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

Recently 3d$^7$ Co$^{2+}$-based magnets with strong spin-orbit coupling effects have been proposed to show dominant Kitaev magnetic exchange interactions and a suppression of Heisenberg interactions. This makes them good candidates to observe Kitaev quantum spin liquid behavior on honeycomb lattices. Here we present physical properties and the phase diagram of a prominent Co$^{2+}$-based Kitaev quantum spin liquid candidate \NCTO{} for magnetic fields along the $a$, $a^*$ and $c$-axis. We measured magnetization, specific heat, dielectric constant, thermal expansion, and magnetocaloric effect to 14 T for most measurements and 60 T for the magnetization along the $a^*$- and the $c$-axis. At zero field we find no electric polarization, which is more consistent with a zigzag antiferromagnetic state than a 3Q model that was proposed. In applied fields sharp features in the dielectric constant indicate the involvement of electric dipoles strongly coupled to magnetism that locally break spatial inversion symmetry, though without net electric polarization. We observe three successive field-induced phase transitions before saturation is reached for fields along $a$ and $a^*$, and two new temperature-induced phase boundaries in the spin-polarized phase. The lowest-field boundary is strongly hysteretic whereas the others are all second-order-like. The highest-field phase boundary has a mean field-like dome shape in temperature-field space. Above the highest-field phase transition at 9.5 T for fields along $a$, $a^*$, the magnetocaloric effect shows a drop in spin entropy continuing up at least 20 T, consistent with a spin gap opening. Above 9.5 T the magnetization vs magnetic field approaches saturation, though subject to an additional linear Van-Vleck-like contribution continuing to at least 60 T...