Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Fermionic Entanglement Entropy of Causal Diamonds in Two-Dimensional Minkowski Space

Felix Finster, Magdalena Lottner, Albert Much, Simone Murro

TL;DR

The paper analyzes the fermionic Renyi entanglement entropy of a causal diamond in two-dimensional Minkowski space for a quasi-free Dirac field with ultraviolet regularization. It extends matrix-valued symbol techniques to truncated pseudo-differential operators to derive a logarithmically enhanced area law: the entropy scales as the logarithm of the inverse regularization parameter with a universal coefficient that depends only on the Renyi index. The analysis decomposes into physical preliminaries, operator-norm bounds, and a detailed spectral study of truncated operators, ultimately showing that off-diagonal matrix effects are negligible in the limit. This work provides a rigorous relativistic analogue of area-law behavior for entanglement in a simple spacetime geometry and yields explicit universal constants for the entropy growth.

Abstract

The fermionic Rényi entanglement entropy is studied for causal diamonds in two-dimensional Minkowski spacetime. Choosing the quasi-free state describing the Minkowski vacuum with an ultraviolet regularization, a logarithmically enhanced area law is derived.

The Fermionic Entanglement Entropy of Causal Diamonds in Two-Dimensional Minkowski Space

TL;DR

The paper analyzes the fermionic Renyi entanglement entropy of a causal diamond in two-dimensional Minkowski space for a quasi-free Dirac field with ultraviolet regularization. It extends matrix-valued symbol techniques to truncated pseudo-differential operators to derive a logarithmically enhanced area law: the entropy scales as the logarithm of the inverse regularization parameter with a universal coefficient that depends only on the Renyi index. The analysis decomposes into physical preliminaries, operator-norm bounds, and a detailed spectral study of truncated operators, ultimately showing that off-diagonal matrix effects are negligible in the limit. This work provides a rigorous relativistic analogue of area-law behavior for entanglement in a simple spacetime geometry and yields explicit universal constants for the entropy growth.

Abstract

The fermionic Rényi entanglement entropy is studied for causal diamonds in two-dimensional Minkowski spacetime. Choosing the quasi-free state describing the Minkowski vacuum with an ultraviolet regularization, a logarithmically enhanced area law is derived.
Paper Structure (12 sections, 8 theorems, 111 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Physical Preliminaries
  3. The Dirac Field in Two-Dimensional Minkowski Space
  4. The Quantized Dirac Field and its Vacuum State
  5. The Entanglement Entropy of a Causal Diamond
  6. Schatten-von-Neumann Bounds for Pseudo-Differential Operators
  7. Singular Values and Schatten-von Neumann Classes
  8. Non-Smooth Spectral Functions
  9. Spectral Analysis of Truncated Pseudo-Differential Operators
  10. An Area Law for the Causal Diamond
  11. The Diagonal Terms
  12. The Off-Diagonal Terms

Key Result

Theorem 1.1

(Entanglement entropy of a causal diamond) For any $\varkappa >0$, the Rényi entanglement entropy of a causal diamond in two-dimensional Minkowski space satisfies the relation In particular, for $\varkappa =1$, it holds

Figures (1)

  • Figure 1: A causal diamond.

Theorems & Definitions (13)

  • Theorem 1.1
  • Example 3.2
  • Proposition 3.3
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • Proposition 4.4
  • Lemma 5.1
  • proof
  • Proposition 5.2
  • ...and 3 more