Field crossover exponent ($\Delta =\beta \delta =\beta +\gamma$)

The field crossover exponent describes the field scale $h\sim |t{|}^{\Delta }$ at which the zero-field critical regime changes. In the two-variable scaling hypothesis for critical phenomena, its value is the ratio $y_h/y_t$ of magnetic and thermal scaling dimensions. It is distinct from either $y_h$ or the scaling dimension of the order parameter. The Widom scaling relation and differentiation of the homogeneous free-energy density give the displayed identities when ordinary scaling applies.