Measuring Renyi Entanglement Entropy in Quantum Monte Carlo Simulations

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 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 Neel ground state obeys the expected area law for systems up to linear size L = 32.