Fig. 1: Phase diagram and corundum structure.

a Phase diagram of \({({{{{{{{{\rm{V}}}}}}}}}_{1-x}{{{{{M}}}}}_{x})}_{2}\,{{{{{{{{\rm{O}}}}}}}}}_{3}\), with M = Cr, Ti, as a function of the doping concentration x and pressure, from ref. ²⁹. AFM and PM stand for antiferromagnetic and paramagnetic, respectively. All transition lines (solid black lines) are first order, with the one separating PM metal from PM insulator that terminates into a second-order critical point (black dot). The paramagnetic metal and insulating phases of Cr-doped V₂O₃ are commonly referred to as α and β phases, respectively, and shown in the figure. b Non-primitive hexagonal unit cell of the high-temperature corundum phase. The V–V nearest neighbour distance within the a_H−b_H plane, ℓ₀, and along c_H, d₀, are also shown. Also shown is the reference frame we use throughout the work, with a_H∥x and c_H∥z. c Sketch of the electronic Mott transition in V₂O₃. In the atomic limit, the two conduction electrons of V³⁺ occupy the t_2g orbital of the cubic-field split 3d-shell. Because of the additional trigonal distortion, the t_2g is further split into a lower \({e}_{g}^{\pi }\) doublet and higher a_1g singlet. The two electrons thus sit into the \({e}_{g}^{\pi }\) orbital, in a spin triplet configuration because of Hund's rules. In the solid the atomic levels corresponding to removing or adding one electron broaden into lower and upper Hubbard bands, LHB and UHB, respectively. The LHB has prevailing \({e}_{g}^{\pi }\) character while the UHB has dominant a_1g character⁴¹. Note that we do not show, for simplicity, the multiplet structure that the Hubbard bands must have because of Coulomb exchange splitting. In the metal phase, overlapping quasiparticle bands appear at the Fermi level.