Note on the Bloch theorem

The Bloch theorem in ordinary quantum mechanics means the absence of the total electric current in equilibrium. In the present paper, we analyze the possibility that this theorem remains valid within quantum field theory relevant for the description of both high-energy physics and condensed matter physics phenomena. First of all, we prove that the total electric current in equilibrium is the topological invariant for the gapped fermions that are subject to periodical boundary conditions; i.e., it is robust to the smooth modification of such systems. This property remains valid when the interfermion interactions due to the exchange by bosonic excitations are taken into account perturbatively. We give the proof of this statement to all orders in perturbation theory. Thus, we prove the weak version of the Bloch theorem and conclude that the total current remains zero in any system, which is obtained by smooth modification of the one with the gapped charged fermions, periodical boundary conditions, and vanishing total electric current. We analyze several examples, in which the fermions are gapless. In some of them, the total electric current vanishes. At the same time, we propose the counterexamples of the equilibrium gapless systems, in which the total electric current is nonzero.