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arXiv:2311.03448 · Physical review. B./Physical review. B

Integral algorithm of exponential observables for interacting fermions in quantum Monte Carlo simulations

Xu Zhang, Gaopei Pan, Bin-Bin Chen, Kai Sun, Zi Yang Meng

Published May 21, 2024• 18 citations
arXivHTMLPDF

Abstract

Exponential observables, formulated as $ln\ensuremath{\langle}{e}^{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}}\ensuremath{\rangle}$ where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ is an extensive quantity, play a critical role in the study of quantum many-body systems, examples of which include the free energy and entanglement entropy. Given that ${e}^{X}$ becomes exponentially large (or small) in the thermodynamic limit, the accurate computation of the expectation value of this exponential quantity presents a significant challenge. In this paper, we propose a comprehensive algorithm to quantify these observables in interacting fermion systems, utilizing the determinant quantum Monte Carlo method. We have applied this algorithm to the two-dimensional square-lattice half-filled Hubbard model and $\ensuremath{\pi}$-flux t-V model. In the Hubbard model case at the strong-coupling limit, our method showcases a significant accuracy improvement on free energy compared to conventional methods that are derived from the internal energy, and in the t-V model, we indicate that the free energy offers a precise determination of the second-order phase transition. We also illustrate that this approach delivers highly efficient and precise measurements of the $n\mathrm{th}$ R\'enyi entanglement entropy. Even more noteworthy is that this improvement comes without incurring increases in computational complexity. This algorithm effectively suppresses exponential fluctuations and can be easily generalized to other models.

DOI
10.1103/physrevb.109.205147

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Exponential observables, formulated as ln\ensuremath⟨eX\ifmmode^\elseˆ\fi\ensuremath⟩ln\ensuremath{\langle}{e}^{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}}\ensuremath{\rangle}ln\ensuremath⟨eX\ifmmode^\elseˆ\fi​\ensuremath⟩ where X\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}X\ifmmode^\elseˆ\fi​ is an extensive quantity, play a critical role in the study of quantum many-body systems, examples of which include the free energy and entanglement entropy. Given that eX{e}^{X}eX becomes exponentially large (or small) in the thermodynamic limit, the accurate computation of the expectation value of this exponential quantity presents a significant challenge. In this paper, we propose a comprehensive algorithm to quantify these observables in interacting fermion systems, utilizing the determinant quantum Monte Carlo method. We have applied this algorithm to the two-dimensional square-lattice half-filled Hubbard model and \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux t-V model. In the Hubbard model case at the strong-coupling limit, our method showcases a significant accuracy improvement on free energy compared to conventional methods that are derived from the internal energy, and in the t-V model, we indicate that the free energy offers a precise determination of the second-order phase transition. We also illustrate that this approach delivers highly efficient and precise measurements of the nthn\mathrm{th}nth R'enyi entanglement entropy. Even more noteworthy is that this improvement comes without incurring increases in computational complexity. This algorithm effectively suppresses exponential fluctuations and can be easily generalized to other models.
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.
We introduce a Monte Carlo determinantal algorithm that allows to efficiently sample dynamical quantities, such as the self-energy, of fermionic quantum systems in their thermodynamic limit. The method expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results of this algorithm for the lattice Hubbard model with those obtained from diagrammatic Monte Carlo and the recently introduced connected determinant approach, we show that we can reach higher perturbation orders and greater accuracy for the same computational effort.
We introduce and compare three different Monte Carlo determinantal algorithms that allow one to compute dynamical quantities, such as the self-energy, of fermionic systems in their thermodynamic limit. We show that the most efficient approach expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results for the two-dimensional Hubbard model with those obtained from state-of-the-art diagrammatic Monte Carlo, we show that we can reach higher perturbation orders and greater accuracy for the same computational effort.
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.
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.
In the absence of a fermion sign problem, auxiliary-field (or determinantal) quantum Monte Carlo (DQMC) approaches have long been the numerical method of choice for unbiased, large-scale simulations of interacting many-fermion systems. More recently, the conceptual scope of this approach has been expanded by introducing ingenious schemes to compute entanglement entropies within its framework. On a practical level, these approaches, however, suffer from a variety of numerical instabilities that have largely impeded their applicability. Here we report on a number of algorithmic advances to overcome many of these numerical instabilities and significantly improve the calculation of entanglement measures in the zero-temperature projective DQMC approach, ultimately allowing us to reach similar system sizes as for the computation of conventional observables. We demonstrate the applicability of this improved DQMC approach by providing an entanglement perspective on the quantum phase transition from a magnetically ordered Mott insulator to a band insulator in the bilayer square lattice Hubbard model at half filling.