Exact Solution to a Class of Generalized Kitaev Spin-$1/2$ Models in Arbitrary Dimensions

We construct a class of exactly solvable generalized Kitaev spin-$1/2$ models in arbitrary dimensions, which is beyond the category of quantum compass models. The Jordan-Wigner transformation is employed to prove the exact solvability. An exactly solvable quantum spin-$1/2$ models can be mapped to a gas of free Majorana fermions coupled to static $Z_2$ gauge fields. We classify these exactly solvable models according to their parent models. Any model belonging to this class can be generated by one of the parent models. For illustration, a two dimensional ($2D$) tetragon-octagon model and a three dimensional ($3D$) $xy$ bond model are studied.