Inversion-asymmetric antiferromagnets with odd-parity spin-polarization pattern, referred to as odd-parity magnets, have been proposed as a new venue for spintronics. These odd-parity magnets require commensurate ordering to ensure an effective time-reversal symmetry, which guarantees a strictly antisymmetric spin polarization of the electronic states. Recently, nonsymmorphic centrosymmetric crystals have been identified as a broad class of materials which could exhibit unit-cell doubling magnetism with odd-parity spin polarization. Here we investigate the stability of these states against incommensuration. We first demonstrate that the symmetry conditions which permit a $p$-wave magnetic state with a $p$-wave spin-polarization pattern also permit the existence of a nonrelativistic Lifshitz invariant in the phenomenological Ginzburg-Landau free energy. This implies magnetism with an incommensurate ordering vector, independent of its microscopic origin. $f$- and $h$-wave magnetic states with $f$- and $h$-wave spin polarizations are also prone to incommensurability, especially when they have an itinerant origin. Here the symmetry which allows the odd-parity spin polarization also guarantees the existence of van Hove saddle points off the time-reversal invariant momenta, which promote incommensurate spin fluctuations in quasi-two-dimensional electronic systems. Finally, we study the effect of weak spin-orbit coupling in locally noncentrosymmetric materials and find that it favors antiferromagnetic phases with in-plane magnetic moments. However, the inclusion of the spin-orbit coupling also introduces a new mechanism for driving incommensuration. Our results imply that odd-parity magnetic states are likely to be preceded by an incommensurate phase, or emerge directly from the normal state via a first-order transition. These conclusions are consistent with the phase diagram of several candidate materials.

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• Antiferromagnetism
• Spintronics
• Discrete symmetries in condensed matter
• Heisenberg model
• Hubbard model
• Landau-Ginzburg theory