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Entanglement and Renormalization Group Irreversibility of Quantum Field Theory in AdS

Nicolás Abate, Ignacio Salazar, Gonzalo Torroba

Abstract

We study nonperturbative aspects of quantum field theory (QFT) in rigid anti de Sitter (AdS) spacetime using quantum information theoretic methods. While irreversibility of renormalization group (RG) flows is well established in flat space, it is not obvious whether it persists in AdS, where negative curvature and an asymptotic timelike boundary significantly modify infrared dynamics. Using strong subadditivity and AdS invariance, we derive an entropic second-order differential inequality for the difference between the vacuum entanglement entropy of a QFT and that of its ultraviolet fixed point, evaluated for spherical bulk regions. This inequality allows us to define RG charges that measure the relevant number of degrees of freedom, and we prove the irreversibility of the RG in $2,\,3,$ and $4$ spacetime dimensions. We further analyze free scalar and fermion theories in AdS, developing lattice formulations adapted to the geometry and computing entanglement entropies and RG charges. In AdS$_2$, we obtain analytic results for a massive Dirac fermion and compare them with numerical lattice calculations. These examples illustrate the general irreversibility theorem and clarify the distinction between conformal and massive theories in AdS.

Entanglement and Renormalization Group Irreversibility of Quantum Field Theory in AdS

Abstract

We study nonperturbative aspects of quantum field theory (QFT) in rigid anti de Sitter (AdS) spacetime using quantum information theoretic methods. While irreversibility of renormalization group (RG) flows is well established in flat space, it is not obvious whether it persists in AdS, where negative curvature and an asymptotic timelike boundary significantly modify infrared dynamics. Using strong subadditivity and AdS invariance, we derive an entropic second-order differential inequality for the difference between the vacuum entanglement entropy of a QFT and that of its ultraviolet fixed point, evaluated for spherical bulk regions. This inequality allows us to define RG charges that measure the relevant number of degrees of freedom, and we prove the irreversibility of the RG in and spacetime dimensions. We further analyze free scalar and fermion theories in AdS, developing lattice formulations adapted to the geometry and computing entanglement entropies and RG charges. In AdS, we obtain analytic results for a massive Dirac fermion and compare them with numerical lattice calculations. These examples illustrate the general irreversibility theorem and clarify the distinction between conformal and massive theories in AdS.
Paper Structure (19 sections, 133 equations, 12 figures)

This paper contains 19 sections, 133 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Quantum field theory in AdS
  3. Review of AdS geometry
  4. CFTs and renormalization group flows in AdS
  5. Entanglement structure of CFTs in AdS
  6. Irreversibility formula in AdS
  7. Warm-up: the C-theorem in AdS$_2$
  8. Proof of the irreversibility formula
  9. RG charges and C, F and A theorems
  10. Dirac fermion in AdS$_2$
  11. Review of the replica calculation and analytic continuation
  12. Entanglement entropy in terms of the Painlevé VI solution
  13. Lattice calculations
  14. Lattice calculations for free scalar fields
  15. Scalar in AdS$_d$
  16. ...and 4 more sections

Figures (12)

  • Figure 1: Penrose diagram for AdS$_d$ for $d=2$ (left) and $d>2$ (right).
  • Figure 2: Penrose diagrams of the Lorentzian cylinder (left) and Minkowski (right) spacetimes. Both are related via the conformal transformation \ref{['eq:rpmdef']}. Moreover, AdS is conformally equivalent to a portion of these spacetimes, as we show in grey.
  • Figure 3: Entangling regions with arbitrary boundary $\beta=\gamma(\Omega)$ on the null cone are employed to show the Markov property in AdS. The case $\beta=\beta_0$ gives a spherical region of proper area proportional to $(\ell\tan\beta_0)^{d-2}$.
  • Figure 4: Geometric setup used in the derivation of the infinitesimal boosted SSA inequality.
  • Figure 5: Running C-function for massive Dirac fermion in the hyperbolic disk, evaluated using the Painlevé VI equation.
  • ...and 7 more figures