Nontopological fractional fermion number in the Jackiw-Rossi model

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