Can a spin chain relate combinatorics to number theory?

Abstract

The Motzkin spin chain is a spin-1 frustration-free model introduced by Shor & Movassagh. The ground state is constructed by mapping of random walks on upper half of the square lattice to spin configurations. It has unusually large entanglement entropy [quantum fluctuations]. We simplify the model by removing one of the local equivalence moves of the Motzkin 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. We solve the periodic free Motzkin chain by generalized functional Bethe Ansatz method. We construct a T − Q relation with an additional parameter to formulate the energy spectrum. This new parameter is related to the roots of unity and can be described by the Möbius function of the number theory. We observe further patterns of the number theory.

The spin-one Motzkin chain is gapped for any area weight $t<1$

“…For example, Motzkin spin chains have been used in quantum error-correcting codes [BCS ¸B19], with their generalizations being "good quantum codes" [MO20]. Moreover, they generate exact holographic tensor network representations [AEK21] and have connections to number theory [HSK22].…”