Ferrimagnetism from quantum fluctuations in Kitaev materials

Ferrimagnetism appears in the temperature-field phase diagrams of several candidate Kitaev materials, such as the honeycomb cobaltates ${\mathrm{Na}}₂{\mathrm{Co}}₂{\mathrm{TeO}}₆$ and ${\mathrm{Na}}₃{\mathrm{Co}}₂{\mathrm{SbO}}₆$. In a number of instances, however, the exact nature of the corresponding ground states remains the subject of ongoing debate. We show that general symmetry considerations can rule out candidate states that are incompatible with the observed ferrimagnetic behavior. In particular, we demonstrate that a ferrimagnetic response cannot be reconciled with a collinear zigzag ground state, owing to the combined time-reversal and translational symmetry inherent to that configuration. Instead, the observed behavior is fully compatible with the symmetries of noncollinear multi-$\mathbf{q}$ states, such as the triple-$\mathbf{q}$ discussed in the context of ${\mathrm{Na}}₂{\mathrm{Co}}₂{\mathrm{TeO}}₆$. We exemplify this general result by computing the ferrimagnetic response of an extended Heisenberg-Kitaev-Gamma model with explicit sublattice symmetry breaking within linear spin-wave theory. If the model realizes a triple-$\mathbf{q}$ ground state, the calculated magnetization curve is well consistent with the low-temperature behavior observed in ${\mathrm{Na}}₂{\mathrm{Co}}₂{\mathrm{TeO}}₆$. In this case, a finite magnetization remains in the zero-temperature limit as a consequence of quantum fluctuations, even if the $g$ factors on the different sublattices are identical. For a zigzag ground state, by contrast, the total magnetization vanishes both at zero and finite temperatures, independent of possible sublattice-dependent $g$ factors, as expected from the symmetry analysis. The implications of our general result for other Kitaev materials exhibiting ferrimagnetic behavior are also briefly discussed.