In the first part of this talk we will present a path integral derivation of a general relation between the ground state entanglement Hamiltonian and the physical stress tensor for a Conformal Field Theory (CFT). For spherical entangling surfaces in a CFT, this leads to first law-like relation between variations of entanglement entropy (EE) and energy as well as a set of constraint equations for the EE variation.

Via AdS/CFT, these equations can be recast as Perturbative Einstein's Equations in the bulk dual.

In the second part, we will present results on the entanglement Hamiltonian (EH) of chiral fermions living on a spatial circle. In particular we focus on the effects of periodic vs. anti periodic boundary conditions on the EH. We will relate the calculation of the fermion EH to the solution of a Riemann Hilbert Problem, and propose a generalization of Riemann Hilbert Problem for spinor bundles in higher dimensions.