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arXiv:1604.00388 · Journal of High Energy Physics

Dynamics of the area law of entanglement entropy

Stefan Leichenauer, Mudassir Moosa, Michael Smolkin

Published Sep 1, 2016• 16 citations
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Abstract

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.

DOI
10.1007/jhep09(2016)035

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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.
We explore entanglement entropy of a cap-like region for a generic quantum field theory residing in the Bunch-Davies vacuum on de Sitter space. Entanglement entropy in our setup is identical with the thermal entropy in the static patch of de Sitter, and we derive a simple relation between the vacuum expectation value of the energy-momentum tensor trace and the RG flow of entanglement entropy. In particular, renormalization of the bare couplings and logarithmic divergence of the entanglement entropy are interrelated in our setup. We confirm our findings by recovering known universal contributions for a free field theory deformed by a mass operator as well as obtain correct universal behaviour at the fixed points. Simple examples of entanglement entropy flows are elaborated in d=2,3,4. Inthreedimensionswefindthatwhiletherenormalizedentanglemententropy is stationary at the fixed points, it is not monotonic. We provide a computational evidence that the universal ‘area law’ for a conformally coupled scalar is different from the known result in the literature, and argue that this difference survives in the limit of flat space. Finally, we carry out the spectral decomposition of entanglement entropy flow and discuss its application to the F-theorem.
Understanding the dependence of entanglement entropy on the renormalized mass in quantum field theories can provide insight into phenomena such as quantum phase transitions, since the mass varies in a singular way near the transition. Here we perturbatively calculate the entanglement entropy in interacting scalar field theory, focusing on the dependence on the field’s mass. We study λϕ 4 and g ϕ 3 theories in their ground state. By tracing over a half space, using the replica trick and position space Green’s functions on the cone, we show that spacetime volume divergences cancel and renormalization can be consistently performed in this conical geometry. We establish finite contributions to the entanglement entropy up to two-loop order, involving a finite area law. The resulting entropy is simple and intuitive: the free theory result in d = 3 (that we included in an earlier publication) Δ S ∼ A m 2 ln ( m 2 ) is altered, to leading order, by replacing the bare mass m by the renormalized mass m r evaluated at the renormalization scale of zero momentum.
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 .
Subdominant contributions to the entanglement entropy of quantum fields include logarithmic corrections to the area law characterized by universal coefficients that are independent of the ultraviolet regulator and capture detailed information on the geometry around the entangling surface. We determine two universal coefficients of the entanglement entropy for a massive scalar field in a static closed universe $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{3}$ perturbatively and verify the results numerically. The first coefficient describes a well-known generic correction to the area law independent of the geometry of the entangling surface and background. The second coefficient describes a curvature-dependent universal term with a nontrivial dependence on the intrinsic and extrinsic geometries of the entangling surface and curvature of the background. The numerical calculations confirm the analytical results to a high accuracy. The first and second universal coefficients are determined numerically with a relative error with respect to the analytical values of the orders ${10}^{\ensuremath{-}4}$ and ${10}^{\ensuremath{-}2}$, respectively.
The time evolution of entanglement tracks how information propagates in interacting quantum systems. We study entanglement entropy in CFT2{\mathrm{CFT}}_{2}CFT2​ with a time-dependent Hamiltonian. We perturb by operators with time-dependent source functions and use the replica trick to calculate higher-order corrections to entanglement entropy. At first order, we compute the correction due to a metric perturbation in AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}AdS3​/CFT2​ and find agreement on both sides of the duality. Past first order, we find evidence of a universal structure of entanglement propagation to all orders. The central feature is that interactions entangle unentangled excitations. Entanglement propagates according to ``entanglement diagrams,'' proposed structures that are motivated by accessory spacetime diagrams for real-time perturbation theory. To illustrate the mechanisms involved, we compute higher-order corrections to free fermion entanglement entropy. We identify an unentangled operator, one which does not change the entanglement entropy to any order. Then, we introduce an interaction and find it changes entanglement entropy by entangling the unentangled excitations. The entanglement propagates in line with our conjecture. We compute several entanglement diagrams. We provide tools to simplify the computation of loop entanglement diagrams, which probe UV effects in entanglement propagation in CFT and holography.
Entanglement entropy satisfies a first law-like relation, which equates the first order perturbation of the entanglement entropy for the region A A to the first order perturbation of the expectation value of the modular Hamiltonian, \delta S_{A}=\delta \langle K_A \rangle δ S A = δ ⟨ K A ⟩ . We propose that this relation has a finer version which states that, the first order perturbation of the entanglement contour equals to the first order perturbation of the contour of the modular Hamiltonian, i.e. \delta s_{A}(\textbf{x})=\delta \langle k_{A}(\textbf{x})\rangle δ s A ( 𝐱 ) = δ ⟨ k A ( 𝐱 ) ⟩ . Here the contour functions s_{A}(\textbf{x}) s A ( 𝐱 ) and k_{A}(\textbf{x}) k A ( 𝐱 ) capture the contribution from the degrees of freedom at \textbf{x} 𝐱 to S_{A} S A and K_A K A respectively. In some simple cases k_{A}(\textbf{x}) k A ( 𝐱 ) is determined by the stress tensor. We also evaluate the quantum correction to the entanglement contour using the fine structure of the entanglement wedge and the additive linear combination (ALC) proposal for partial entanglement entropy (PEE) respectively. The fine structure picture shows that, the quantum correction to the boundary PEE can be identified as a bulk PEE of certain bulk region. While the shows that the quantum correction to the boundary PEE comes from the linear combination of bulk entanglement entropy. We focus on holographic theories with local modular Hamiltonian and configurations of quantum field theories where the applies.