Measuring Renyi Entanglement Entropy with Quantum Monte Carlo

Abstract: We develop a quantum Monte Carlo procedure, in the valence bond basis, to measure the Renyi entanglement entropy of a many-body ground state as the expectation value of a unitary {\it Swap} operator acting on two copies of the system. An improved estimator involving the ratio of {\it Swap} operators for different subregions enables simulations to converge the entropy in a time polynomial in the system size. We demonstrate convergence of the Renyi entropy to exact results for a Heisenberg chain. Finally, we calculate the scaling of the Renyi entropy in the two-dimensional Heisenberg model and confirm that the N\'eel groundstate obeys the expected area law for systems up to linear size L=28.

• The paper introduces a novel Quantum Monte Carlo algorithm designed to accurately measure the Renyi entanglement entropy, specifically S2, in quantum many-body systems in 1D and 2D.
• The method utilizes an innovative estimator involving a unitary Swap operator within the valence bond basis, providing a scalable framework previously limited in higher dimensions.
• Results validate the QMC approach against established methods in 1D and successfully demonstrate the area law for entanglement entropy in 2D systems, supporting broader applicability.

Measuring Renyi Entanglement Entropy with Quantum Monte Carlo

The paper presents a robust quantum Monte Carlo (QMC) algorithm for calculating the Renyi entanglement entropy, specifically $S_2$, in quantum many-body systems. This work addresses the challenge of measuring entanglement entropy in higher dimensional systems beyond 1D, where scalable simulation methods were previously limited. By leveraging the valence bond basis and introducing an innovative estimator involving a unitary Swap operator, the authors offer a scalable framework applicable to both 1D and 2D systems.

Summary of Methodology and Results

The primary advancement in this paper is the development of a QMC procedure to measure the Renyi entanglement entropy $S_2$. The entropies are calculated as the expectation value of a Swap operator acting on two non-interacting replicas of the system. The approach centers on calculating $S_2$ as $S_2(\rho_A) = -\ln(\langle Swap_A \rangle)$, where $\rho_A$ is the reduced density matrix, and $\langle Swap_A \rangle$ is accessible through the QMC simulation in the valence bond basis.

The authors showcase the potential of this method by applying it to the spin-1/2 Heisenberg model on both 1D and 2D lattices. In 1D, the $S_2$ measured using QMC converges with high fidelity to results previously obtained through density matrix renormalization group (DMRG) simulations. In 2D, the findings corroborate the area law for entanglement entropy by showing that the Renyi entropy scales with the boundary of a subregion $A$, rather than its volume, upholding theoretical expectations from prior studies.

Implications and Challenges

A significant contribution of this research is the demonstration of an algorithm that scales polynomially with system size, suggesting potential applicability to large quantum systems, including potentially those beyond 2D. The methodological innovation of using a Swap operator in conjunction with an improved ratio estimator mitigates the challenge of exponentially small expectation values, which traditionally complicate statistical convergence in QMC simulations.

The work raises questions about future applications, particularly regarding scalability to larger system sizes and other quantum models. The authors speculate that enhancing the region sequences used for the improved ratio estimator could address scalability constraints, making it feasible to tackle even larger quantum systems. Additionally, they suggest potential future integration with loop algorithms to further reduce computational complexity.

Conclusion

The presented QMC method for measuring Renyi entanglement entropy is a noteworthy progression in computational condensed matter physics, offering a practical tool for exploring entanglement properties in quantum many-body systems. The authors present a clear pathway toward addressing the computational challenges inherent in higher-dimensional systems, supporting ongoing research into quantum criticality and topological phases. This approach sets the stage for broader applications in the study of quantum entanglement and is likely to stimulate further advancements in numerical quantum simulations.