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Communications in Mathematical Physics

Asymptotic Analysis of von Neumann Entropy in Conformal Field Theory

Xiaomin Tang, Feng Xu

Published Apr 16, 2022• 1 citation
Publisher

Abstract

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 .

DOI
10.1007/s00220-022-04393-y

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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 SA\mathcal{S}_ASA​ and SB\mathcal{S}_BSB​ 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)U(1)U(1)-current model and the SU(n)SU(n)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 ppp-nuclearity condition for some 0<p<10 < p < 10<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.
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.
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Entanglement entropy in type II1_11​ von Neumann algebra: examples in Double-Scaled SYK
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An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III _{1} 1 , in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II _∞ ∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.
This paper provides an in-depth analysis of entanglement entropy (EE) in quantum field theory (QFT), with a particular focus on its computation using the replica trick and its applications to both conformal and non-conformal systems. Beginning with an introduction to the basics of QFT, the study explains how entanglement entropy quantifies the quantum correlations between subsystems in a pure state, represented by the von Neumann entropy of the reduced density matrix. The replica trick is employed to derive the entanglement entropy, involving path integrals over n-sheeted Riemann surfaces. The paper demonstrates the method’s utility in 1+1-dimensional conformal field theory (CFT), where the central charge governs universal properties of entropy. It further extends the analysis to massive field theories, finite systems, and topological phases, exploring how deviations from conformal symmetry affect entanglement. The work also delves into the AdS/CFT correspondence, showcasing how holographic techniques facilitate entanglement entropy calculations in higher-dimensional systems via the Ryu-Takayanagi formula. In addition, the study investigates entanglement entropy in quantum lattice systems, considering the effects of spatial and thermal fluctuations. Numerical methods are used to compute the scaling of entanglement entropy with interval length for varying system masses and temperatures, highlighting the influence of these parameters on entropy behavior. Fractal lattice structures are explored to uncover unique entropy scaling laws and self-similar entanglement patterns. This study advances the under-standing of EE in diverse quantum systems and establishes a foundation for exploring its role in quantum criticality and fractal geometries. The paper also explores multiscale entanglement entropy (MSE), offering a comprehensive framework for future research in quantum critical phenomena and non-equilibrium systems, shedding light on the complex relationship between quantum entanglement and thermal effects. This work serves as a foundational reference for future theoretical and computational studies in quantum entanglement