Entanglement entropy (EE) plays a central role in the intersection of quantum information science and condensed matter physics. However, scanning the EE for two-dimensional and higher-dimensional systems still remains challenging. To address this challenge, we propose a quantum Monte Carlo scheme capable of extracting large-scale data of Rényi EE with high precision and low technical barrier. Its advantages lie in the following aspects: a single simulation can obtain the continuous variation curve of EE with respect to parameters, greatly reducing the computational cost; the algorithm implementation is simplified, and there is no need to alter the spacetime manifold during the simulation, making the code easily extendable to various many-body models. Additionally, we introduce a formula to calculate the derivative of EE without resorting to numerical differentiation from dense EE data. We then demonstrate the feasibility of using EE and its derivative to find phase transition points, critical exponents, and various phases. Entanglement entropy is a crucial quantity in several research fields, yet its calculation in higher-dimensional systems remains challenging. Here the authors develop a quantum Monte Carlo-based algorithm to efficiently extract large-scale entanglement entropy data and apply it to quantum critical systems.
Entanglement entropy (EE) plays a central role in the intersection of quantum information science and condensed matter physics. However, scanning the EE for two-dimensional and higher-dimensional systems still remains challenging. To address this challenge, we propose a quantum Monte Carlo scheme capable of extracting large-scale data of Rényi EE with high precision and low technical barrier. Its advantages lie in the following aspects: a single simulation can obtain the continuous variation curve of EE with respect to parameters, greatly reducing the computational cost; the algorithm implementation is simplified, and there is no need to alter the spacetime manifold during the simulation, making the code easily extendable to various many-body models. Additionally, we introduce a formula to calculate the derivative of EE without resorting to numerical differentiation from dense EE data. We then demonstrate the feasibility of using EE and its derivative to find phase transition points, critical exponents, and various phases. Entanglement entropy is a crucial quantity in several research fields, yet its calculation in higher-dimensional systems remains challenging. Here the authors develop a quantum Monte Carlo-based algorithm to efficiently extract large-scale entanglement entropy data and apply it to quantum critical systems.
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) is much higher than that of spin/boson (∼βN). Here, N is the total number of sites and β 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.
Entanglement entropy has been a powerful tool for analyzing phases and criticality in pure ground states via quantum Monte Carlo (QMC). However, mixed-state entanglement, relevant to systems with dissipation, finite temperature, and disjoint regions, remains less explored due to the lack of efficient numerical methods. In this work, we present a practical and easy-to-implement QMC method within the reweight-annealing framework, enabling efficient computation of the entanglement R\'enyi negativity by tracking its variation along given parameter paths. This method is scalable, parallelizable, and well suited for high-dimensional and large-scale simulations. Applying it to diverse scenarios, including one- and two-dimensional systems, ground and thermal states, and bipartite and tripartite partitions, not only is the information of the underlying conformal field theory achieved, but the role of entanglement in quantum and thermal phase transitions is revealed.
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 q 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); data collapse of its first derivative yields the correlation-length exponent ν for both the O(3) and Ising universality classes, realized in the spin-21 columnar dimerized Heisenberg model and the bilayer Heisenberg-Ising model. The same simulations extract the Goldstone-mode count NG=2 and the Z2 ground-state degeneracy ln2 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.
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 Z2 topological ordered state and the Kagome 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.
Based on the recently developed resummation-based quantum Monte Carlo method for the 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) is the second-order R\'enyi EE, such that the S(2) for a generic 2D quantum SU(N) spin models can be readily computed with high accuracy. We benchmark our algorithm with the previously proposed estimators of S(2) on 1D and 2D 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) Heisenberg model with continuously varying N. Our ResumEE algorithm is efficient for precisely evaluating the entanglement entropy of SU(N) spin models with continuous N and reliable access to the conformal field theory data for the highly entangled quantum matter.