Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains

We introduce a family of SO($n$)-symmetric spin chains which generalize the transverse-field Ising chain for $n=1$. These spin chains are defined with Gamma matrices and can be exactly solved by mapping to $n$ species of itinerant Majorana fermions coupled to a static ${\mathbb{Z}}_2$ gauge field. Their phase diagrams include a critical point described by the $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n{)}_1$ conformal field theory as well as two distinct gapped phases. We show that one of the gapped phases is a trivial phase and the other realizes a symmetry-protected topological phase when $n\ge 2$. These two gapped phases are proved to be related to each other by a Kramers-Wannier duality. Furthermore, other elegant structures in the transverse-field Ising chain, such as the infinite-dimensional Onsager algebra, also carry over to our models.