Non-Interacting Motzkin Chain - Periodic Boundary Conditions

TL;DR

This paper studies a Motzkin spin chain with periodic boundary conditions, revealing its integrability when restricted to height-preserving moves and its relation to non-interacting spin-1/2 XXX chains, with interactions introduced by height-changing moves.

Contribution

It demonstrates the integrability of the Motzkin chain under certain conditions and relates it to well-known spin chains, providing new insights into its spectrum and degeneracies.

Findings

Model becomes integrable with height-preserving moves.

Full Hilbert space resembles two non-interacting XXX chains.

Including height-changing moves introduces interactions.

Abstract

The Motzkin spin chain is a spin-1 model introduced in \cite{shor} as an example of a system exhibiting a high degree of quantum fluctuations whose ground state can be mapped to Motzkin paths that are generated with local equivalence moves. This model is difficult to solve in general but keeping just the height preserving local equivalence moves we show that the model becomes integrable which when projected to certain subspaces of the full Hilbert space is isomorphic to the spin-$\frac{1}{2}$ XXX chain. In fact in the full Hilbert space the system is akin to two non-interacting spin-$\frac{1}{2}$ XXX chains making the spectrum the same as the latter with the change coming in the degeneracy of the states. We then show that including the height-changing local-equivalence move is the same as introducing interactions in the above system.