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Dynamical entanglement entropy with angular momentum and U(1) charge

Pawel Caputa, Gautam Mandal, Ritam Sinha

TL;DR

This work analyzes time-dependent entanglement entropy in a 1+1D CFT with angular momentum and U(1) charge, showing saturation to the grand canonical entropy at late times and providing a holographic match via spinning BTZ black holes with a U(1) connection. It develops both CFT and holographic computations of EE for spin and charge, including a proof-of-principle RT derivation in 1+1D CFT through a double-scaling limit and clarifies the role of the bulk CS boundary term in incorporating charge. A key result is that EE growth for small intervals obeys a first-law-like relation and that saturation follows from a quantum ergodicity mechanism, with universal implications across dimensions and for massive theories. The paper also discusses charged, massive scalar checks, higher-spin extensions, and the emergence of GGEs, linking dynamical EE to fundamental ergodicity and holographic dualities.

Abstract

We consider time-dependent entanglement entropy (EE) for a 1+1 dimensional CFT in the presence of angular momentum and U(1) charge. The EE saturates, irrespective of the initial state, to the grand canonical entropy after a time large compared with the length of the entangling interval. We reproduce the CFT results from an AdS dual consisting of a spinning BTZ black hole and a flat U(1) connection. The apparent discrepancy that the holographic EE does not a priori depend on the U(1) charge while the CFT EE does, is resolved by the charge-dependent shift between the bulk and boundary stress tensors. We show that for small entangling intervals, the entanglement entropy obeys the first law of thermodynamics, as conjectured recently. The saturation of the EE in the field theory is shown to follow from a version of quantum ergodicity; the derivation indicates that it should hold for conformal as well as massive theories in any number of dimensions.

Dynamical entanglement entropy with angular momentum and U(1) charge

TL;DR

This work analyzes time-dependent entanglement entropy in a 1+1D CFT with angular momentum and U(1) charge, showing saturation to the grand canonical entropy at late times and providing a holographic match via spinning BTZ black holes with a U(1) connection. It develops both CFT and holographic computations of EE for spin and charge, including a proof-of-principle RT derivation in 1+1D CFT through a double-scaling limit and clarifies the role of the bulk CS boundary term in incorporating charge. A key result is that EE growth for small intervals obeys a first-law-like relation and that saturation follows from a quantum ergodicity mechanism, with universal implications across dimensions and for massive theories. The paper also discusses charged, massive scalar checks, higher-spin extensions, and the emergence of GGEs, linking dynamical EE to fundamental ergodicity and holographic dualities.

Abstract

We consider time-dependent entanglement entropy (EE) for a 1+1 dimensional CFT in the presence of angular momentum and U(1) charge. The EE saturates, irrespective of the initial state, to the grand canonical entropy after a time large compared with the length of the entangling interval. We reproduce the CFT results from an AdS dual consisting of a spinning BTZ black hole and a flat U(1) connection. The apparent discrepancy that the holographic EE does not a priori depend on the U(1) charge while the CFT EE does, is resolved by the charge-dependent shift between the bulk and boundary stress tensors. We show that for small entangling intervals, the entanglement entropy obeys the first law of thermodynamics, as conjectured recently. The saturation of the EE in the field theory is shown to follow from a version of quantum ergodicity; the derivation indicates that it should hold for conformal as well as massive theories in any number of dimensions.

Paper Structure

This paper contains 20 sections, 92 equations, 4 figures.

Table of Contents

  1. Introduction and Summary
  2. Entanglement Entropies with spin: CFT
  3. Thermofield double
  4. Time-dependent EE
  5. Finite interval
  6. Single CFT, arbitrary state
  7. Information loss
  8. EE with spin: holographic calculation
  9. The importance of double scaling:
  10. Conclusion of this section
  11. EE for a charged state: CFT
  12. Holographic EE with charge: BTZ plus CS U(1)
  13. A puzzle
  14. Higher Spin
  15. Universal limits
  16. ...and 5 more sections

Figures (4)

  • Figure 1: The lower branch cuts on left and right represent the (holomorphic and antiholomorphic coordinates) of the entangling interval in the first copy of the CFT. The upper branch cuts represent the second CFT.
  • Figure 2: Saturation of the Entanglement Entropy for different values of $\Omega$
  • Figure 3: A sketch of the 'proof' of the Ryu-Takayanagi formula. The use of AdS/CFT map in the left vertical arrow is justified in the limit of $n=1+ \epsilon$ (see text).
  • Figure 4: Plot of $S[\rho_{A,\beta}]/l$ (the LHS of (\ref{['reasonable']}) vs $\beta\mu$, for $\beta m=10$. The plot marked with squares has $l/\beta=10$; the plot marked with triangles has $l/\beta=25$. The solid line corresponds to the RHS of (\ref{['reasonable']}), viz. the thermal entropy density, which is obtained using standard formulae, and Mathematica. It is clear that (\ref{['reasonable']}) holds to a good accuracy. The agreement is better for small $\beta\mu$ than for large $\beta\mu$; however, this could be due to some numerical instability.