Demonstrating the wormhole mechanism of the entanglement spectrum via a perturbed boundary

TL;DR

This paper investigates the entanglement spectrum in topological states, revealing limitations of the Li-Haldane conjecture and proposing a wormhole mechanism as a fundamental explanation for the entanglement spectrum.

Contribution

It introduces the wormhole mechanism as a new fundamental principle explaining the entanglement spectrum, extending beyond the Li-Haldane conjecture.

Findings

Counterexamples to the Li-Haldane conjecture are found.

The wormhole mechanism explains the formation of the entanglement spectrum.

The Li-Haldane conjecture is a special case within the wormhole framework.

Abstract

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.