Integrable boundary terms for the finite open-chain deformation

Establish the boundary terms needed to realize an integrable finite open chain at the strong-deformation point of the graph-dressed PXP model satisfying the hard-boson integrability condition.

Background

The paper studies a next-nearest-neighbor density deformation of the PXP Hamiltonian, together with onsite fields, to test whether graph-edge entanglement certification persists away from the reference model. At interaction strength λ/J=1\lambda/J=1 and zero onsite field, the bulk coefficients satisfy a known hard-boson integrability condition. However, the authors use this point only as a strong-deformation control because they have not established the boundary terms required for an integrable finite open chain. Determining those boundary terms would clarify whether the finite open-chain realization is genuinely integrable and would enable a more controlled comparison between integrable and nonintegrable dynamics.

References

The $\lambda/J=1$, $h_i=0$ point also needs care: its bulk coefficients satisfy the known hard-boson integrability condition $w2=UV+V2$ with $|w|=J$, $U=0$ and $V=\lambda$. We use this point as a strong-deformation control, with no claim of robustness to generic nonintegrable perturbations; we have not established the boundary terms needed for an integrable finite open chain.

— Sparse and weak-measurement certification of graph-edge entanglement in PXP scar wavepackets  (2609.24139 - Wang et al., 21 Sep 2026) in Section 4.3, “Perturbations, alternative inputs and detector effects”