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arXiv:2412.13777

Reduced density matrix and entanglement Hamiltonian for a free real scalar field on an interval

Baranov, Mikhail A.

Published Dec 18, 2024• 1 citation
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Abstract

An exact result for the reduced density matrix on a finite interval for a $1+1$ 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 $M$ of the field, find the leading $\sim1/\ln M$ non-local corrections. In the opposite $M\to\infty$ case, it is argued that, up to terms $O(M^{-1/2})$, the entanglement Hamiltonian is local with the density being the Hamiltonian density spatially modulated by a triangular-shape function.

DOI
10.48550/arxiv.2412.13777

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Abstract We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular Hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement Hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O (1) terms in the entanglement entropy.
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