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arXiv:2112.15178 · npj Quantum Materials

Measuring Rényi entanglement entropy with high efficiency and precision in quantum Monte Carlo simulations

Jiarui Zhao, Bin-Bin Chen, Yan-Cheng Wang, Zheng Yan, Meng Cheng, Zi Yang Meng

Published Jun 29, 2022• 61 citations
arXivHTMLPDF

Abstract

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 $${{\mathbb{Z}}}_{2}$$ Z2 topological ordered state and the Kagome $${{\mathbb{Z}}}_{2}$$ 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.

DOI
10.1038/s41535-022-00476-0

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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 develop a nonequilibrium increment method to compute the Rényi entanglement entropy and investigate its scaling behavior at the deconfined critical (DQC) point via large-scale quantum Monte Carlo simulations. To benchmark the method, we first show that, at a conformally invariant critical point of O(3) transition, the entanglement entropy exhibits universal scaling behavior of area law with logarithmic corner corrections, and the obtained correction exponent represents the current central charge of the critical theory. Then we move on to the deconfined quantum critical point, where we still observe similar scaling behavior, but with a very different exponent. Namely, the corner correction exponent is found to be negative. Such a negative exponent is in sharp contrast with the positivity condition of the Rényi entanglement entropy, which holds for unitary conformal field theories (CFTs). Our results unambiguously reveal fundamental differences between DQC and quantum critical points described by unitary CFTs.
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 Rényi entanglement entropy in quantum many-body systems can be viewed as the difference in free energy between partition functions with different trace topologies. We introduce an external field λ that controls the partition function topology, allowing us to define a notion of nonequilibrium work as λ is varied smoothly. Nonequilibrium fluctuation theorems of the work provide us with statistically exact estimates of the Rényi entanglement entropy. This framework also naturally leads to the idea of using quench functions with spatially smooth profiles, providing us a way to average over lattice scale features of the entanglement entropy while preserving long distance universal information. We use these ideas to extract universal information from quantum Monte Carlo simulations of SU(N) spin models in one and two dimensions. The vast gain in efficiency of this method allows us to access unprecedented system sizes up to 192×96 spins for the square lattice Heisenberg antiferromagnet.
Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. \nThrough measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. \nWe describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. \nThis description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. \n \nUsing the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. \nThe scaling near the critical point also shows signatures identifying the universality class of the model. \nImproved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. \nWe explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. \nThis same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. \nAll of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool.
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 $\ensuremath{\langle}{e}^{\ensuremath{-}{S}^{(2)}}\ensuremath{\rangle}$, 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.
The Shannon-Rényi (SR) entropy characterizes the probability distribution of many-body wave functions in a chosen basis and is directly accessible from single-shot measurement snapshots in cold-atom experiments, yet its theoretical evaluation in interacting systems carries exponential cost. Here we construct a field-theoretic spacetime manifold for the SR entropy in which qqq replicas are glued at the imaginary-time boundaries, recasting it as a ratio of generalized partition functions amenable to large-scale quantum Monte Carlo simulation. On this basis we develop a reweight-annealing framework with two complementary strategies: a virtual-parameter scheme interpolating between unglued and fully glued limits via binomial-weighted partial gluing, and a real-parameter scheme using a physical coupling constant as the annealing variable within stochastic series expansion. Both reduce the computational cost from exponential to polynomial. Most importantly, we establish for the first time a finite-size scaling theory for the SR entropy near a quantum critical point, Sq=a(g)L2+Sq0((g−gc)L1/ν)S_q=a(g)L^2+S_q^0\left((g-g_c)L^{1/ν}\right)Sq​=a(g)L2+Sq0​((g−gc​)L1/ν); data collapse of its first derivative yields the correlation-length exponent ννν for both the O(3) and Ising universality classes, realized in the spin-12\frac{1}{2}21​ columnar dimerized Heisenberg model and the bilayer Heisenberg-Ising model. The same simulations extract the Goldstone-mode count NG=2N_G=2NG​=2 and the Z2Z_2Z2​ ground-state degeneracy ln⁡2\ln 2ln2 from subleading terms. These results elevate the SR entropy to a standalone, order-parameter-free diagnostic of quantum criticality with the same scaling status as the entanglement entropy, and close the loop between theory and quantum simulation by enabling quantitative benchmarking of experimental snapshots from state-of-the-art quantum simulators.