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The 3$d^7$ Co$^{2+}$-based insulating magnet \NCTO{} has recently beenreported to have strong Kitaev interactions on a honeycomb lattice, and is thusbeing considered as a Kitaev quantum spin liquid candidate. However, due to theexistence of other types of interactions, a spontaneous long-range magneticorder occurs. This order is suppressed by applied magnetic fields leading to asuccession of phases and ultimately saturation of the magnetic moments. Theprecise phase diagram, the nature of the phases, and the possibility that oneof the field-induced phases is a Kitaev quantum spin liquid phase are still amatter of debate. Here we measured an extensive set of physical properties tobuild the complete temperature-field phase diagrams to magnetic saturation at10 T for magnetic fields along the $a$- and $a^*$-axes, and a partial phasediagram up to 60 T along $c$. We probe the phases using magnetization, specificheat, magnetocaloric effect, magnetostriction, dielectric constant, andelectric polarization, which is a symmetry-sensitive probe. With thesemeasurements we identify all the previously incomplete phase boundaries andfind new high-field phase boundaries. We find strong magnetoelectric couplingin the dielectric constant and moderate magnetostrictive coupling at severalphase boundaries. Furthermore, we detect the symmetry of the magnetic orderusing electrical polarization measurements under magnetic fields. Based on ouranalysis, the absence of electric polarization under zero or finite magneticfield in any of the phases or after...

Electronic and magnetic phase diagrams of Kitaev quantum spin liquid candidate Na$_2$Co$_2$TeO$_6$