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Imaginary part of timelike entanglement entropy

Wu-zhong Guo, Jin Xu

TL;DR

The paper investigates the imaginary part of timelike entanglement entropy (EE) in quantum field theory by exploiting the twist-operator framework and analytic continuation from Euclidean to Lorentzian signature.A central result is the universal first-order temporal-derivative commutator of twist operators, which yields a universal imaginary contribution proportional to the central charge in several states (vacuum, thermal, AdS3 duals), while spatial derivatives do not introduce new imaginary parts.For general states, the imaginary part receives state-dependent corrections from operators with fractional conformal dimensions, necessitating higher-order temporal derivatives and a refined formula linking the timelike EE to spacelike EE and its derivatives; these ideas extend to higher dimensions with strip geometries, where imaginary parts vanish in even dimensions and survive in odd ones.The work also connects the imaginary part to the spectra of the reduced density matrix, offering a spectral perspective on timelike EE and outlining potential holographic and entanglement-spectra implications for future research.

Abstract

In this paper, we explore the imaginary part of the timelike entanglement entropy. In the context of field theory, it is more appropriate to obtain the timelike entanglement entropy through the Wick rotation of the twist operators. It is found that, in certain special cases, the imaginary part of the timelike entanglement entropy is related to the commutator of the twist operator and its first-order temporal derivative. To evaluate these commutators, we employ the operator product expansion of the twist operators, revealing that the commutator is generally universal across most scenarios. However, in more general cases, the imaginary part of the timelike entanglement entropy proves to be more complex. We compute the commutator of the twist operators along with its higher-order temporal derivatives. Utilizing these results, we derive a modified formula for the imaginary part of the timelike entanglement entropy. Furthermore, we extend this formula to the case of strip subregion in higher dimensions. Our analysis shows that for the strip geometry, the imaginary part of the timelike entanglement entropy is solely related to the commutators of the twist operator and its first-order temporal derivative. The findings presented in this paper provide valuable insights into the imaginary part of timelike entanglement entropy and its physical significance.

Imaginary part of timelike entanglement entropy

TL;DR

The paper investigates the imaginary part of timelike entanglement entropy (EE) in quantum field theory by exploiting the twist-operator framework and analytic continuation from Euclidean to Lorentzian signature.A central result is the universal first-order temporal-derivative commutator of twist operators, which yields a universal imaginary contribution proportional to the central charge in several states (vacuum, thermal, AdS3 duals), while spatial derivatives do not introduce new imaginary parts.For general states, the imaginary part receives state-dependent corrections from operators with fractional conformal dimensions, necessitating higher-order temporal derivatives and a refined formula linking the timelike EE to spacelike EE and its derivatives; these ideas extend to higher dimensions with strip geometries, where imaginary parts vanish in even dimensions and survive in odd ones.The work also connects the imaginary part to the spectra of the reduced density matrix, offering a spectral perspective on timelike EE and outlining potential holographic and entanglement-spectra implications for future research.

Abstract

In this paper, we explore the imaginary part of the timelike entanglement entropy. In the context of field theory, it is more appropriate to obtain the timelike entanglement entropy through the Wick rotation of the twist operators. It is found that, in certain special cases, the imaginary part of the timelike entanglement entropy is related to the commutator of the twist operator and its first-order temporal derivative. To evaluate these commutators, we employ the operator product expansion of the twist operators, revealing that the commutator is generally universal across most scenarios. However, in more general cases, the imaginary part of the timelike entanglement entropy proves to be more complex. We compute the commutator of the twist operators along with its higher-order temporal derivatives. Utilizing these results, we derive a modified formula for the imaginary part of the timelike entanglement entropy. Furthermore, we extend this formula to the case of strip subregion in higher dimensions. Our analysis shows that for the strip geometry, the imaginary part of the timelike entanglement entropy is solely related to the commutators of the twist operator and its first-order temporal derivative. The findings presented in this paper provide valuable insights into the imaginary part of timelike entanglement entropy and its physical significance.

Paper Structure

This paper contains 21 sections, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Timelike and spacelike entanglement entropy relation
  3. Timelike entanglement entropy via analytical continuation
  4. The imaginary term for different separations
  5. Commutators of twist operators
  6. The commutator of twist operators in vacuum state
  7. Commutators of twist operator and its first-order derivative
  8. Vacuum state
  9. Thermal state
  10. States dual to pure AdS$_3$
  11. Results for general states
  12. Universal result for commutator involving first-order temporal derivative
  13. The derivative of the commutator with respect to the spatial coordinate $x$
  14. Results for commutators involving higher order derivative
  15. General formula for the imaginary part of timelike EE
  16. ...and 6 more sections

Figures (1)

  • Figure 1: (a) gives the timelike separation $t>t'$. (b) gives the timelike separation $t<t'$. (c) gives the spacelike separation, where the light cones of its two points have no intersection on the Cauchy surface $t=t_0$. (d) gives the spacelike separation, where the light cones of its two points intersect on the Cauchy surface $t=t_0$.