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arXiv:1810.06110 · Journal of High Energy Physics

Scrambling time from local perturbations of the rotating BTZ black hole

Andrius Štikonas

Published Feb 1, 2019• 23 citations
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Abstract

Abstract In this paper, we investigate the entanglement entropy of the rotating BTZ black hole perturbed by a massive back-reacting free falling particle. Then, mutual information between two finite intervals in two asymptotic regions of rotating BTZ is derived. It allows us to find the scrambling time, the time scale in which mutual information vanishes. We give a dual large c CFT description in terms of a thermofield double state with different temperatures for left and right moving modes that is perturbed by a local operator. Exact matching between gravity and CFT results is obtained.

DOI
10.1007/jhep02(2019)054

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Abstract In this paper, we investigate the entanglement entropy of the rotating BTZ black hole perturbed by a massive back-reacting free falling particle. Then, mutual information between two finite intervals in two asymptotic regions of rotating BTZ is derived. It allows us to find the scrambling time, the time scale in which mutual information vanishes. We give a dual large c CFT description in terms of a thermofield double state with different temperatures for left and right moving modes that is perturbed by a local operator. Exact matching between gravity and CFT results is obtained.
In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) proposals allow us to find the same quantities using calculations in general relativity. We will introduce main concepts of holography, quantum information and conformal field theory that will be used to derive the results of this thesis. In the first part of the thesis, we explicitly compute entanglement entropy of the rotating BTZ black hole by directly applying HRT proposal and finding lengths of spacelike geodesics. Rényi entropy of thermal state perturbed by a local quantum quench is computed by mapping correlators on two glued cylinders to the plane for field theory containing a single free boson and for 2d CFTs in the large c limit. We consider Thermofield Double State (TFD) which is an entangled state in direct product of two 2D CFTs. It is conjectured to be holographically equivalent to the eternal BTZ black hole. TFD state is perturbed by a local quench in one CFT and mutual information between two intervals in two CFTs is computed. We find when mutual information vanishes and interpret this as scrambling time, i.e. time scale required for the system to thermalize. This field theory result is modelled with a massive free falling particle in the BTZ black hole. We have computed the back-reaction of the particle on the metric of BTZ and used RT proposal to find holographic entanglement entropy. Finally, we generalize this calculation to the case of rotating BTZ with inner and outer horizons. It is dual to the CFT with different temperatures for left and right moving modes. We calculate mutual information and scrambling time and find exact agreement between results in the gravity and those in the CFT.
We compute the mutual information between finite intervals in two non-compact 2d CFTs in the thermofield double formulation after one of them has been locally perturbed by a primary operator at some time tω in the large c limit. We determine the time scale, called the scrambling time, at which the mutual information vanishes and the original entanglement between the thermofield double gets destroyed by the perturbation. We provide a holographic description in terms of a free falling particle in the eternal BTZ black hole that exactly matches our CFT calculations. Our results hold for any time tω . In particular, when the latter is large, they reproduce the bulk shock-wave propagation along the BTZ horizon description.
We investigate the spreading of entanglement, and entanglement memory effects, in two-dimensional conformal field theory (CFT) propagating on evaporating black hole backgrounds.Memory effects leading to late-time spikes in mutual information for widely separated intervals are well known in CFTs admitting a quasiparticle description.In this work we examine the effect of black hole scrambling on late time mutual information spikes for disjoint intervals in free fermion CFT prepared in a thermofield double state.Latetime entanglement revival is driven by island-induced purification of modes in the union of the intervals.We show across two distinct 2D gravity models, Jackiw-Teitelboim (JT) gravity and the Russo-Susskind-Thorlacius (RST) model, that parametrically dialing up black hole scrambling time smooths out and suppresses entanglement spikes until they disappear at a critical scale, interpolating between free quasiparticle and maximal scrambling pictures.At the critical point, the interval lengths are exponential in black hole scrambling time.We further find a very closely related effect manifest as an entanglement dip for a single interval in a single-sided evaporating RST black hole.
We investigate the spreading of entanglement, and entanglement memory effects, in two-dimensional conformal field theory (CFT) propagating on evaporating black hole backgrounds. Memory effects leading to late-time spikes in mutual information for widely separated intervals are well known in CFTs admitting a quasiparticle description. In this work we examine the effect of black hole scrambling on late time mutual information spikes for disjoint intervals in free fermion CFT prepared in a thermofield double state. Late-time entanglement revival is driven by island-induced purification of modes in the union of the intervals. We show across two distinct 2D gravity models, Jackiw-Teitelboim (JT) gravity and the Russo-Susskind-Thorlacius (RST) model, that parametrically dialing up black hole scrambling time smooths out and suppresses entanglement spikes until they disappear at a critical scale, interpolating between free quasiparticle and maximal scrambling pictures. At the critical point, the interval lengths are exponential in black hole scrambling time. We further find a very closely related effect manifest as an entanglement dip for a single interval in a single-sided evaporating RST black hole.
We study local quenches in 1+1 dimensional conformal field theories at large-c by operators carrying higher spin charge. Viewing such states as solutions in Chern-Simons theory, representing infalling massive particles with spin-three charge in the BTZ back-ground, we use the Wilson line prescription to compute the single-interval entanglement entropy (EE) and scrambling time following the quench. We find that the change in EE is finite (and real) only if the spin-three charge q is bounded by the energy of the perturbation E, as |q|/c < E 2 /c 2. We show that the Wilson line/EE correlator deep in the quenched regime and its expansion for small quench widths overlaps with the Regge limit for chaos of the out-of-time-ordered correlator. We further find that the scrambling time for the two-sided mutual information between two intervals in the thermofield double state increases with increasing spin-three charge, diverging when the bound is saturated. For larger values of the charge, the scrambling time is shorter than for pure gravity and controlled by the spin-three Lyapunov exponent 4π/β. In a CFT with higher spin chemical potential, dual to a higher spin black hole, we find that the chemical potential must be bounded to ensure that the mutual information is a concave function of time and entanglement speed is less than the speed of light. In this case, a quench with zero higher spin charge yields the same Lyapunov exponent as pure Einstein gravity.
We study the perturbation due to rotating shockwaves in Banados-Teitelboim-Zanelli geometries at late times and analyze the change in mutual information between the two subsystems belonging to the dual CFTL{\mathrm{CFT}}_{L}CFTL​ and CFTR{\mathrm{CFT}}_{R}CFTR​. We find that the scrambling of mutual information is in general governed by the Lyapunov index ${\ensuremath{\lambda}}_{L}$ which is bounded by $\ensuremath{\kappa}=\frac{2\ensuremath{\pi}}{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}\ensuremath{\ge}\frac{2\ensuremath{\pi}}{\ensuremath{\beta}}$, where $\ensuremath{\mu}={r}_{\ensuremath{-}}/{r}_{+}$ and L\mathcal{L}L is the angular momentum of the shockwave. For the special case of L=1\mathcal{L}=1L=1 we find the mutual information analytically and show that it is characterized by ${\ensuremath{\lambda}}_{L}=\ensuremath{\kappa}/2$ with the scrambling time for large black holes given as ${t}_{*}=\frac{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu})}{\ensuremath{\pi}}\mathrm{log}S$.