False Onsager relations

Recent research suggests that when a system has a "false time-reversal violation" the Onsager reciprocity relations hold despite the presence of a magnetic field. The purpose of this work is to clarify that the Onsager relations may well be violated in the presence of a "false time -reversal violation": that rather guarantees the validity of distinct relations, which we dub "false Onsager relations". We also point out that for quantum systems "false time-reversal violation" is omnipresent and comment that, per se, this has in general no consequence in regard to the validity of Onsager relations, or the more general non-equilibrium fluctuation relations, in the presence of a magnetic field. Our arguments are illustrated with the Heisenberg model of a magnet in an external magnetic field.