Conductivity Enhancement in a Diffusive Fermi Liquid due to Bose-Einstein Condensation of Magnons

We theoretically study the conductivity of a disordered 2D metal when it is coupled to ferromagnetic magnons with a quadratic spectrum and a gap $\Delta$. In the diffusive limit, a combination of disorder and magnon-mediated electron interaction leads to a sharp metallic correction to the Drude conductivity as the magnons approach criticality, i.e., $\Delta \to 0$. The correction is non-singular and is distinctively weaker than, for example, the logsquared correction obtained when disordered electrons couple to diffusive spin fluctuations near a Hertz-Millis transition. The possibility of verifying this prediction in an S = 1/2 easy-plane ferromagnetic insulator K$_2$CuF$_4$ under an external magnetic field is proposed. Our results show that the onset of a magnon BEC in an insulator can be detected via electrical transport measurements on the proximate metal.