  We show from detailed magnetometry across 2-300 K that the
BiFeO$_3$-Bi$_2$Fe$_4$O$_9$ nanocomposite offers a unique spin morphology where
superspin glass (SSG) and dilute antiferromagnet in a field (DAFF) coexist at
the interface between ferromagnetic Bi$_2$Fe$_4$O$_9$ and antiferromagnetic
BiFeO$_3$. The coexisting SSG and DAFF combine to form a local spin texture
which gives rise to a path-dependent exchange bias below the spin freezing
temperature ($\sim$29K). The exchange bias varies depending on the protocol or
path followed in tracing the hysteresis loop. The exchange bias has been
observed below blocking temperature (T$_B$$\sim$60K) of the superparamagnetic
Bi$_2$Fe$_4$O$_9$. The conventional exchange bias (CEB) increases
nonmonotonically as temperature decreases. The magnitude of both exchange bias
(H$_E$) and coercivity (H$_C$) increase with decrease in temperature and are
found to be asymmetric below 20K depending on the path followed in tracing the
hysteresis loop and bias field. The local spin texture at the interface between
ferromagnetic and antiferromagnetic particles generates a nonswitchable
unidirectional anisotropy along the negative direction of the applied field.
The influence of this texture also shows up in "asymmetric" jumps in the
hysteresis loop at 2 K which smears off at higher temperature. The role of the
interface spin texture in yielding the path dependency of exchange bias is thus
clearly delineated.
