  We study the approximability of computing the partition function for
ferromagnetic two-state spin systems. The remarkable algorithm by Jerrum and
Sinclair showed that there is a fully polynomial-time randomized approximation
scheme (FPRAS) for the special ferromagnetic Ising model with any given uniform
external field. Later, Goldberg and Jerrum proved that it is #BIS-hard for
Ising model if we allow inconsistent external fields on different nodes. In
contrast to these two results, we prove that for any ferromagnetic two-state
spin systems except the Ising model, there exists a threshold for external
fields beyond which the problem is #BIS-hard, even if the external field is
uniform.
