$SL(2,\hspace{0.17em}R)$ lattices as information processors

Black holes past their Page times should act as efficient scramblers and information mirrors. The information of the infalling bits are rapidly encoded by the old black hole in the Hawking quanta, but it should take time that is exponential in the Page time entropy to decode the interior. Motivated by the features of fragmentation instability of near-extremal black holes, we construct a simple phenomenological model of the black hole as a lattice of interacting nearly ${\mathrm{AdS}}_2$ throats with gravitational hair charges propagating over the lattice. We study the microstate solutions and their response to shocks. The energy of the shocks are almost wholly absorbed by the total Arnowitt-Deser-Misner mass of the ${\mathrm{AdS}}_2$ throats, but the information of their locations and time ordering come out in the hair oscillations, which decouple from the final microstate to which the full system quickly relaxes. We discuss the Hayden-Preskill protocol of decoding infalling information. We also construct generalizations of our model involving a lattice of ${\mathrm{AdS}}_2$ throats networked via wormholes and their analogs in the form of tensor networks of Sachdev-Ye-Kitaev spin states.

Figure 1

Schematic representation of our model, comprised of a lattice of ${\mathrm{AdS}}_2$ black holes carrying $SL(2,R)$ charges ${\mathcal{Q}}_i$ and interacting with mobile $SL(2,R)$ charges ${\mathsf{Q}}_i$ representing gravitational hair. Although we have shown a two-dimensional lattice in this figure, the lattice should actually have the dimensionality of the black hole horizon (in this case it represents the horizon of a four-dimensional black hole). In this paper, we will study a chain with a periodic boundary condition explicitly.

Figure 2

Plot of $\tau (u)$ vs $u$ given by (38) for various choices of values of the $SL(2,R)$ charges allowed by (43) and $\beta =2\pi$.

Figure 3

Evolution of ${\mathcal{E}}_{\mathcal{Q}}$ and ${\mathcal{E}}_{\mathsf{Q}}$ after a shock in a lattice with five sites (${\mathrm{AdS}}_2$ black holes). Here $\lambda =1$ and $\sigma =0.01$. The shock with $e=0.4$ has been injected into the first site at $u=0$. The variables $M_i$ and ${\mathcal{Q}}_i$ of the initial microstate solution have been chosen randomly as described in the text with the total initial mass $M=20$ and $Q\approx 5.05$. Also $\alpha =1$. Note that ${\mathcal{E}}_{\mathcal{Q}}=M$, the sum of the ADM masses of the ${\mathrm{AdS}}_2$ throats, and ${\mathcal{E}}_{\mathsf{Q}}$, the energy in the hair charges, are separately conserved (except at the moment of shock injection) to a very good approximation. Also approximately all the energy injected in the shock goes to ${\mathcal{E}}_{\mathcal{Q}}$.

Figure 4

The evolution of the $SL(2,R)$ charges ${\mathcal{Q}}_i$ and $M_i$ in our five-site model after the first site is shocked at $u=0$. Here $\lambda$, $\sigma$, $\alpha$ and $e$ (the shock energy) are set as in Fig. 3. We have chosen the initial conditions randomly corresponding to a typical initial microstate as exhibited above. The corresponding evolution of the energies ${\mathcal{E}}_{\mathcal{Q}}$ and ${\mathcal{E}}_{\mathsf{Q}}$ are as in Fig. 3. It is clear from the above plots that all $M_i$ and thus ${\mathcal{Q}}_i$ relax eventually to constant values with ${\mathcal{Q}}_i^0$ converging to the same value at each lattice site. This implies that the full system relaxes to a final microstate solution.

Figure 5

The dependence of the relaxation time on $M$ and $Q$. Note there is about 1%–2% variation over choice of initial microstates. This is not shown above.

Figure 6

Mixing indicates the dynamics of our model which mixes up all the degrees of freedom with the external shocks and decoding is the procedure that Bob uses to extract information about the shocks. We will consider the situation where the hair oscillations (${\mathsf{Q}}_i^{\mathrm{rad}}$) are initially zero to simplify our decoding protocol. After the model relaxes to a new microstate with some hair oscillations on top, Bob applies his decoding procedure on the decoupled hair ${\mathsf{Q}}^{\mathrm{rad}}$ and recovers some classical information about the shocks, namely their positions and time ordering.

Figure 7

Amplitudes of the Fourier transform of $q_i^{\prime }$ for each site obtained after the full system relaxes to the final microstate. The peaks coincide with the numerically computed normal mode frequencies to a very good approximation. The finite width of these peaks is due to finiteness of the time window and also numerical noise.

Figure 8

${\mathrm{AdS}}_2$ throats in a three-site chain can be networked via wormholes as depicted above. Each ${\mathrm{AdS}}_2$ throat has a pantlike structure bounded by two geodesics carrying charges ${\mathcal{Q}}_{a,i}$ and ${\mathcal{Q}}_{b,i}$ at the $i$th site such that ${\mathcal{Q}}_i={\mathcal{Q}}_{a,i}+{\mathcal{Q}}_{b,i}$ and ${\mathcal{Q}}_{a,i}={\mathcal{Q}}_{b,i-1}$ (also for ${\mathcal{Q}}_{b,i}={\mathcal{Q}}_{a,i+1}$) for $i=1$, 2, 3. The other edges are simply lightlike geodesics—the red dotted lines mark light rays also. The pants are glued across the geodetic junctions with equal $SL(2,R)$ charges. It is clear that any lightlike geodesic starting from a boundary can traverse the entire network—note that the time shifts while traversing the geodetic junctions—the impact point at the other edge is marked with a cross. Here we show an example of the transmission of a shock through the network with perfectly transmitting conditions at the junctions and perfectly reflecting conditions at the boundaries.

Figure 9

A tensor network of a chain of SYK spin states with periodic boundary conditions is illustrated above. We partition the $N_i$ spins into two blocks $N_{a.i}$ and $N_{b,i}$ such that $N_{a,i}=N_{b,i-1}$ (and thus $N_{b,i}=N_{a,i+1}$) for $i=1$, 2, 3. We then maximally entangle the blocks of spins with the corresponding blocks of the neighboring sites as shown above. Finally, Euclidean evolution by the SYK Hamiltonians over each site (see text) symmetrizes over the $N_i$ spins at each site and different ways of pairing $2N_i$ Majorana fermions (with appropriate sign factors) at each site into $N_i$ pairs each of which is projected onto a spin state.

Figure 10

Plot of the $q_i^{\prime }$ after a single shock at site 1. The amplitude of the oscillations can be clearly seen to grow with time.

Figure 11

The envelope of $q_i^{\prime }$ is fit to an exponential (red curve).

Figure 12

$q_i^{\prime }$ after the drift is subtracted out.

Figure 13

Plot of the phase against the frequency for site 1. The gaps in the plot are due to interpolation. Note that the phase is computed modulo $2\pi$.