  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 R\'enyi 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 R\'enyi entropies obtained via
the BW ansatz properly describe the scaling properties predicted by conformal
field theory. Remarkably, the BW R\'enyi 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.
