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Towards entanglement entropy with UV cutoff in conformal nets

Yul Otani, Yoh Tanimoto

TL;DR

The paper develops a continuum, intrinsic definition of entanglement entropy for chiral components of conformal field theories within the algebraic QFT framework. By leveraging the split property and conformal nuclearity, it constructs a regularized entropy using a distance parameter δ and a conformal energy cutoff E tied to the discrete spectrum of the conformal Hamiltonian $L_0$, proving finiteness of the resulting quantity as δ→0. The main technical achievement is an explicit, δ-independent upper bound for the regularized entropy $H_I^E(oldsymbol{ extomega})$ that depends on the dimensions of $L_0$-eigenspaces and the energy cutoff, giving a principled way to quantify entanglement with a UV cutoff in the continuum. The work clarifies the relationship between nuclearity, the split property, and entanglement in AQFT, and discusses connections and contrasts with lattice results and potential extensions to higher dimensions and central charge considerations.

Abstract

We consider the entanglement entropy for a spacetime region and its spacelike complement in the framework of algebraic quantum field theory. For a Möbius covariant local net satisfying a certain nuclearity property, we consider the von Neumann entropy for type I factors between local algebras and introduce an entropic quantity. Then we implement a cutoff on this quantity with respect to the conformal Hamiltonian and show that it remains finite as the distance of two intervals tends to zero. We compare our definition to others in the literature.

Towards entanglement entropy with UV cutoff in conformal nets

TL;DR

The paper develops a continuum, intrinsic definition of entanglement entropy for chiral components of conformal field theories within the algebraic QFT framework. By leveraging the split property and conformal nuclearity, it constructs a regularized entropy using a distance parameter δ and a conformal energy cutoff E tied to the discrete spectrum of the conformal Hamiltonian , proving finiteness of the resulting quantity as δ→0. The main technical achievement is an explicit, δ-independent upper bound for the regularized entropy that depends on the dimensions of -eigenspaces and the energy cutoff, giving a principled way to quantify entanglement with a UV cutoff in the continuum. The work clarifies the relationship between nuclearity, the split property, and entanglement in AQFT, and discusses connections and contrasts with lattice results and potential extensions to higher dimensions and central charge considerations.

Abstract

We consider the entanglement entropy for a spacetime region and its spacelike complement in the framework of algebraic quantum field theory. For a Möbius covariant local net satisfying a certain nuclearity property, we consider the von Neumann entropy for type I factors between local algebras and introduce an entropic quantity. Then we implement a cutoff on this quantity with respect to the conformal Hamiltonian and show that it remains finite as the distance of two intervals tends to zero. We compare our definition to others in the literature.

Paper Structure

This paper contains 13 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Möbius covariant local nets
  4. Von Neumann entropy
  5. Nuclear maps
  6. Nuclearity conditions and the split property
  7. Towards cutoff entropy for a chiral net
  8. The energy function
  9. Regularization by distance
  10. The auxiliary functional $\theta_{\delta,+}$ and its entropy $S_1(\theta_{\delta,+})$
  11. Implementing cutoff
  12. Lifting the regularization by distance
  13. Conclusions and final remarks

Key Result

Proposition 2.1

For a parameter $p$ with $0<p<1$, there is a constant $c_p>0$ such that where the left-hand side equals $0$ at $t = 0$ by convention. Moreover, the optimal value is $c_p = \frac{1}{(1-p)e}$.

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 19 more