  We calculate the analytic form of the vacuum modular Hamiltonian for a two
interval region and the algebra of a current $j(x)=\partial \phi(x)$
corresponding to a chiral free scalar $\phi$ in $d=2$. We also compute
explicitly the mutual information between the intervals. This model shows a
failure of Haag duality for two intervals that translates into a loss of a
symmetry property for the mutual information usually associated with modular
invariance. Contrary to the case of a free massless fermion, the modular
Hamiltonian turns out to be completely non local. The calculation is done
diagonalizing the density matrix by computing the eigensystem of a correlator
kernel operator. These eigenvectors are obtained by a novel method that
involves solving an equivalent problem for an holomorphic function in the
complex plane where multiplicative boundary conditions are imposed on the
intervals. Using the same technique we also re-derive the free fermion modular
Hamiltonian in a more transparent way.
