Circulating currents and magnetic moments in quantum rings

In circuits containing closed loops the operator for the current is determined by charge conservation up to an arbitrary divergenceless current. In this work we propose a formula to calculate the magnetically active circulating current $I_{\rm ring}$ flowing along a quantum ring connected to biased leads. By {\em gedanken experiments} we argue that $I_{\rm ring}$ can be obtained from the response of the gran-canonical energy of the ring to an external magnetic flux. The results agree with those of the conventional approach in the case of isolated rings. However, for connected rings $I_{\rm ring}$ cannot be obtained as a linear combination of bond currents.