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arXiv:2303.00772 · Physical review. B./Physical review. B

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

Zenan Liu, Rui-Zhen Huang, Zheng Yan, Dao‐Xin Yao

Published Mar 12, 2024• 13 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.109.094416

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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.
Based on the path integral formulation of the reduced density matrix, we develop a scheme to overcome the exponential growth of computational complexity in reliably extracting low-lying entanglement spectrum from quantum Monte Carlo simulations. We test the method on the Heisenberg spin ladder with long entangled boundary between two chains and the results support the Li and Haldane's conjecture on entanglement spectrum of topological phase. We then explain the conjecture via the wormhole effect in the path integral and show that it can be further generalized for systems beyond gapped topological phases. Our further simulation results on the bilayer antiferromagnetic Heisenberg model with 2D entangled boundary across the (2 + 1)D O(3) quantum phase transition clearly demonstrate the correctness of the wormhole picture. Finally, we state that since the wormhole effect amplifies the bulk energy gap by a factor of β, the relative strength of that with respect to the edge energy gap will determine the behavior of low-lying entanglement spectrum of the system.
We propose a physical picture based on the wormhole effect of the path-integral formulation to explain the mechanism of the entanglement spectrum (ES), such that our picture not only explains the topological state with bulk-edge correspondence of the energy spectrum and ES (the Li and Haldane conjecture), but is generically applicable to other systems independent of their topological properties. We point out it is ultimately the relative strength of bulk energy gap (multiplied with inverse temperature $\ensuremath{\beta}=1/T$) with respect to the edge energy gap that determines the behavior of the low-lying ES of the system. Depending on the circumstances, the ES can resemble the energy spectrum of the virtual edge, but can also represent that of the virtual bulk. We design models both in one and two dimensions to successfully demonstrate the bulklike low-lying ES at finite temperatures, in addition to the edgelike case conjectured by Li and Haldane at zero temperature. Our results support the generality of viewing the ES as the wormhole effect in the path integral and the different temperature dependence for the edge and bulk of ES.
Beyond the Li-Haldane-Poilblanc conjecture, we find the entanglement Hamiltonian (EH) is actually not closely similar to the original Hamiltonian on the virtual edge. Unexpectedly, the EH has some relevant long-range interacting terms which hugely affect the physics. Without loss of generality, we study a spin-1/2 Heisenberg bilayer to obtain the entanglement information between the two layers through our newly developed quantum Monte Carlo scheme, which can simulate large-scale EH. Although the entanglement spectrum carrying the Goldstone mode seems like a Heisenberg model on a single layer, which is consistent with Li-Haldane-Poilblanc conjecture, we demonstrate that there actually exists a finite-temperature phase transition of the EH. The results violate the Mermin-Wagner theorem, which means there should be relevant long-range terms in the EH. It reveals that the Li-Haldane-Poilblanc conjecture ignores necessary corrections for the EH which may lead totally different physics.
Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.
Abstract Symmetry-protected topological phases are short-range entangled quantum states characterized by anomalous edge behavior, a manifestation of the bulk-boundary correspondence for topological phases. Moreover, the Li-Haldane conjecture posits that the entanglement spectrum exhibits the same anomaly as the physical edge spectrum, thereby serving as an entanglement-based fingerprint for identifying topological phases. In this work, we extend the entanglement-based diagnostic tools by demonstrating that the edge anomaly is manifested not only in the entanglement spectrum but also in the reduced density matrix itself, a phenomenon we refer to as the mixed-state anomaly. For the two-dimensional SPT phase, we derive a topological contribution in the disorder parameter as a consequence of this mixed-state anomaly. Furthermore, with additional time-reversal symmetry, the anomalous reduced density matrix induces a spontaneous-symmetry-breaking type long-range order in the presence of the disorder operator.
The entanglement spectrum (ES) is a powerful tool for probing topological phases. While its behavior in gapped systems is well understood, its properties in gapless regimes remain unclear. In this work, we employ a quantum Monte Carlo method to study the ES of a two-dimensional square-octagon lattice Heisenberg model at quantum criticality and in the Néel phase. We find that the ES exhibits an M-shape magnon mode with a distinct sublinear dispersion, deviating from the conventional linear magnon. This behavior, similar to that of a one-dimensional long-range Heisenberg chain, reveals the emergence of relevant long-range interactions in the entanglement Hamiltonian. We demonstrate that the mechanism underlying short- and long-range interactions in the entanglement Hamiltonian can be interpreted as the confinement/deconfinement of worldlines in the path integral formulation. Our results reveal that gapless modes can fundamentally change the entanglement Hamiltonian and its spectrum, thereby offering insight into this general phenomenon.