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arXiv:2112.05886 · Nature Communications

Unlocking the general relationship between energy and entanglement spectra via the wormhole effect

Zheng Yan, Zi Yang Meng

Published Apr 24, 2023• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41467-023-37756-7

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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.
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.
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.
Bipartite entanglement measures are surprisingly useful tools to investigate quantum phases of correlated electrons. Here, I analyze the entanglement spectrum of gapped two-leg quantum Heisenberg ladders on a periodic ribbon partitioned into two identical periodic chains. The entanglement spectrum closely reflects the low-energy gapless spectrum of each individual edge. This extends the conjecture initially drawn for fractional quantum Hall systems to the field of quantum magnetism, stating a direct correspondence between the low-energy entanglement spectrum of a partitioned system and the true spectrum of the virtual edges. A mapping of the reduced density matrix to a thermodynamic density matrix is also proposed via the introduction of an effective temperature.
The reduced density matrix (RDM) plays a key role in quantum entanglement and measurement, as it allows the extraction of almost all physical quantities related to the reduced degrees of freedom. However, restricted by the degrees of freedom in the environment, the total system size is often limited, let alone the subsystem. To address this challenge, we propose a quantum Monte Carlo scheme with a low technical barrier, enabling precise extraction of the RDM. To demonstrate the power of the method, we present the fine levels of the entanglement spectrum (ES), which is the logarithmic eigenvalues of the RDM. We clearly show the ES for a 1D ladder with a long entangled boundary, and that for the 2D Heisenberg model with a tower of states. Furthermore, we put forward an efficient way to restore the entanglement Hamiltonian in operator-form from the sampled RDM data. Our simulation results, utilizing unprecedentedly large system sizes, establish a practical computational framework for determining entanglement quantities based on the RDM, such as the ES, particularly in scenarios where the environment has a huge number of degrees of freedom. The reduced density matrix enables extraction of entanglement-related quantities but computing it for large quantum systems is challenging. Mao et al. combine quantum Monte Carlo with exact diagonalization to efficiently extract the reduced density matrix and entanglement spectrum in large 1D and 2D systems.
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.