Abstract In this work we study an unusual phase transition of the Baxter–Wu model in the presence of an external magnetic field. The model is pure Baxter–Wu, which means that only three-spin interactions are taken into account. We construct a phase diagram on the temperature–field plane based mainly on the singularities of the specific heat. These singularities are more clearly observed than those of the magnetic susceptibility which are used in existing works. We establish a discontinuity in the critical exponents when the field is changed from zero to negative.