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arXiv:2401.07244 · Physical review. B./Physical review. B

Incremental SWAP operator for entanglement entropy: Application for exponential observables in quantum Monte Carlo simulation

X. Zhou, Zi Yang Meng, Yang Qi, Yuan Da Liao

Published Apr 2, 2024• 20 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.165106

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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.
Researchers in physical science aim to uncover universal features in strongly interacting many-body systems, often hidden in complicated observables like entanglement entropy (EE). The non-local nature of EE makes it challenging to compute numerically, necessitating the development of an unbiased and convenient algorithm. In this paper, we use quantum Monte Carlo to reveal that the coefficient of variation in direct EE calculations increases exponentially with system size, leading to inaccuracies. To address this issue, we develop a power incremental algorithm and a technique for straightforwardly calculating the universal term of EE, successfully evaluating the EE of a 2D Hubbard model. Our numerical results demonstrate the consistency of the universal coefficient of EE from sharp corners at the Gross-Neveu quantum critical point and for free Dirac fermions. Our method can also be applied to other unstable observables, such as partition functions, entanglement spectra, and negativity, thereby fostering computational and theoretical progress.
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.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.
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.
The subleading corner logarithmic corrections in entanglement entropy (EE) are crucial for revealing the universal characteristics of quantum critical points (QCPs), but they are challenging to detect. Motivated by recent developments in the stable computation of EE in (2+12+12+1)-dimensional [(2+12+12+1)D] quantum many-body systems, we have developed a method for directly measuring the corner contribution in EE with lower computational cost. The cornerstone of our approach is to measure the subtracted corner entanglement entropy (SCEE), defined as the difference between the EEs of subregions with the same boundary length for smooth and cornered boundaries during a sign-problem-free quantum Monte Carlo simulation. Our improved method inherently eliminates not only the area-law term of EE but also the subleading log corrections arising from Goldstone modes, leaving the universal corner contribution as the leading term of the SCEE with greatly improved data quality. Utilizing this advanced approach, we calculate the SCEE of the bilayer Heisenberg model on both square and honeycomb lattices across their (2+12+12+1)D OOO(3) QCPs with different opening angles on the entanglement boundary and obtain accurate values of the corresponding universal corner log coefficients. These findings will encourage further theoretical investigations to access controlled universal information for interacting conformal field theories at (2+12+12+1) dimensions.
Entanglement entropy has become an important tool for identifying phases of matter and phase transitions in quantum systems using quantum Monte Carlo methods. While such approaches have achieved great success in systems composed of interacting spins/bosons, their application to interacting fermionic systems remains limited due to substantially higher computational cost. Here we show a more efficient Monte Carlo algorithm for fermionic systems that enables systematic exploration of entanglement entropy across a wide range of physical parameters. Based on the incremental technique along physical parameters, the method significantly reduces computational effort while maintaining accuracy. Using this approach, we study a two-dimensional square lattice Hubbard model, revealing the phase diagram that includes the Fermi surface and Goldstone modes. We further apply the method to the Gross-Neveu criticality and find that the scaling of the entanglement entropy follows a universal form quantified by the critical exponent ν. Further investigation reveals that the leading coefficient of the entanglement entropy decreases monotonically, rather than developing a local maximum as O(N) transition point. Entanglement entropy is a powerful tool for identifying quantum phases and phase transitions but calculating it in interacting fermionic systems remains challenging. Here, the authors introduce an efficient Monte Carlo method that reduces computational cost and reveals universal scaling across Gross-Neveu fermionic phase transitions.