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Higher spin entanglement entropy from CFT

Shouvik Datta, Justin R. David, Michael Ferlaino, S. Prem Kumar

TL;DR

This work perturbatively computes the finite-temperature entanglement entropy (EE) for a single interval in two-dimensional CFTs with ${ m W}$-algebra symmetry, deformed by a chemical potential for the spin-$3$ current, up to ${ m O}(μ^2)$. Using holomorphic conformal perturbation theory and the replica trick, the authors derive a universal form for the EE correction that is identical for free fermions with ${ m W}_{1+∞}$ symmetry and free bosons with ${ m W}_∞[1]$ symmetry, and they show perfect agreement with the holomorphic holographic entanglement entropy proposal via Wilson lines in SL$(3,R) imes{ m SL}(3,R)$ Chern–Simons theory. The EE result is complemented by explicit computations of the corresponding Rényi entropies, which differ between the fermion and boson theories, and by a holographic check that reproduces the same ${ m O}(μ^2)$ correction. The paper also analyzes a U(1) coset of ${ m W}_{1+∞}$ and a truncation to ${ m W}_N$ algebras, finding the EE to be universal at leading order in $μ$ for certain realizations but noting unresolved issues in twist-field constructions for the coset. Overall, the results provide nontrivial tests of higher-spin holographic entanglement entropy proposals and suggest a universal structure for ${ m W}$-algebra CFTs perturbed by spin-$3$ chemical potentials, with rich directions for future work in more general ${ m W}_N$ theories and interacting cosets.

Abstract

We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential $μ$ for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in $μ$ in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, R) x SL(3, R) Chern-Simons theory. Our result suggests that the order $μ^2$ correction to the entanglement entropy may be universal for W-algebra CFTs with spin-three chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS_3.

Higher spin entanglement entropy from CFT

TL;DR

This work perturbatively computes the finite-temperature entanglement entropy (EE) for a single interval in two-dimensional CFTs with -algebra symmetry, deformed by a chemical potential for the spin- current, up to . Using holomorphic conformal perturbation theory and the replica trick, the authors derive a universal form for the EE correction that is identical for free fermions with symmetry and free bosons with symmetry, and they show perfect agreement with the holomorphic holographic entanglement entropy proposal via Wilson lines in SL Chern–Simons theory. The EE result is complemented by explicit computations of the corresponding Rényi entropies, which differ between the fermion and boson theories, and by a holographic check that reproduces the same correction. The paper also analyzes a U(1) coset of and a truncation to algebras, finding the EE to be universal at leading order in for certain realizations but noting unresolved issues in twist-field constructions for the coset. Overall, the results provide nontrivial tests of higher-spin holographic entanglement entropy proposals and suggest a universal structure for -algebra CFTs perturbed by spin- chemical potentials, with rich directions for future work in more general theories and interacting cosets.

Abstract

We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, R) x SL(3, R) Chern-Simons theory. Our result suggests that the order correction to the entanglement entropy may be universal for W-algebra CFTs with spin-three chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS_3.

Paper Structure

This paper contains 31 sections, 138 equations.

Table of Contents

  1. Introduction and Summary
  2. Higher spin chemical potentials and thermodynamics
  3. 'Holomorphic' perturbation theory and free energy at ${\cal O}(\mu^2)$
  4. Entanglement entropy and conformal perturbation theory
  5. Entanglement entropy at ${\cal O}(\mu^2)$
  6. Universal result for ${\cal O}(\mu^2)$ EE at finite temperature
  7. Free fermions: ${\cal W}_{1+\infty}$ algebra at $\lambda=0$
  8. Bosonization
  9. Rényi and entanglement entropies at ${\cal O}(\mu^2)$
  10. First order correction in $\mu$:
  11. Second order correction:
  12. Free bosons: ${\cal W}_\infty[1]$ symmetry
  13. Correlators involving bosonic twists
  14. Rényi and Entanglement entropies at ${\cal O}(\mu^2)$
  15. Holographic entanglement entropy
  16. ...and 16 more sections