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) = ∂φ(x) corresponding to a chiral free scalar φ 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. arXiv:1809.00026v2 [hep-th]

On entanglement Hamiltonians of an interval in massless harmonic chains

“…In [19] the sums (13) have been rewritten by decomposing the contribution coming from the i-th row of the matrices M and N , and this leads to (16)…”

“…The quadratic operators H M and H N can be decomposed in different ways, as discussed in §2 and in Appendix A. For the sake of simplicity, in the following we describe the continuum limit for the decomposition given by (14) and (15), but the procedure can be easily adapted to the ones given by (16) and (17) or by (18) and (19). In Appendix A we discuss another decomposition, inspired by the numerical analysis performed in [22].…”

“…Considering the decompositions (16) and (17), by adapting the procedure described in §3.2, we find the combinations given by…”

Correlation function of modular Hamiltonians

“…For a spherical region in a conformal field theory, an integral form could be found by conformal transformation of the Rindler wedge [15]. In two dimensional free field theory, a bilocal form of modular Hamiltonian can also be obtained for several disjoint intervals [16,17]. For a conformal field theory in a state which has a gravitational dual, a 1/G N expansion of modular Hamiltonian has also been proposed [18,19] from bulk,Ĥ bdy =Â ext 4G N +Ĥ bulk + · · · + o(G N ), (1.1) where the first term is an area operator of Ryu-Takayanagi [20] surface,Ĥ bulk and other higher order terms are bulk modular Hamiltonian of the corresponding bulk region.…”

Entanglement entropy of linearized gravitons in a sphere

“…There is however a subleading −1/2 log(log(R/ )) term in the mutual information for the l = 0 mode that is not present in the entropy (with the usual lattice regularization)[10]. This cames from superselection sectors for the d = 2 scalar[13,32] 4. For black hole backgrounds another contribution is expected proportional to the c anomaly coefficient 5.…”