In this colloquium, we review recent worldwide experimental advances to generate various kinds of spin-orbital couplings (SOC)
in cold atom systems in an optical lattice and also experiments on spin-orbital coupled materials in 4d or 5d transition metal oxides.
After a brief introduction to magnetisms in quantum Ferromagnetic Heisenberg and anti-ferromagnetic Heisenberg model,
we will discuss quantum Random Heisenberg model whose simplified version called
Sachdev-Ye-Kitaev ( SYK) model can stimulate quantum chaos and quantum information propagations in a black-hole through AdS_2/CFT_1 correspondence.
The impacts of SYK on 2d quantum gravity and Liouville Conformal field theory are stressed.
After briefly reviewing anti-ferromagnetic Heisenberg model in a geometrically frustrated lattice
and a spin-bond correlated quantum spin model called Kitaev model,
we will discuss quantum Ferro- and Anti-ferromagnet Heisenberg model subject to Spin-orbit couplings which host
In-commensurate non-coplanar Skyrmion crystal phases, devil staircases, Cantor sets and multi-fractals.
Implications on current or near future cold atom systems and on 4d or 5d transition metal oxides will be discussed.