Mott insulators with large and active (or multiflavor) local Hilbert spaces widely occur in quantum materials and ultracold atomic systems, and are dubbed “multiflavor Mott insulators”. For these multiflavor Mott insulators, the spin-only description with the quadratic spin interactions is often insufficient to capture the major physical processes. In the situation with active orbitals, the Kugel-Khomskii superexchange model was then proposed. We briefly review this historical model and discuss the modern developments beyond the original spin-orbital context. These include and are not restricted to the 4d/5d transition metal compounds with the spin-orbit-entangled J = 3/2 quadruplets, the rare-earth magnets with two weakly-separated crystal field doublets, breathing magnets and/or the cluster and molecular magnets, et al. We explain the microscopic origin of the emergent Kugel-Khomskii physics in each realization with some emphasis on the J = 3/2 quadruplets, and refer the candidate multiflavor Mott insulators as “J = 3/2 Mott insulators”. For the ultracold atoms, we review the multiflavor Mott insulator realization with the ultracold alkaline and alkaline-earth atoms on the optical lattices. Despite a large local Hilbert space from the atomic hyperfine spin states, the system could naturally realize a large symmetry group such as the Sp(N) and SU(N) symmetries. These ultracold atomic systems lie in the large-N regime of these symmetry groups and are characterized by strong quantum fluctuations. The Kugel-Khomskii physics and the exotic quantum ground states with the “baryon-like” physics can appear in various limits. We conclude with our vision and outlook on this subject.

具有大且活跃（或多风味）局部希尔伯特空间的莫特绝缘体广泛出现在量子材料和超冷原子系统中，被称为“多风味莫特绝缘体”。对于这些多味莫特绝缘体，具有二次自旋相互作用的仅自旋描述通常不足以捕获主要的物理过程。在活性轨道的情况下，提出了Kugel-Khomskii超交换模型。我们简要回顾这个历史模型，并讨论原始自旋轨道背景之外的现代发展。这些包括但不限于具有自旋轨道纠缠J = 3/2 四重态的 4 d /5 d过渡金属化合物 、具有两个弱分离晶体场双峰的稀土磁体、呼吸磁体和/或簇磁体和分子磁体等。我们在每个实现中解释了新兴 Kugel-Khomskii 物理的微观起源，重点强调J  = 3/2 四元组，并将候选多味莫特绝缘体称为“ J  = 3/2 莫特绝缘体”。对于超冷原子，我们回顾了光学晶格上超冷碱金属和碱土原子的多味莫特绝缘体的实现。尽管原子超精细自旋态具有较大的局部希尔伯特空间，但该系统可以自然地实现较大的对称群，例如 Sp( N ) 和 SU( N ) 对称性。这些超冷原子系统位于这些对称群的大N区域中，并具有强烈​​的量子涨落的特征。库格尔-霍姆斯基物理和具有“类重子”物理的奇异量子基态可以出现在各种极限中。最后我们总结了我们对此主题的愿景和展望。