Ferrimagnetism from quantum fluctuations in Kitaev materials

TL;DR

Ferrimagnetism in Kitaev-inspired honeycomb magnets can arise from quantum fluctuations only in noncollinear triple-$\mathbf{q}$ ground states when sublattice symmetry is broken, while collinear zigzag order cannot develop a net moment due to a combined translation-time-reversal symmetry. The authors develop a symmetry-based framework and perform linear spin-wave analyses of extended Heisenberg-Kitaev (HK) and Heisenberg-Kitaev-Gamma (HKΓ) models to quantify magnetization corrections, showing zero-temperature ferrimagnetism for triple-$\mathbf{q}$ states and a finite-temperature compensation point without sublattice-$g$-factor tuning. Zigzag states remain non-ferrimagnetic under the relevant symmetries, whereas triple-$\mathbf{q}$ states produce ferrimagnetism when off-diagonal terms like $\Gamma$ are present; the compensation point $T_\star$ emerges from differential thermal population of magnon modes. The results provide a quantum-mechanical explanation for the low-temperature ferrimagnetism observed in Na$_2$Co$_2$TeO$_6$ and offer a general symmetry-grounded framework applicable to other Kitaev materials.

Abstract

Ferrimagnetism appears in the temperature-field phase diagrams of several candidate Kitaev materials, such as the honeycomb cobaltates Na$_2$Co$_2$TeO$_6$ and Na$_3$Co$_2$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 Na$_2$Co$_2$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 Na$_2$Co$_2$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.

Figures (8)

• Figure 1: Illustration of spin-space transformation mapping two inequivalent spins (large black arrows). A spin at site $i\mu$ (blue dot), where $i$ denotes the magnetic unit cell and $\mu$ the magnetic sublattice, is mapped to a spin at site $j\nu$ (red dot) in a two steps: First, the spin is rotated in spin space via the operation $\mathcal{A}$, with the resulting transformed spin shown as a gray arrow. Second, the spin transformation is accompanied by a lattice transformation $\{\mathcal{R} | \mathbf t\}$, consisting of a lattice rotation $\mathcal{R}$ and a translation $\mathbf t$, mapping site $i\mu$ to $j\nu$.
• Figure 2: Two-site magnetic unit cell in the (a) ferromagnetic and (b) Néel ground states. Here, the magnetic sublattices ($A$ and $B$) coincide with the crystallographic sublattices, represented by the filled and open circles.
• Figure 3: Four-site magnetic unit cell in the (a) zigzag and (b) stripy ground states. $A$, $B$, $C$, and $D$ denote the four magnetic sublattices, with the two crystallographic sublattices indicated by filled and open circles. $\mathbf t_{+}$ and $\mathbf t_-$ are lattice translation vectors.
• Figure 4: (a) Magnon spectrum in the $z$-zigzag ground state of the Heisenberg-Kitaev model with nearest-neighbor couplings $(J, K) = (\cos \phi, \sin \phi)$ at $\phi = 5\pi/8$, and next-nearest-neighbor couplings $(J_{2A}, J_{2B}) = (0.2, -0.1)$, from linear spin-wave theory. The momentum path through the extended Brillouin zone is shown in the central inset of (b), with the green star indicating the Bragg peak associated with the $z$-zigzag order. Pseudo-Goldstone modes appear at $\mathbf{M}$ due to an accidental SU(2) degeneracy of the classical ground state. (b) Sublattice magnetizations $\mathbf{m}_\mu$ along the [001] direction for $\mu = A, B, C, D$ in the $z$-zigzag ground state for $S=1/2$, using the same model parameters as in (a). Extrapolated values in the thermodynamic limit $1/L \to 0$ are shown in the right inset.
• Figure 5: Magnitude of the secondary order parameter $|\mathbf m_\text{sec}|$, defined in Eq. \ref{['eq:unsat-magn']}, as a function of the Heisenberg-Kitaev angle $\phi$, with nearest-neighbor couplings $(J,K) = (\cos\phi, \sin\phi)$ and fixed next-nearest-neighbor couplings $(J_{2A},J_{2B})=(0.2,-0.1)$ at zero temperature, from linear spin-wave theory for $S=1/2$. As dictated by symmetry, there is no uncompensated moment in the zigzag and stripy phases, nor at the Heisenberg points ($\phi = 0, \pi$). The colors indicate the four magnetically-ordered phases of the nearest-neighbor model, while the white regions mark potential new phases near its phase boundaries (gray dashed lines), induced by the next-nearest-neighbor couplings.