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arXiv:2307.10602 · npj Quantum Information

Universal term of entanglement entropy in the π-flux Hubbard model

Yuan Da Liao

Published Apr 15, 2025• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41534-025-01010-3

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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.
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 is a unique probe to reveal universal features of strongly interacting many-body systems. In two or more dimensions these features are subtle, and detecting them numerically requires extreme precision, a notoriously difficult task. This is especially challenging in models of interacting fermions, where many such universal features have yet to be observed. In this Letter we tackle this challenge by introducing a new method to compute the Rényi entanglement entropy in auxiliary-field quantum Monte Carlo simulations, where we treat the entangling region itself as a stochastic variable. We demonstrate the efficiency of this method by extracting, for the first time, universal subleading logarithmic terms in a two-dimensional model of interacting fermions, focusing on the half-filled honeycomb Hubbard model at T=0. We detect the universal corner contribution due to gapless fermions throughout the Dirac semi-metal phase and at the Gross-Neveu-Yukawa critical point, where the latter shows a pronounced enhancement depending on the type of entangling cut. Finally, we observe the universal Goldstone mode contribution in the antiferromagnetic Mott insulating phase.
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.
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.
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.
Characterizing universal entanglement features in higher-dimensional quantum matter is a central goal of quantum information science and condensed matter physics. While the subleading corner terms in two-dimensional quantum systems encapsulate essential universal information of the underlying conformal field theory, our understanding of these features remains remarkably limited compared to their one-dimensional counterparts. We address this challenge by investigating the entanglement dynamics of fermionic systems along the imaginary-time evolution. We uncover a pioneering non-equilibrium scaling law where the corner entanglement entropy grows linearly with the logarithm of imaginary time, dictated solely by the universality class of the quantum critical point. Through unbiased Quantum Monte Carlo simulations, we verify this scaling in the interacting Gross-Neveu-Yukawa model, demonstrating that universal data can be accurately recovered from the early stages of relaxation. Our findings significantly circumvent the computational bottlenecks inherent in reaching full equilibrium convergence. This work establishes a direct link between the fundamental theory of non-equilibrium critical phenomena and the high-precision determination of universal entanglement properties on both classical and quantum platforms, paving the way for probing the rich entanglement structure of quantum critical systems.