Odd-parity Magnetism from the Generalized Bloch Theorem

Abstract: In the non-relativistic limit, helimagnetic order is always associated with odd-parity magnetism. That is, for single-particle states the expectation value of the electronic spin is odd in crystal momentum, which implies direct control of the spin by means of electric fields. However, the theoretical description of helimagnets is hindered by the fact that the spiral pitch may require large super cells or even be incommensurate with the lattice. In the this letter we show that such issues may be remedied by use of the Generalized Bloch theorem. It allows one to describe (by models or first principles) the system in terms of the primitive unit cell, from which all relevant properties can be obtained by downfolding in reciprocal space. We exemplify the procedure using MnI$_2$ and NiI$_2$, which are known type II multiferroics having spiral order and the helimagnetic metal MnTe$_2$. We analyze how the magnitude of spin splitting depends on orbital composition of bands, and we show that spin splitting is maximized for states having large odd-orbital ($p$-type) character. It is straightforward to generalize the framework to handle response functions for helimagnets using only the primitive unit cell and the present downfolding procedure thus strongly facilitate theoretical progress in the field.