We compute the vacuum fermion current in $(2+1)$ dimensional Jackiw-Rossi model by using the $1/m$ 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 \textit{ not} topological.
Categories: hep-th cond-mat.mes-hall