Spin waves in Na 2 Co 2 TeO 6 studied by high-frequency/high-field ESR: Successes and failures of the triple- q model

The Kitaev candidate material ${\mathrm{Na}}₂{\mathrm{Co}}₂{\mathrm{TeO}}₆$ is proposed to be proximate to a quantum spin liquid state but a suitable spin model and the nature of its ground states are still under debate. Our high-frequency/high-field electron spin resonance spectroscopy studies of ${\mathrm{Na}}₂{\mathrm{Co}}₂{\mathrm{TeO}}₆$ single crystals under in-plane and out-of-plane magnetic fields elucidate the ground state by investigating its low-energy spin wave excitations. Several excitation modes are observed in the low-field phase and in the phases induced by $B\ensuremath{\parallel}{a}^*$. In addition, the spectra exhibit a frequency-independent feature at the phase boundary connected to the putative quantum phase transition. For magnetic fields applied along the $c$ axis, the observation of three distinct spin wave modes in the antiferromagnetic (AFM) ground state reveals a previously unresolved splitting of the zero-field excitation gap into $\mathrm{\ensuremath{\Delta}}=211\phantom{\rule{4pt}{0ex}}\mathrm{GHz}$ and ${\mathrm{\ensuremath{\Delta}}}₂=237\phantom{\rule{4pt}{0ex}}\mathrm{GHz}$. The softening of one of these modes evidences a field-induced phase transition at ${B}_{\mathrm{c}1}=4.7\phantom{\rule{4pt}{0ex}}\mathrm{T}$, which is corroborated by a clear anomaly in the isothermal magnetization. Spin wave calculations based on the extended Heisenberg-Kitaev model exclude a zigzag ground state of the AFM phase. A triple-$\mathbf{q}$ spin configuration correctly predicts two spin wave modes but fails to reproduce the softening mode. Our analysis shows that the triple-$\mathbf{q}$ ground state model of ${\mathrm{Na}}₂{\mathrm{Co}}₂{\mathrm{TeO}}₆$ is incomplete and suggests the relevance of interlayer interactions.