Measuring Rényi entanglement entropy with high efficiency and precision in quantum Monte Carlo simulations

Entanglement entropy (EE), a measure of quantum entanglement in many-body systems, holds crucial information about quantum phases and phase transitions. Its scaling form reveals universal properties, making its computation highly significant across condensed matter, quantum information, and high-energy physics. In (2+1)d systems, the entanglement entropy (both von Neumann and Rényi) typically scales as S ≈ α|A| − γln(|A|) + O(1/l), where α is a non-universal area law coefficient, |A| is the area of the region A, and γ (or γ when γ=0) is a universal coefficient related to the system's intrinsic properties (e.g., central charge in CFT or topological entanglement entropy (TEE) in topological phases). Extracting these universal coefficients requires challenging finite-size scaling, demanding efficient computational methods. Quantum Monte Carlo (QMC) methods have shown promise, but their efficiency and stability have been limited, particularly in systems with multi-spin interactions or frustrations. This paper presents a novel nonequilibrium increment method that combines nonequilibrium measurements based on the Jarzynski equality with the replica-swapping ‘increment trick’ to overcome these challenges. The method leverages the divide-and-conquer approach of the nonequilibrium process, leading to improved simulation speed and data quality.

Previous studies have explored the scaling of Rényi entanglement entropy in various (2+1)d systems using QMC. These studies included systems exhibiting spontaneous O(N) symmetry breaking, critical points, and Z₂ topological order. However, computational challenges, mainly arising from the need for replica calculations and complex connectivity in sampling, have hampered the accurate extraction of universal corrections beyond the leading area law, especially in systems with multi-spin interactions or frustrations. Existing QMC methods often struggle to reach sufficiently large system sizes or achieve the necessary precision to resolve the universal logarithmic corrections and topological entanglement entropy. This paper aims to address these limitations.

The core of the proposed method lies in a novel nonequilibrium increment estimator for Rényi entanglement entropy. This approach combines two key techniques: 1. **Nonequilibrium measurements based on Jarzynski equality:** This technique avoids direct measurement of small observables in equilibrium by instead measuring the work done during a nonequilibrium process, thus providing more stable and efficient results. This is based on Jarzynski's equality, relating the free energy difference to the average of the exponential of the work done in a nonequilibrium process. In this case, the work done is directly related to the Rényi entanglement entropy. 2. **Increment trick:** To further boost efficiency, the nonequilibrium process is divided into multiple smaller, independent parts using the increment trick. Each part involves a smaller nonequilibrium change in a parameter controlling the connectivity of replicas in the calculation. By applying Jarzynski's equality to each segment and summing the contributions, a more precise and computationally efficient estimation of the Rényi entanglement entropy is obtained. This parallel approach drastically reduces computational time and enhances the stability of the results, especially for large systems and complex models. The algorithm is detailed in the paper, outlining how the parameter controlling the connectivity of replicas is tuned during the Monte Carlo simulation and how the contributions from each segment are combined to obtain the final result. The paper compares this new approach with a previously proposed nonequilibrium method. The comparison, demonstrated using a 2D antiferromagnetic Heisenberg model, shows the increment method's superior convergence and reduced error bars for the same computational cost.

The authors demonstrate the efficacy of their method using three representative examples of (2+1)d quantum many-body lattice models: 1. **Heisenberg antiferromagnet:** For this system with spontaneously broken continuous symmetry, the method accurately extracts the logarithmic correction coefficient (γ), consistent with theoretical predictions and previous results. Importantly, it achieves this using a standard finite-temperature simulation, avoiding the need for specialized techniques used in previous works. 2. **O(3) quantum critical point:** The J₁-J₂ Heisenberg model, realizing an O(3) quantum critical point, was studied to showcase the method's ability to handle models with broken translation symmetry. By carefully selecting the entangling region, the logarithmic correction coefficient was determined, again consistent with theoretical expectations and existing studies, but with significantly improved accuracy and larger system sizes. 3. **Topological ordered states:** The method's application to systems with topological order is demonstrated on two models: a Z₂ toric code toy model and the more complex Kagome Z₂ quantum spin liquid model. For both, the topological entanglement entropy (TEE) was accurately determined. This is a significant advancement, as previous QMC studies using the Levin-Wen prescription struggled to obtain the correct TEE value. The authors' method, particularly when applied to the Kagome model, overcomes the limitations of earlier work by reaching significantly larger system sizes, allowing the TEE to clearly emerge. The key here is that the algorithm naturally samples the minimum entropy states (MES), crucial for obtaining the correct TEE on the torus geometry.

The accurate extraction of entanglement entropy, particularly the universal logarithmic corrections and topological entanglement entropy, provides deep insights into quantum phases and critical phenomena. The nonequilibrium increment method introduced here addresses long-standing computational challenges, enabling the efficient and precise measurement of Rényi entanglement entropy in complex (2+1)d systems. The successful application of this method to diverse models—including those with spontaneous symmetry breaking, conformal field theory descriptions, and topological order—demonstrates its versatility and power. This opens up new avenues for studying a wide range of quantum many-body systems. The ability to reach larger system sizes with better precision than previous methods allows for more definitive identification of topological order and quantum critical behavior.

This paper presents a significant advancement in the computational tools available for studying entanglement entropy in quantum many-body systems. The nonequilibrium increment method offers a practical and efficient approach for obtaining accurate Rényi entanglement entropy in challenging (2+1)d systems. Future work could focus on applying this method to other intriguing systems, such as fermionic models, gauge field theories, and those exhibiting non-Fermi liquid behavior, furthering our understanding of these complex phases of matter. The improved computational efficiency should also allow exploration of more intricate entanglement measures and the study of dynamic aspects of entanglement.

While the nonequilibrium increment method significantly improves upon previous QMC approaches, some limitations remain. The accuracy of the TEE calculation depends on reaching sufficiently large system sizes to allow the universal contribution to emerge clearly. While the paper demonstrates success in several cases, this might not always be achievable for all systems or desired levels of precision. Moreover, the method's computational cost, although improved, still scales with system size, potentially limiting its application to extremely large systems. Further algorithmic optimizations may be possible to improve efficiency further.