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arXiv:0903.5284 · Classical and Quantum Gravity

Reduced density matrix and internal dynamics for multicomponent regions

Horacio Casini, Marina Huerta

Published Sep 7, 2009• 188 citations
arXivHTMLPDF

Abstract

We find the density matrix corresponding to the vacuum state of a massless Dirac field in two dimensions reduced to a region of the space formed by several disjoint intervals. We calculate explicitly its spectral decomposition. The imaginary power of the density matrix is a unitary operator implementing an internal time flow (the modular flow). We show that in the case of more than one interval, this evolution is non-local, producing both advance in the causal structure and 'teleportation' between the disjoint intervals. However, it only mixes the fields on a finite number of trajectories, one for each interval. As an application of these results we compute the entanglement entropy for the massive multi-interval case in the small mass limit.

DOI
10.1088/0264-9381/26/18/185005

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We find the density matrix corresponding to the vacuum state of a massless Dirac field in two dimensions reduced to a region of the space formed by several disjoint intervals. We calculate explicitly its spectral decomposition. The imaginary power of the density matrix is a unitary operator implementing an internal time flow (the modular flow). We show that in the case of more than one interval, this evolution is non-local, producing both advance in the causal structure and 'teleportation' between the disjoint intervals. However, it only mixes the fields on a finite number of trajectories, one for each interval. As an application of these results we compute the entanglement entropy for the massive multi-interval case in the small mass limit.
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An exact result for the reduced density matrix on a finite interval for a 1+11+11+1 dimensional free real scalar field in the ground state is presented. In the massless case, the Williamson decomposition of the appearing kernels is explicitly performed, which allows to reproduce the known result for the entanglement (modular) Hamiltonian and, for a small mass MMM of the field, find the leading ∼1/ln⁡M\sim1/\ln M∼1/lnM non-local corrections. In the opposite M→∞M\to\inftyM→∞ case, it is argued that, up to terms O(M−1/2)O(M^{-1/2})O(M−1/2), the entanglement Hamiltonian is local with the density being the Hamiltonian density spatially modulated by a triangular-shape function.
A bstract We study the massless Dirac field on the line in the presence of a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. Considering this system in its ground state, we derive the modular Hamiltonians of the subregion given by the union of two disjoint equal intervals at the same distance from the defect. The absence of energy dissipation at the defect implies the existence of two phases, where either the vector or the axial symmetry is preserved. Besides a local term, the densities of the modular Hamiltonians contain also a sum of scattering dependent bi-local terms, which involve two conjugate points generated by the reflection and the transmission. The modular flows of each component of the Dirac field mix the trajectory passing through a given initial point with the ones passing through its reflected and transmitted conjugate points. We derive the two-point correlation functions along the modular flows in both phases and show that they satisfy the Kubo-Martin-Schwinger condition. The entanglement entropies are also computed, finding that they do not depend on the scattering matrix.
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We revisit the calculation of multi-interval modular Hamiltonians for free fermions using a Euclidean path integral approach. We show how the multi-interval modular flow is obtained by gluing together the single interval modular flows. Using this relation, we obtain an exact expression for the multi-interval modular Hamiltonian and entanglement entropy in agreement with existing results. An essential ingredient in our derivation is the introduction of the modular action. This determines the non-local field theory describing the free fermion reduced density matrix, and makes manifest it’s non-local conformal symmetry and U(1) Kacs-Moody symmetry.