Entanglement Entropy in Algebraic Quantum Field Theory

Yul Otani

Published Mar 23, 2017• 2 citations

Abstract

We consider the problem of defining the entanglement entropy for chiral nets in the framework of algebraic quantum field theory.Considering a Möbius covariant local net with the split property, we give a sensible definition for the entropy H E I of a state restricted to a local algebra of an open connected non-dense interval I ⊂ S 1 , with a given "conformal energy cutoff E".Considering the vacuum state restricted to any such interval, we prove that the latter is finite, and we give some upper bound estimates in terms of the dimensions of eigenspaces of the conformal Hamiltonian.This thesis is based on a joint work with Yoh Tanimoto.

DOI: 10.15083/00076078

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Entanglement Entropy in Algebraic Quantum Field Theory

Yul Otani

Published Mar 23, 2017• 2 citations

Publisher

Abstract

DOI: 10.15083/00076078

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In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated with any pair of causally disjoint spacetime regions

\mathcal{S}_A

and

\mathcal{S}_B

with positive relative distance. Among them, the canonical entanglement entropy is defined as the von Neumann entropy of a canonical intermediate type I factor. In this work, we show that the canonical entanglement entropy of the vacuum state is finite for a broad class of conformal nets including the

U(1)

-current model and the

SU(n)

-loop group models. Since previous studies suggest that this finiteness property is related to nuclearity properties of the system, we show that the mutual information is finite in any local QFT satisfying a modular

p

-nuclearity condition for some

0 < p < 1

. A similar finiteness result is established for another notion of entanglement entropy introduced in this paper. We conclude with remarks for future work in this direction.

Abstract Given a QFT net

{{\mathcal {A}}}

A of local von Neumann algebras

{{\mathcal {A}}}(O)

A ( O ) , we consider the von Neumann entropy

S_{{\mathcal {A}}}(O, {\widetilde{O}})

S A ( O , O ~ ) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras

{{\mathcal {A}}}(O)\subset {{\mathcal {A}}}({\widetilde{O}})

A ( O ) ⊂ A ( O ~ ) (split property). This canonical entanglement entropy

S_{{\mathcal {A}}}(O, {\widetilde{O}})

S A ( O , O ~ ) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). The finiteness property is derived by an explicit formula of entropy and an observation that the operators in the definition are closely related to Hankel operators. In this paper we give further analysis of this entropy using a variety of techniques that have been developed in different context, and in particular we show that there is an upper bound given by a positive constant multiply by

|\ln \eta |

| ln η | , where

\eta

η is the cross ratio of the underlying system, when

\eta \rightarrow 0

η → 0 .

We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on

{\mathrm{\mathbb{R}}}^{1,d-1}

. We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C T appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient

\frac{\sigma }{C_T}=\frac{\pi^2}{24}

in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.

h

. The excitation entanglement entropy, unlike the entanglement entropy, is a finite quantity (independent of the cutoff), and hence a good physical observable. We show that the excitation entanglement entropy for any given interval is uniquely specified by a local second order differential equation sourced by the one point function of the energy momentum tensor computed in the excited background state, and two boundary and smoothness conditions. We analyze low and high temperature behavior of the excitation entanglement entropy and show that ${\mathrm{\ensuremath{\Delta}}}_{h}S$ grows as a function of temperature. We prove an ``integrated positivity'' for the excitation entanglement entropy, that although ${\mathrm{\ensuremath{\Delta}}}_{h}S$ can be positive or negative, its average value is always positive. We also discuss the mutual and multipartite information and (strong) subadditivity inequality in the presence of generic excitations and their conformal descendants.