Weak spin-orbit interactions induce exponentially flat mini- bands in magnetic metals without inversion symmetry

– In metallic magnets like MnSi the interplay of two very weak spin-orbit coupling effects can strongly modify the Fermi surface. In the absence of inversion symmetry even a very small Dzyaloshinsky-Moriya interaction of strength δ ≪ 1 distorts a ferromagnetic state into a chiral helix with a long pitch of order 1/δ. We show that additional small spin-orbit coupling terms of order δ in the band structure lead to the formation of exponentially flat mini-bands with a bandwidth of order e √ δ parallel to the direction of the helix. These flat mini-bands cover a rather broad belt of width √ δ on the Fermi surface where electron motion parallel to the helix practically stops. We argue that these peculiar band-structure effects lead to pronounced features in the anomalous skin effect. Introduction. – MnSi is a textbook example [1] of a magnetic metal where weak spinorbit (SO) coupling leads to a helical modulation of the ferromagnetic order [2,3]. While this physics has been well understood for almost 25 years [2], the interest in MnSi was recently renewed [4–6] after it was discovered [4] that moderate pressures p suppress the long range helical order and drive the system for p > pc ≈ 14.6kbar into a novel state characterized by an anomalous resistivity, ρ ∼ T , which is observed over almost three decades [4] in temperature T and in a huge pressure range [5]. This has been taken as evidence for the existence of a genuine non-Fermi liquid phase in this extremely clean cubic system. Recent neutron scattering results [6] for p > pc suggest that while true long-range order is lost in this phase, a peculiar partial helical order survives on intermediate time and length scales. This motivates us to study the motion of electrons in a helical state in more detail. Within this paper we try to answer a simple question: What is the generic band-structure of electrons in the magnetically ordered state of a metal without inversion symmetry? We will show that even in the case of weak spin-orbit interaction (as realized in MnSi) the answer to this question is surprisingly complex: the interplay of two weak spin-orbit effects of similar strength leads to a pronounced restructuring of the Fermi surface. For a large portion of the Fermi surface the electron motion parallel to the helix turns out to be almost completely frozen. In the following we will briefly review how the Dzyaloshinsky-Moriya interaction leads to helical order in magnets lacking inversion symmetries. We then calculate the band structure within the magnetically ordered phase first neglecting all spin-orbit effects besides the