We present a method for improving measurements of the entanglement Rényi entropies in quantum Monte Carlo simulations by relating them with measurements of participation Rényi entropies. Exploiting the capability of building improved estimators for the latter allows to obtain very good estimates for entanglement Rényi entropies. When considering a full system instead of a bipartition, the method can be further ameliorated providing access to the thermodynamic Rényi entropies with high accuracy. We also explore a recently-proposed method for the reconstruction of the entanglement spectrum from entanglement Rényi entropies and finally show how potential entanglement Hamiltonians may be tested for their validity using a comparison with thermal Rényi entropies.

We explore the Rényi entanglement entropies of a one-dimensional (line) subsystem of length L embedded in two-dimensional L × L square lattice for quantum spin models whose ground-state breaks a continuous symmetry in the thermodynamic limit. Using quantum Monte Carlo simulations, we first study the J1 − J2 Heisenberg model with antiferromagnetic nearest-neighbor J1 > 0 and ferromagnetic second-neighbor couplings J2 ≤ 0. The signature of SU(2) symmetry breaking on finite size systems, ranging from L = 4 up to L = 40 clearly appears as a universal additive logarithmic correction to the Rényi entanglement entropies: lq ln L with lq 1, independent of the Rényi index and values of J2. We confirm this result using a high precision spin-wave analysis (with restored spin rotational symmetry) on finite lattices up to 10 5 × 10 5 sites, allowing to explore further non-universal finite size corrections and study in addition the case of U(1) symmetry breaking. Our results fully agree with the prediction lq = nG/2 where nG is the number of Goldstone modes, by Metlitski and Grover [arXiv:1112.5166].