-order transition for ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$ would imply that the model of Wille, Berera, and de Fontaine [Phys. Rev. Lett. 60, 1065 (1988)] needs to be modified to be directly applicable to this material.