Anomalous Fermi liquid phase in metallic Skyrmion crystals

In non-centrosymmetric crystals such as MnSi, magnetic order can take the form of a skyrmion crystal (SkX) . In this phase, conduction electrons coupled to the local magnetic moments acquire a Berry's phase, leading to an emergent electromagnetism. Motivated by experimental reports of a non-Fermi liquid phase in MnSi, in which resistivity is observed to scale as $\Delta \rho \sim T^{3/2}$, here we examine the effect of coupling phonons of an incommensurate SkX to electrons. Despite the formal similarity to a system consisting of a Fermi surface coupled to an electromagnetic field, the Berry phase fluctuations do not lead to non-Fermi liquid behavior. Instead, we propose a different mechanism in which electrons scatter off columnar fluctuation in a three-dimensional SkX. When the effects of lattice induced anisotropy are neglected, these fluctuations are ultra-soft and induce an `anomalous Fermi liquid' in which Landau quasiparticles survive but with an anomalous $\Delta \rho(T)\sim T^{7/4}$ resistivity perpendicular to the columns, and a Fermi liquid resistivity along them.