Entanglement Hamiltonian of quantum critical chains and conformal field theories

Abstract

We consider a lattice version of the Bisognano-Wichmann (BW) modular Hamiltonian as an ansatz for the bipartite entanglement Hamiltonian of the quantum critical chains. Using numerically unbiased methods, we check the accuracy of the BW-ansatz by both comparing the BW Rényi entropy to the exact results, and by investigating the size scaling of the norm distance between the exact reduced density matrix and the BW one. Our study encompasses a variety of models, scanning different universality classes, including transverse field Ising, Potts and XXZ chains. We show that the Rényi entropies obtained via the BW ansatz properly describe the scaling properties predicted by conformal field theory. Remarkably, the BW Rényi entropies faithfully capture also the corrections to the conformal field theory scaling associated to the energy density operator. In addition, we show that the norm distance between the discretized BW density matrix and the exact one asymptotically goes to zero with the system size: this indicates that the BW-ansatz can be also employed to predict properties of the eigenvectors of the reduced density matrices, and is thus potentially applicable to other entanglement-related quantities such as negativity.