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arXiv:2303.14326 · Physical review. B./Physical review. B

Stable computation of entanglement entropy for two-dimensional interacting fermion systems

Gaopei Pan, Yuan Da Liao, Weilun Jiang, Jonathan D’Emidio, Yang Qi, Zi Yang Meng

Published Aug 28, 2023• 24 citations
arXivHTMLPDF

Abstract

There is no doubt that the information hidden in entanglement entropy (EE), for example, the $n\mathrm{th}$ 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.

DOI
10.1103/physrevb.108.l081123

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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., SnA=11\ensuremath−nlnTr(\ensuremathρAn){S}_{n}^{A}=\frac{1}{1\ensuremath{-}n}ln\mathrm{Tr}({\ensuremath{\rho}}_{A}^{n})SnA​=1\ensuremath−n1​lnTr(\ensuremathρAn​), where \ensuremathρA=TrA‾\ensuremathρ{\ensuremath{\rho}}_{A}={\mathrm{Tr}}_{\overline{A}}\ensuremath{\rho}\ensuremathρA​=TrA​\ensuremathρ 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.
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 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.
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.
The density matrix renormalization group (DMRG) method allows an efficient computation of the properties of interacting 1D quantum systems. Two-dimensional (2D) systems, capable of displaying much richer quantum behavior, generally lie beyond its reach except for very small system sizes. Many of the physical properties of 2D systems carry into the quasi-1D case, for which, unfortunately, the standard 2D DMRG algorithm fares little better. By finding the form of the entanglement entropy in quasi-1D systems, we directly identify the reason for this failure. Using this understanding, we explain why a modified algorithm, capable of cleverly exploiting this behavior of the entanglement entropy, can accurately reach much larger system sizes. We demonstrate the power of this method by accurately finding quantum critical points in frustration induced magnetic transitions, which remain inaccessible using the standard DMRG or the Monte Carlo methods. We contrast half-integer and integer spin cases.