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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.

Use $s=|t|$, $x=h/s^{\Delta }$ and separate scaling functions above and below the transition. Up to a nonsingular conversion between physical and dimensionless field, magnetization scaling-function matching givesOn the ordered side take $h\to 0^{+}$ after the thermodynamic limit, selecting one pure thermodynamic phase. A finite nonzero $g_{m,-}(0^{+})$ impliesThe critical isotherm must have a finite nonzero limit at fixed small $h$ as $s\to 0$. Thus $g_m(x)\sim C\mathrm{sgn}(x)|x{|}^{1/\delta }$ at large $|x|$. Cancellation of the residual power of $s$ requiresThe absolute value and branch choice matter: spontaneous zero-field magnetization is below $T_c$, whereas the symmetry-unbroken thermodynamic phase has $m(t,0)=0$. The field crossover exponent is not the scaling dimension of the magnetization field.