Cross derivative of the Gibbs free energy: A universal and efficient method for phase transitions in classical spin models

With an auxiliary weak external magnetic field, we reexamine the fundamental thermodynamic function, Gibbs free energy $G(T,h)$, to study phase transitions in classical spin lattice models. A cross derivative, i.e., the second-order partial derivative of $G(T,h)$ with respect to both temperature and field, is calculated to precisely locate the critical temperature, which also reveals the nature of a transition. The strategy is efficient and universal, as exemplified by the five-state clock model, two-dimensional (2D) and 3D Ising models, and the $XY$ model, no matter if a transition is trivial or exotic with complex excitations. More importantly, other conjugate pairs could also be integrated into a similar cross derivative if necessary, which would greatly enrich our vision and means to investigate phase transitions both theoretically and experimentally.