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

The 3$d^7$ Co$^{2+}$-based insulating magnet \NCTO{} has recently been reported to have strong Kitaev interactions on a honeycomb lattice, and is thus being considered as a Kitaev quantum spin liquid candidate. However, due to the existence of other types of interactions, a spontaneous long-range magnetic order occurs. This order is suppressed by applied magnetic fields leading to a succession of phases and ultimately saturation of the magnetic moments. The precise phase diagram, the nature of the phases, and the possibility that one of the field-induced phases is a Kitaev quantum spin liquid phase are still a matter of debate. Here we measured an extensive set of physical properties to build the complete temperature-field phase diagrams to magnetic saturation at 10 T for magnetic fields along the $a$- and $a^*$-axes, and a partial phase diagram up to 60 T along $c$. We probe the phases using magnetization, specific heat, magnetocaloric effect, magnetostriction, dielectric constant, and electric polarization, which is a symmetry-sensitive probe. With these measurements we identify all the previously incomplete phase boundaries and find new high-field phase boundaries. We find strong magnetoelectric coupling in the dielectric constant and moderate magnetostrictive coupling at several phase boundaries. Furthermore, we detect the symmetry of the magnetic order using electrical polarization measurements under magnetic fields. Based on our analysis, the absence of electric polarization under zero or finite magnetic field in any of the phases or after...