Entropy and modular Hamiltonian for a free chiral scalar in two intervals

Abstract

We calculate the analytic form of the vacuum modular Hamiltonian for a two interval region and the algebra of a current $j(x)=\partial \varphi (x)$ corresponding to a chiral free scalar $\varphi$ 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.

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