Leading order magnetic field dependence of conductivities in anomalous hydrodynamics

TL;DR

This paper demonstrates that the magnetic field dependence of longitudinal conductivity in anomalous hydrodynamics is frame dependent at first order, and that previous results are not physically meaningful unless the magnetic field is incorporated into the equilibrium state.

Contribution

It clarifies the frame dependence of magnetic field effects in anomalous hydrodynamics and shows how to properly include magnetic fields to obtain physical results.

Findings

Magnetic field dependence in longitudinal conductivity is frame dependent.

Previous literature results are not frame invariant and thus not physical.

Including magnetic fields in the equilibrium state removes unphysical frame dependence.

Abstract

We show that literature results claimed for the magnetic field dependence of the longitudinal conductivity in anomalous first-order hydrodynamics are frame dependent at this derivative order. In particular, we focus on $(3+1)$-dimensional hydrodynamics in the presence of a constant ${\cal O}(\partial)$ magnetic field with a $U(1)$ chiral anomaly and demonstrate that, for constitutive relations up to and including order one in derivatives, the anomaly drops out of the longitudinal conductivity. In particular, magnetic field dependent terms that were previously found in the literature only enter the non-zero frequency thermoelectric conductivities through explicitly frame dependent pieces indicating that they are not physical. This issue can be avoided entirely by incorporating the magnetic field into the fluid's equilibrium state.