Entanglement Hamiltonian for the massless Dirac field on a segment with an inhomogeneous background

We study the entanglement Hamiltonian of an interval for the free massless Dirac field in an inhomogeneous background on a finite segment and in the ground state. We consider a class of metrics that are Weyl equivalent to the flat metric through a Weyl factor that depends only on the spatial coordinate, with the same boundary condition imposed at both endpoints of the segment. The explicit form of the entanglement Hamiltonian is written as the sum of a local and a bilocal term. The weight function of the local term allows us to study a contour function for the entanglement entropies. For the model obtained from the continuum limit of the rainbow chain, the analytic expressions are compared with exact numerical results from the lattice, showing an excellent agreement.