  We analyse the two purely isospin-violating decays $\Lambda_b\rightarrow
\Sigma^0 \phi$ and $\Lambda_b\rightarrow \Sigma^0 J/\psi$, proceed merely via
the exchange topologies, in the framework of perturbative QCD approach.
Assuming $\Sigma^0$ baryon belongs to the idealized isospin triplet with quark
components of $usd$, the branching ratios of the two decay modes are predicted
to be tiny, of the order $10^{-8}-10^{-9}$, leading to a difficulty in
observing them. We then extend our study to include the $\Sigma-\Lambda$
mixing.It is found that the mixing has significant effect on the
$\Lambda_b\rightarrow \Sigma$ decays, especially it can greatly increase the
rate of the $J/\psi$ process, by as much as two orders of magnitude, yield
$10^{-7}$, which should be searchable in the future. We also estimate a set of
asymmetry observables with and without the mixing effect, which will be tested
in coming experiments.
