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arXiv:1406.2716 · Physical Review Letters

Entanglement Entropy Flow and the Ward Identity

Vladimir Rosenhaus, Michael Smolkin

Published Dec 24, 2014• 30 citations
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Abstract

We derive differential equations for the flow of entanglement entropy as a function of the metric and the couplings of the theory. The variation of the universal part of entanglement entropy under a local Weyl transformation is related to the variation under a local change in the couplings. We show that this relation is, in fact, equivalent to the trace Ward identity. As a concrete application of our formalism, we express the entanglement entropy for massive free fields as a two-point function of the energy-momentum tensor.

DOI
10.1103/physrevlett.113.261602

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We derive differential equations for the flow of entanglement entropy as a function of the metric and the couplings of the theory. The variation of the universal part of entanglement entropy under a local Weyl transformation is related to the variation under a local change in the couplings. We show that this relation is, in fact, equivalent to the trace Ward identity. As a concrete application of our formalism, we express the entanglement entropy for massive free fields as a two-point function of the energy-momentum tensor.
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We study the evolution of the universal area law of entanglement entropy when the Hamiltonian of the system undergoes a time dependent perturbation. In particular, we derive a general formula for the time dependent first order correction to the area law under the assumption that the field theory resides in a vacuum state when a small time-dependent perturbation of a relevant coupling constant is turned on. Using this formula, we carry out explicit calculations in free field theories deformed by a time dependent mass, whereas for a generic QFT we show that the time dependent first order correction is governed by the spectral function defining the two-point correlation function of the trace of the energy-momentum tensor. We also carry out holographic calculations based on the HRT proposal and find qualitative and, in certain cases, quantitative agreement with the field theory calculations.
In this note we do the analysis of entanglement entropy more carefully when the non-conformal theory flows to a non-trivial IR fixed point. In particular we emphasize the role of the trace of the energy-momentum tensor in these calculations. We also compare the current technique for evaluating the entanglement entropy, particularly the Green's function method for gaussian theories, with the dilaton effective action approach and show that they compute identical quantities. As a result of this, the dilaton effective action approach can be thought of as an extension of Green's function technique to interacting theories.
For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv:1509.00113 .
It was shown recently that boundary terms of conformal anomalies recover the universal contribution to the entanglement entropy and also play an important role in the boundary monotonicity theorem of odd-dimensional quantum field theories. Motivated by these results, we investigate relationships between boundary anomalies and the stress tensor correlation functions in conformal field theories. In particular, we focus on how the conformal Ward identity and the renormalization group equation are modified by boundary central charges. Renormalized stress tensors induced by boundary Weyl invariants are also discussed, with examples in spherical and cylindrical geometries.