Measuring Renyi Entanglement Entropy with Quantum Monte Carlo Matthew B. Hastings, 1 Iv´ an Gonz´ alez, 2 Ann B. Kallin, 3 and Roger G. Melko 3 $^{1}$Microsoft Research, Station Q, CNSI Building, University of California, Santa Barbara, CA, 93106 $^{2}$Centro de Supercomputaci´ on de Galicia, Avda. de Vigo s/n, E-15705 Santiago de Compostela, Spain $^{3}$Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada (Dated: October 22, 2018) 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 Swap operator acting on two copies of the system. An improved estimator involving the ratio of Swap operators for different subregions enables convergence of the entropy in a simulation time polyno- mial 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 = 32. The measurement of entanglement entropy provides new tools to determine universal properties of interact- ing quantum many-body systems in condensed-matter physics. For example, conformally invariant systems in one dimension (1D) display universal logarithmic scaling of entanglement entropy [1]. Many quantum systems in two dimensions (2D) and higher are predicted to have “area law” scaling of entanglement entropy, with either universal additive logarithmic corrections for critical sys- tems [2, 3] or universal additive constant corrections for topologically ordered systems [4, 5]. Scalable simulation methods have become crucial for the study of groundstate, finite-temperature, and criti- cal properties of quantum many-body systems. Unfortu- nately, there are no known scalable simulation methods to calculate entanglement entropy in D > 1. Calculating the von Neumann entropy ( S$_{1}$ , defined below) requires calculating the reduced density matrix, which can be done in Lanczos or density matrix renormalization group (DMRG). However, scalable simulation techniques in di- mensions D > 1, are so far essentially restricted to quan- tum Monte Carlo (QMC), which does not provide access to the reduced density matrix, but only approximates it via a Metropolis importance-sampling algorithm. We resolve this problem and show that a generalized Renyi entropy ( S$_{2}$ , defined below) can be related to the expectation value of a unitary Swap operator in the va- lence bond basis, directly accessible in QMC. Using QMC simulations, we calculate S$_{2}$ for a spin 1 / 2 Heisenberg model on 1D and 2D lattices. In 1D, we demonstrate that S$_{2}$ calculated with QMC converges to the exact re- sult obtained with DMRG simulations. In 2D, we show that the leading-order term in the scaling of S$_{2}$ goes like the boundary of a subregion A , confirming the area law expected from recent studies [6]. The generalized Renyi entanglement entropies are de- fined by where ρ$_{A}$ is the reduced-density matrix of a subregion A entangled with its complement B , and n > 0. In the limit n → 1, one recovers the familiar von Neumann en- tropy, S$_{1}$ ( ρ$_{A}$ ) = − Tr( ρ$_{A}$ ln ρ$_{A}$ ). Recently, the generalized Renyi entropies have attracted considerable attention in the condensed matter community, due to their ability to encode information about the whole “entanglement spec- trum” of ρ$_{A}$ , together allowing the set of S$_{n}$ to contain much more information than S$_{1}$ alone [7]. For example, the concept of topological entanglement entropy has re- cently been generalized to the family of Renyi entropies, where it has been shown to be equal to the logarithm of the total quantum dimension, S$_{top}$ ∝ log D , independent of n [8]. Universal corrections to the scaling of S$_{n}$ have also been calculated in a field-theoretic treatment of the O ( N ) model [9]. Further, any two Renyi entropies S$_{n}$ and S$_{m}$ obey the inequality S$_{n}$ ≥ S$_{m}$ for n < m , making S$_{2}$ a useful lower bound on S$_{1}$ . Valence bond basis and the Swap operator. – We briefly review the notation of Sandvik, and refer the reader to Refs. [10–12] for additional details on the valence bond FIG. 1: (color online) A six-site chain, with two non- interacting copies (top and bottom) before (a) and after (b) the Swap A operation. The region A consists of three light- colored sites on the left; the complement region B of three dark sites on the right. The curved lines denote singlets in the state | V r 〉 , which is a product of two different valence bond states, one per copy. The ground state of the entire system is a linear combination of similar | V r 〉 . S$_{n}$ ( ρ$_{A}$ ) = 1 1 − n ln [ Tr ( ρ n A )] , (1) arXiv:1001.2335v2 [cond-mat.str-el] 3 Mar 2010 (a) (b) | V$_{r}$ 〉 Swap$_{A}$ | V$_{r}$ 〉