  The spin Hall magnetoresistance (SMR) of epitaxial BiFeO$_3$ thin films is
investigated. SMR consistent with ferromagnetic interfacial states for
BiFeO$_3$ films fabricated on (001) SrTiO$_3$ (R' BFO) and LaAlO$_3$ (T' BFO)
substrates is found, albeit with different temperature dependencies. For T'
BFO, the SMR is enhanced at room temperature, and decays with reduced
temperatures. By contrast, R' BFO shows a monotonic decrease in SMR response
with increasing temperature, mirroring the trend of a weak ferromagnet. Density
functional theory shows that this difference originates from the coupling of
the applied magnetic field to oxygen octahedral rotation (R') and spin (T')
degrees of freedom.
