Ferrimagnetism from triple-q order in Na$_2$Co$_2$TeO$_6$

The candidate Kitaev magnet Na$_2$Co$_2$TeO$_6$ exhibits a characteristic ferrimagnetic response at low temperatures, with a finite residual magnetization that changes sign at a compensation point located at around half the ordering temperature. We argue that the behavior can be naturally understood to arise in this material as a consequence of a noncollinear triple-$\mathbf{q}$ magnetic ground state. Using large-scale classical Monte Carlo simulations, we study the finite-temperature response of the pertinent honeycomb Heisenberg-Kitaev-$\Gamma$-$\Gamma'$ model in weak training fields. Our model features all symmetry-allowed nearest-neighbor exchange interactions, as well as sublattice-dependent next-nearest-neighbor interactions, consistent with the reported crystal structure of the material. We also consider a six-spin ring exchange perturbation, which allows us to tune between the two different magnetic long-range orders that have been suggested for this material in the literature, namely, a collinear single-$\mathbf{q}$ zigzag state and a noncollinear triple-$\mathbf{q}$ state. We demonstrate that the experimentally-observed ferrimagnetic response of Na$_2$Co$_2$TeO$_6$ can be well described within our modeling if the magnetic ground state features noncollinear triple-$\mathbf{q}$ order. The observation of a compensation point, where the residual magnetization reverses sign, suggests a sublattice $g$-factor anisotropy, with a larger out-of-plane $g$-factor on the sublattice with stronger antiferromagnetic intrasublattice exchange. By contrast, a classical Heisenberg-Kitaev-$\Gamma$-$\Gamma'$-type model with collinear zigzag ground state is insufficient even in principle to describe the observed behavior. Our results illustrate the unconventional physics of noncollinear magnetic long-range orders hosted by frustrated magnets with bond-dependent interactions.