Crossover and universality breaking in the dilute Baxter-Wu model

The critical behavior of the Baxter-Wu model belongs to the universality class of the four-state Potts model. While the introduction of annealed vacancies does not alter the criticality of the four-state Potts model, the dilute Baxter-Wu model has remained the subject of several competing scenarios. Here we investigate the phase diagram of the spin-$1$ Baxter-Wu model in the presence of a crystal field using transfer-matrix calculations and large-scale Monte Carlo simulations. Our results provide strong evidence for continuously varying critical exponents at finite dilution and reveal a crossover to first-order behavior. Along the line of continuous transitions, the central charge remains close to $c=1$, while the scaling dimensions systematically deviate from the spin-$1/2$ limit as the crystal field increases, eventually giving way to a first-order regime at strong fields. These findings resolve previous ambiguities and establish a consistent picture of the critical behavior of the dilute spin-$1$ Baxter-Wu model.