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arXiv:2303.16935 · Journal of High Energy Physics

Irreversibility, QNEC, and defects

Horacio Casini, Ignacio Salazar Landea, Gonzalo Torroba

Published Jul 3, 2023• 35 citations
arXivHTMLPDF

Abstract

Abstract We first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we study the quantum null energy condition (QNEC) on the light-cone for a CFT. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions d ≤ 4. The F-theorem on defects (d = 3) is a new result using information-theoretic methods. For d ≥ 4 we also establish the monotonicity of the relative entropy coefficient proportional to Rd−4. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections.

DOI
10.1007/jhep07(2023)004

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Abstract We study conformal field theories with boundaries, and their boundary renormalization group (RG) flows, using methods from quantum information theory. Positivity of the relative entropy, together with unitarity and Lorentz invariance, give rise to bounds that characterize the irreversibility of such flows. This generalizes the recently proved entropic g-theorem to higher dimensions. In 2 + 1 dimensions with a boundary, we prove the entropic b-theorem — the decrease of the two-dimensional Weyl anomaly under boundary RG flows. In higher dimensions, the bound implies that the leading area coefficient of the entanglement entropy induced by the defect decreases along the flow. Our proof unifies these properties, and provides an information-theoretic interpretation in terms of the distinguishability between the short distance and long distance states. Finally, we establish a sum rule for the change in the area term in theories with boundaries, which could have implications for models with localized gravity.
We establish the irreversibility of renormalization group flows on a pointlike defect inserted in a d-dimensional Lorentzian conformal field theory. We identify the impurity entropy g with the quantum relative entropy in two equivalent ways. One involves a null deformation of the Cauchy surface, and the other is given in terms of a local quench protocol. Positivity and monotonicity of the relative entropy imply that g decreases monotonically along renormalization group flows, and provides a clear information-theoretic meaning for this irreversibility.
Energy conditions play an important role in constraining the dynamics of quantum field theories as well as gravitational theories. For example, in semi-classical gravity, the achronal averaged null energy condition (AANEC) can be used to prove that it is always slower to traverse through a wormhole than to travel around via its exterior. Such conditions prevent causality violations that would lead to paradoxes. Recent advances have been made in proving previously conjectured energy conditions directly in quantum field theory (QFT) as well as in uncovering new ones. This thesis will be an exploration of various energy inequalities in conformal field theories (CFTs) as well as semi-classical quantum gravity. At the core of this work lies the recently proved quantum null energy condition (QNEC). The QNEC bounds the null energy flowing past a point by a certain second shape derivative of entanglement entropy. We will demonstrate that the QNEC represents a deep connection between causality, energy and entanglement in quantum field theories. We explore this connection first in the context of holographic CFTs. Calculations in holographic theories will lead us to conjecture that the so-called diagonal QNEC is saturated in all interacting QFTs. We will then provide further, independent evidence that this conjecture holds for all CFTs with a twist gap by explicitly calculating shape derivatives of entanglement entropy using defect CFT techniques.
Abstract We explore a C-theorem in defect conformal field theories (DCFTs) that unify all the known conjectures and theorems until now. We examine as a candidate C-function the additional contributions from conformal defects to the sphere free energy and the entanglement entropy across a sphere in a number of examples including holographic models. We find the two quantities are equivalent, when suitably regularized, for codimension-one defects (or boundaries), but differ by a universal constant term otherwise. Moreover, we find in a few field theoretic examples that the sphere free energy decreases but the entanglement entropy increases along a certain renormalization group (RG) flow triggered by a defect localized perturbation which is assumed to have a trivial IR fixed point without defects. We hence propose a C-theorem in DCFTs stating that the increment of the regularized sphere free energy due to the defect does not increase under any defect RG flow. We also provide a proof of our proposal in several holographic models of defect RG flows.
The Quantum Null Energy Condition (QNEC) relates energy to the second variation of entropy in relativistic quantum field theory. We use the QNEC inequality to bound entanglement entropy in quenches. At early times the entanglement entropy grows quadratically in time, and the QNEC provides an upper bound on the prefactor. We demonstrate that the bound is tight, by showing that it is saturated in certain quench protocols: boundary state quenches in conformal field theories in any dimensions. In higher than two dimensions we compute entanglement entropy using AdS/CFT. Our results are the first purely field theoretic applications of the QNEC.