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arXiv:2310.01490 · Physical review. B./Physical review. B

Resummation-based quantum Monte Carlo for entanglement entropy computation

Menghan Song, Ting-Tung Wang, Zi Yang Meng

Published Sep 10, 2024• 7 citations
arXivHTMLPDF

Abstract

Based on the recently developed resummation-based quantum Monte Carlo method for the $\text{SU}(N)$ spin and loop-gas models, we developed an algorithm, dubbed ResumEE, to compute the entanglement entropy (EE) with greatly enhanced efficiency. Our ResumEE exponentially speeds up the computation of the exponentially small value of the $\ensuremath{\langle}{e}^{\ensuremath{-}{S}^{(2)}}\ensuremath{\rangle}$, where ${S}^{(2)}$ is the second-order R\'enyi EE, such that the ${S}^{(2)}$ for a generic 2D quantum $\text{SU}(N)$ spin models can be readily computed with high accuracy. We benchmark our algorithm with the previously proposed estimators of ${S}^{(2)}$ on 1D and 2D $\text{SU}(2)$ Heisenberg spin systems to reveal its superior performance and then use it to detect the entanglement scaling data of the N\'eel-to-VBS transition on 2D $\text{SU}(N)$ Heisenberg model with continuously varying $N$. Our ResumEE algorithm is efficient for precisely evaluating the entanglement entropy of $\text{SU}(N)$ spin models with continuous $N$ and reliable access to the conformal field theory data for the highly entangled quantum matter.

