Non-topological fractional fermion number in the Jackiw-Rossi model

We compute the vacuum fermion current in $(2+1)$ dimensional Jackiw-Rossi model by using the $1\mathrm{/}m$ expansion. The current is expressed through a weighted $\eta$-function with a matrix weight. In the presence of such a weight, the usual proof of topological nature of $\eta (0)$ is not longer applicable. Direct computations confirm the following surprising result: the fermion number induced by vortices in the Jackiw-Rossi model is $\mathit{\text{not}}$ topological.