A “Fourth” Checkpoint for Second-Order Transitions

In order that a given phase transition can theoretically be second order, it needs to pass checkpoints. Already known are three checkpoints, viz., the checkpoint of third-degree term, the checkpoint of commensurate-incommensurate gap index and the checkpoint of intermediate-rank lineal phase. The present paper proposes a “fourth” checkpoint or the checkpoint of positive-definiteness and incompatible inequalities as different from and independent of the three above-mentioned ones. Exhibited are examples of transitions that pass all of the first, second and third checkpoints but not the fourth, and examples of transitions that pass also the fourth.