  The modular (or entanglement) Hamiltonian correspondent to the
half-space-bipartition of a quantum state uniquely characterizes its
entanglement properties. However, in the context of lattice models, its
explicit form is analytically known only for the Ising chain and certain free
theories in one-dimension. In this work, we provide a throughout investigation
of entanglement Hamiltonians in lattice models obtained via the
Bisognano-Wichmann theorem, which provides an explicit functional form for the
entanglement Hamiltonian itself in quantum field theory. Our study encompasses
a variety of one- and two-dimensional models, supporting diverse quantum phases
and critical points, and, most importantly, scanning several universality
classes, including Ising, Potts, and Luttinger liquids. We carry out extensive
numerical simulations based on the density-matrix-renormalization-group method,
exact diagonalization, and quantum Monte Carlo. In particular, we compare the
exact entanglement properties and correlation functions to those obtained
applying the Bisognano-Wichmann theorem on the lattice. We carry out this
comparison on both the eigenvalues and eigenvectors of the entanglement
Hamiltonian, and expectation values of correlation functions and order
parameters. Our results evidence that, as long as the low-energy description of
the lattice model is well-captured by a Lorentz-invariant quantum field theory,
the Bisognano-Wichmann theorem provides a qualitatively and quantitatively
accurate description of the lattice entanglement Hamiltonian. The resulting
framework paves the way to direct studies of entanglement properties utilizing
well-established statistical mechanics methods and experiments.
