Persistent entropy current? A third-law paradox

We consider persistent currents at finite temperature induced by the Aharonov-Bohm effect in a multiply connected quantum system threaded by a magnetic flux. In general, both the energy current $I_E$ and the particle current $I_N$ are nonzero in the limit $T\rightarrow 0$, while the entropy of the system $S(T)\rightarrow 0$ as $T\rightarrow 0$, consistent with the third law of thermodynamics. The conventional definition of the heat current is $I_Q=I_E-\mu I_N$, with the entropy current defined as $I_S=I_Q/T$. We show that generically the persistent heat current defined in this way is nonzero in the limit $T\rightarrow 0$, leading to the paradoxical result that $I_S \rightarrow \infty$ as $T\rightarrow 0$ despite the fact that $S(T)\rightarrow 0$ and $I_N$ is finite. This suggests that the conventional definition of heat current is problematic for a quantum system in thermal equilibrium. A curl-free formula for the entropy current is proposed as a possible way out of this paradox.