Extensive Bethe Scar Towers and Their Entanglement Structure Beyond Integrability

The one-dimensional XXX Heisenberg chain admits Bethe-Ansatz eigenstates, including a class of “free-magnon” solutions consisting of a single finite-momentum excitation dressed by multiple zero-momentum magnons. We show that this entire class survives the breaking of integrability by SU(2)-invariant two-body interactions of arbitrary range. In the nonintegrable regime, each such Bethe-Ansatz seed with crystal momentum k generates an exact tower of nonthermal eigenstates connected by the total spin-raising operator. These Bethe scar ladders generalize the well-known Dicke tower (recovered at k=0) to all momenta.

Unlike Dicke scars, which occupy only one momentum sector, Bethe scars occur at all allowed crystal momenta, resulting in a number of nonthermal eigenstates that grows extensively with system size. We further derive closed-form expressions that predict the entanglement entropy of each level of the tower, revealing a systematic sub-volume scaling: Bethe scars form a distinct entanglement band that lies above Dicke states yet remains well separated from the thermal continuum. Our results demonstrate that remnants of Bethe physics can still play a role in violating thermalization beyond integrability.