DOI
10.1103/physrevb.110.115117

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Based on the recently developed resummation-based quantum Monte Carlo method for the SU(N)\text{SU}(N)SU(N) spin and loop-gas models, we developed an algorithm, dubbed ResumEE, to compute the entanglement entropy (EE) with greatly enhanced efficiency. Our ResumEE exponentially speeds up the computation of the exponentially small value of the \ensuremath⟨e\ensuremath−S(2)\ensuremath⟩\ensuremath{\langle}{e}^{\ensuremath{-}{S}^{(2)}}\ensuremath{\rangle}\ensuremath⟨e\ensuremath−S(2)\ensuremath⟩, where S(2){S}^{(2)}S(2) is the second-order R'enyi EE, such that the S(2){S}^{(2)}S(2) for a generic 2D quantum SU(N)\text{SU}(N)SU(N) spin models can be readily computed with high accuracy. We benchmark our algorithm with the previously proposed estimators of S(2){S}^{(2)}S(2) on 1D and 2D SU(2)\text{SU}(2)SU(2) Heisenberg spin systems to reveal its superior performance and then use it to detect the entanglement scaling data of the N'eel-to-VBS transition on 2D SU(N)\text{SU}(N)SU(N) Heisenberg model with continuously varying NNN. Our ResumEE algorithm is efficient for precisely evaluating the entanglement entropy of SU(N)\text{SU}(N)SU(N) spin models with continuous NNN and reliable access to the conformal field theory data for the highly entangled quantum matter.
Abstract We develop a nonequilibrium increment method in quantum Monte Carlo simulations to obtain the Rényi entanglement entropy of various quantum many-body systems with high efficiency and precision. To demonstrate its power, we show the results on a few important yet difficult (2 + 1)dquantum lattice models, ranging from the Heisenberg quantum antiferromagnet with spontaneous symmetry breaking, the quantum critical point with O(3) conformal field theory (CFT) to the toric code Z2{{\mathbb{Z}}}_{2}Z2​ Z2 topological ordered state and the Kagome Z2{{\mathbb{Z}}}_{2}Z2​ Z2 quantum spin liquid model with frustration and multi-spin interactions. In all these cases, our method either reveals the precise CFT data from the logarithmic correction or extracts the quantum dimension in topological order, from the dominant area law in finite-size scaling, with very large system sizes, controlled errorbars, and minimal computational costs. Our method, therefore, establishes a controlled and practical computation paradigm to obtain the difficult yet important universal properties in highly entangled quantum matter.
We propose a method to efficiently compute the entanglement entropy (EE) of quantum many-body systems. Our approach, called the incremental SWAP operator method, combines the simplicity of the SWAP operator used in projector quantum Monte Carlo simulations with recent advances in precisely computing exponential observables using incremental algorithms. We apply this technique to obtain accurate EE data at reduced computational cost for 1D and 2D antiferromagnetic Heisenberg models with different bipartition schemes. Using the computed EE data, we extract the area law coefficient, universal logarithmic corrections from Goldstone modes, and the geometric constant, finding quantitative agreement with analytical predictions. Moreover, in an unbiased numerical simulation of 2D antiferromagnetic Heisenberg model, we successfully obtain reliable universal logarithmic corrections from sharp corners that match expected theoretical values. The consistency between our numerical results and theoretical calculations demonstrates the power of our approach for accessing challenging universal entanglement properties. The extensions of our method to other quantum spin/boson models and the interacting fermion models are outlined.
The entanglement entropy probing novel phases and phase transitions numerically via quantum Monte Carlo has made great achievements in large-scale interacting spin/boson systems. In contrast, the numerical exploration in interacting fermion systems is rare, even though fermion systems attract more attentions in condensed matter. The fundamental restrictions is that the computational cost of fermion quantum Monte Carlo (∼βN3\sim βN^3∼βN3) is much higher than that of spin/boson (∼βN\sim βN∼βN). Here, NNN is the total number of sites and βββ is the inverse temperature or projection length. To tackle this problem, we propose a fermionic quantum Monte Carlo algorithm based on the incremental technique along physical parameters, which greatly improves the efficiency of extracting entanglement entropy. We benchmark the developed algorithm by calculating the scaling behavior of the entanglement entropy in a two-dimensional square lattice Hubbard model. The obtained phase diagram including Fermi surface and Goldstone modes validates the correctness of the algorithm. Remarkably, our method shows the high-efficiency with respect to the existing algorithms, while keeping the high computation precision. We proceed to apply this algorithm to explore the scaling behavior of the entanglement entropy and particularly its derivative at Gross-Neveu criticality. Our results elucidate that such critical behavior can be quantified by the correlation length exponent.
Entanglement entropy (EE) plays a central role in the intersection of quantum information science and condensed matter physics. However, scanning the EE for two-dimensional and higher-dimensional systems still remains challenging. To address this challenge, we propose a quantum Monte Carlo scheme capable of extracting large-scale data of Rényi EE with high precision and low technical barrier. Its advantages lie in the following aspects: a single simulation can obtain the continuous variation curve of EE with respect to parameters, greatly reducing the computational cost; the algorithm implementation is simplified, and there is no need to alter the spacetime manifold during the simulation, making the code easily extendable to various many-body models. Additionally, we introduce a formula to calculate the derivative of EE without resorting to numerical differentiation from dense EE data. We then demonstrate the feasibility of using EE and its derivative to find phase transition points, critical exponents, and various phases. Entanglement entropy is a crucial quantity in several research fields, yet its calculation in higher-dimensional systems remains challenging. Here the authors develop a quantum Monte Carlo-based algorithm to efficiently extract large-scale entanglement entropy data and apply it to quantum critical systems.
There is no doubt that the information hidden in entanglement entropy (EE), for example, the nthn\mathrm{th}nth order R\'enyi EE, i.e., ${S}_{n}^{A}=\frac{1}{1\ensuremath{-}n}ln\mathrm{Tr}({\ensuremath{\rho}}_{A}^{n})$, where ${\ensuremath{\rho}}_{A}={\mathrm{Tr}}_{\overline{A}}\ensuremath{\rho}$ is the reduced density matrix, can be used to infer the organizing principle of two-dimensional (2D) interacting fermion systems, ranging from spontaneous symmetry-breaking phases and quantum critical points to topologically ordered states. It is far from clear, however, whether EE can be obtained with the precision required to observe these fundamental features---usually in the form of universal finite-sized scaling behavior. Even for the prototypical 2D interacting fermion model---the Hubbard model---to all existing numerical algorithms, the computation of EE has not been successful with reliable data from which the universal scaling regime can be accessed. Here, we explain the reason for these unsuccessful attempts of EE computations in quantum Monte Carlo simulations in the past decades and, more importantly, show how to overcome the conceptual and computational barrier with the incremental algorithm, such that the stable computation of EE in 2D interacting fermion systems can be achieved and universal scaling information can be extracted. Relevance toward experimental 2D interacting fermion systems is discussed.
We develop a quantum Monte Carlo procedure, in the valence bond basis, to measure the Renyi entanglement entropy of a many-body ground state as the expectation value of a unitary Swap operator acting on two copies of the system. An improved estimator involving the ratio of Swap operators for different subregions enables convergence of the entropy in a simulation time polynomial in the system size. We demonstrate convergence of the Renyi entropy to exact results for a Heisenberg chain. Finally, we calculate the scaling of the Renyi entropy in the two-dimensional Heisenberg model and confirm that the Néel ground state obeys the expected area law for systems up to linear size L=32.