Absence of Energy Currents in an Equilibrium State and Chiral Anomalies

A long time ago, Bloch showed that in a system of interacting nonrelativistic particles the net particle-number current must vanish in any equilibrium state. Bloch's argument does not generalize easily to the energy current. We devise an alternative argument which proves the vanishing of the net energy currents in equilibrium states of lattice systems as well as systems of nonrelativistic particles with finite-range potential interactions. We discuss some applications of these results. In particular, we show that neither a one-dimensional (1D) lattice system nor a 1D system of nonrelativistic particles with finite-range potential interactions can flow to a conformal field theory with unequal left-moving and right-moving central charges.