The reduced density matrix of a spatial subsystem can be written as the exponential of the modular Hamiltonian (a.k.a. entanglement Hamiltonian) and its eigenvalues provide the entanglement entropy. Hence, this operator contains a lot of information about the entanglement of the corresponding spatial bipartition. Within Algebraic Quantum Field Theory, the modular Hamiltonian and the modular conjugation are two essential tools of the Tomita-Takesaki modular theory. In the first part of this talk, it is shown how the modular conjugation in a 2D CFT at finite temperature provides the thermal entropy of a single interval of finite length. The gravitational dual interpretation of this result in terms of geodesic bit threads is also discussed, including its extension to higher dimensions for the case where the spatial subsystem is supported in a spherical region. In the second part of the talk, considering the harmonic chain in its ground state and in large mass regime, the entanglement Hamiltonian of two disjoint blocks is explored, showing the it displays geometric transitions as the the distance between two blocks of fixed lengths vary.
