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The Motzkin spin chain is a spin-$1$ frustration-free model introduced byShor & Movassagh. The ground state is constructed by mapping random walks onthe upper half of the square lattice to spin configurations. It has unusuallylarge entanglement entropy [quantum fluctuations]. The ground state of theMotzkin chain can be analytically described by the Motzkin paths. There is noanalytical description of the excited states. The model is not solvable. Wesimplify the model by removing one of the local equivalence moves of theMotzkin paths. The system becomes integrable [similar to the XXX spin chain].We call it free Motzkin chain. From the point of view of quantum integrability,the model is special since its $R$-matrix does not have crossing unitarity. Wesolve the periodic free Motzkin chain by generalizing the functional BetheAnsatz method. We construct a $T-Q$ relation with an additional parameter toformulate the energy spectrum. This new parameter is related to the roots ofunity and can be described by the M\"obius function in number theory. Weobserve further patterns of number theory.

Can a spin chain relate combinatorics to number theory?