0 10 20 30 0.3 0.4 0.5 0.6 0.7 L=32 L=28 L=24 L=20 L=16 L=12 L=8 L=6 L=4 ℓ S 2 /ℓ regions A ℓ cons is ting of ℓ -by- ℓ squares, we can use a more finely grained se quence of regions; e.g. by defining A i to contain i sites s uch that for i = ℓ $^{2}$, A i is an ℓ -by- ℓ square. Incrementing th e number of sites in i by one would result in (at worst) O ( Nm $^{2}$) scaling in the current algorithm. However, we als o note that recently-developed loop algo- rithms result in significant improvement in scaling, allow- ing for the conve rgence of observables such as energy and spin correlation functions for L = 256 and larger [12]. In these schemes, the scaling of the direct swap operator (Eq. (9)) would decrease to O ( m ), while the expected scaling of the im proved ratio estimator would be O ( Nm ) at worst, depen ding on the definitions of the A i regions. We therefore ex pect that the algorithm will require only a polynomial n umber of samples to converge the Renyi entropy on arbit rary system sizes. 4 Acknowledgm ents. – The authors thank A. Sandvik and A. J. Berlinky for useful discussions. This work was made possible by the computing facilities of SHARCNET and CESGA. Support was provided by NSERC of Canada (A.B.K. and R.G.M.). FIG. 4: (color online) Scaling of the Renyi entropy divided by the linear dimension ℓ of the entangled region A for different systems sizes in 2D. All data is calculated with the improved ratio estimator, Eq. (10), with r max = 1. Ref. [6] one expects the scaling of S$_{1}$ in the N´eel state to obey the area law; also recall that S$_{2}$ ≤ S$_{1}$ . The data illustrated is clearly consistent with the area law S$_{2}$/ℓ ∼ const . for ℓ ≪ L , at which point boundary effects likely become important. In particular, multiplicative log corrections (which were apparent in similar system sizes and geometries for the Valence-Bond entanglement entropy [14, 15]) are not present. Discussion. – In this paper we have presented an al- gorithm for measuring the Renyi entropy S$_{2}$ in valence bond basis QMC simulations via the expectation value of a swap operator between two copies of the system. Us- ing an improved ratio estimator, we are able to converge the expectation value of S$_{2}$ to the exact result on a 1D Heisenberg chain. Using the same procedure, we have presented the first measurement of the Renyi entangle- ment entropy in a 2D system, confirming the area law for the N´eel groundstate of the Heisenberg model. The simple double-projector QMC algorithm used in this paper is known to scale approximately as O ( m $^{2}$), [12] assuming m > N . Thus, the current simulation re- sults for the direct swap expectation value, Eq. (9), also have O ( m $^{2}$) scaling, while results obtained using the im- proved ratio estimator, Eq. (10), have O ( Lm $^{2}$) scaling in the current geometries. We note however that these geometries may not be ideal for very large ℓ and L , even in the improved ratio sampling. For example, for L = 32, results for r$_{max}$ = 2 and r$_{max}$ = 1, although remaining converged within error bars, begin to develop larger dis- crepancies when ℓ approaches L . From the current work (which used about 10 CPU- years), the question naturally arises whether it is possible to converge the expectation value of the swap operator on larger system sizes. We expect that this will be possible using a different sequence of regions A i in the improved ratio estimator. 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