In a seminal paper [1], Kugel and Khomskii demonstrated about 50 years ago that, in strongly-correlated systems, orbital-ordering can arise from a purely electronic superexchange interaction. Since then, the Kugel-Khomskii mechanism became a pillar of orbital physics. Besides orbital phase transitions, it can give rise to a plethora of emergent phenomena, from exotic magnetic states, orbital liquid phases, hidden order and much more. Yet, identifying real systems in which orbital-ordering arises purely from this mechanism has been a challenge. In fact, ions with orbital degrees of freedom are also of the Jahn-Teller kind. This poses a chicken-and-egg problem: electronic ordering can give rise to lattice distortions and lattice distortions can give rise to orbital ordering. This problem was eventually solved [2-4], showing how the two effects can be disentangled. From the solution emerged another obstacle: lattice distortions are very efficient in freezing orbital fluctuations via the crystal-field splitting, hampering the effects of superexchange. In most systems, orbital ordering thus occurs at temperatures well above the critical temperature at which super-exchange can drive it. Representative cases for which the opposite is true were recently identified, however [5-7]. This can lead to surprising effects, such as the inversion of magnetic and orbital ordering temperatures. In this talk, after a general introduction to the problem, I will discuss such paradigmatic cases. The understanding gained provides guidelines for finding Kugel-Khomskii materials [8].

[1] K. I. Kugel' and D. I. Khomskii, Zh. Eksp. Teor. Fiz. 64, 1429 (1973).
[2] E. Pavarini, E. Koch, A.I. Lichtenstein, Phys. Rev. Lett. 101, 266405 (2008)
[3] E. Pavarini and E. Koch, Phys. Rev. Lett. 104, 086402 (2010)
[4] H. Sims, E. Pavarini, and E. Koch, Phys. Rev. B 96, 054107 (2017)
[5] X-J. Zhang, E. Koch and E. Pavarini, Phys. Rev. B 105, 115104 (2022)
[6] X-J. Zhang, E. Koch and E. Pavarini, Phys. Rev. B 106, 115110 (2022)
[7] X-J. Zhang, E. Koch and E. Pavarini, Phys. Rev. Lett. 135, 026508 (2025)
[8] E. Koch and E. Pavarini, submitted.