Entanglement Hamiltonian of two disjoint blocks in the harmonic chain

We study the entanglement Hamiltonian of two disjoint blocks in the harmonic chain on the line and in its ground state. In the regime of large mass, the only non-vanishing terms are the on-site and nearest-neighbour ones. Analytic expressions are obtained for their profiles, which are written in terms of piecewise linear functions that can be discontinuous and display sharp transitions as the separation between the blocks changes. In the regime of vanishing mass, where the matrices characterizing the entanglement Hamiltonian contain couplings at all distances, we explore the location of the subdominant terms and some combinations of matrix elements that are useful for the continuum limit, comparing the results with the corresponding ones for a free chiral current. The single-particle entanglement spectrum is also investigated.