Exact Bethe zero modes of the non-integrable alternating sign Heisenberg chain

Exact solutions of quantum lattice models serve as useful guides for interpreting physical phenomena in condensed matter systems. Prominent examples of integrability appear in one dimension, including the Heisenberg chain, where the Bethe ansatz has been widely successful. Recent work has noted that certain non-integrable models harbor quantum many-body scar states, which form a superspin of regular states hidden in an otherwise chaotic spectrum. We consider one of the simplest examples of a non-integrable model, the alternating ferromagnetic-antiferromagnetic Heisenberg chain, and show the presence of exponentially many zero-energy states. While the existence of a high degeneracy of zero modes in other models has been previously reported, we highlight features of the alternating chain that allow a treatment with the Bethe ansatz. Surprisingly for a non-integrable system, we can derive explicit expressions for zero-energy eigenfunctions for several magnons including solutions with fractionalized particle momentum. We conjecture a picture of magnon pairing which may generalize to many particles. Our work opens the avenue to describe a large subspace of eigenstates of non-integrable models using the Bethe ansatz, with potential for stabilizing ordered states at high energies.