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We introduce a family of SO($n$)-symmetric spin chains which generalize thetransverse-field Ising chain for $n=1$. These spin chains are defined withGamma matrices and can be exactly solved by mapping to $n$ species of itinerantMajorana fermions coupled to a static $\mathbb{Z}_2$ gauge field. Their phasediagrams include a critical point described by the $\mathrm{Spin}(n)_{1}$conformal field theory as well as two distinct gapped phases. We show that oneof the gapped phases is a trivial phase and the other realizes asymmetry-protected topological phase when $n \geq 2$. These two gapped phasesare proved to be related to each other by a Kramers-Wannier duality.Furthermore, other elegant structures in the transverse-field Ising chain, suchas the infinite-dimensional Onsager algebra, also carry over to our models.

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