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The famous Li-Haldane conjecture relates the energy and bi-partiteentanglement spectrum in many topological phases. This correspondence isusually satisfied in chiral ordered states or systems with a symmetry-protectedgapless boundary. Although a lot of theoretical and numerical works haveconfirmed the conjecture in topological states with bulk-boundarycorrespondence, the cases with gapped boundary are widely unknown. Via newlydeveloped quantum Monte Carlo (QMC) scheme, we extract the large-scaleentanglement spectrum and study its relation with energy spectrum in theAffleck-Kennedy-Lieb-Tasaki (AKLT) phase with a tunnable boundary on thesquare-octagon lattice. We find the low-lying entanglement spectrum does notalways show similar behaviors as the energy spectrum on the tunnable boundary.The breakdown of the correspondence originates from the perturbation in theenvironment and the coupling between the subsystem and its environment. Thiscan be understood by the wormhole effect in the path integral representation ofreduced density matrix. Our study generalizes the established correspondencebetween gapless/gapped boundary energy spectrum and entanglement spectrum, andfurther demonstrates there exists a deep connection between physical andentanglement degrees of freedom in quantum many-body systems.

Probing Li-Haldane conjecture with a perturbed boundary