  The Li-Haldane conjecture is one of the most famous conjectures in physics
and opens a new research area in the quantum entanglement and topological
phase. Although a lot of theoretical and numerical works have confirmed the
conjecture in topological states with bulk-boundary correspondence, the cases
with gapped boundary and the systems in high dimension are widely unknown. What
is the valid scope of the Li-Haldane conjecture? Via the newly developed
quantum Monte Carlo scheme, we are now able to extract the large-scale
entanglement spectrum (ES) and study its relation with the edge energy spectrum
generally. Taking the two-dimensional Affleck-Kennedy-Lieb-Tasaki model with a
tunable boundary on the square-octagon lattice as an example, we find several
counterexamples which cannot be explained by the Li-Haldane conjecture; e.g.,
the low-lying entanglement spectrum does not always show similar behaviors as
the energy spectrum on the virtual boundary, and sometimes the ES resembles the
energy spectrum of the edge even if it is gapped. Finally, we demonstrate that
the newly proposed wormhole mechanism on the path integral of a reduced density
matrix is the formation principle of the general ES. We find that the
Li-Haldane conjecture is a particular case in some limit of the wormhole
picture while all the examples of the conjecture we have studied can totally be
explained within the wormhole mechanism framework. Our results provide
important evidence for demonstrating that the wormhole mechanism is the
fundamental principle to explain the ES.
