Crossover from random-exchange to random-field critical behavior in Ising models

We compute the crossover exponent $\phi$ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as Fe$_x$Zn$_{1-x}$F$_2$ and Mn$_x$Zn$_{1-x}$F$_2$, when applying an external magnetic field. Our result $\phi=1.42(2)$ for the crossover exponent is in good agreement with the available experimental estimates.