ANALOGUE QUANTUM BLACK HOLES AYAN MUKHOPADHYAY IIT MADRAS YITP - PowerPoint PPT Presentation

ANALOGUE QUANTUM BLACK HOLES AYAN MUKHOPADHYAY IIT MADRAS YITP WORKSHOP “STRINGS AND FIELDS 2020”, Nov 16-20, 2020

PLAN 1. Introduction: Why a physical model and not just circuits? 2. A simple and successful model 3. Discussion: A deeper QIT analysis of the model 4. Conclusions and Outlook BASED ON KIBE, AM, SOLOVIEV, SWAIN; 2006.08644 (PHYS. REV. D 102 (2020) 8, 086008) ONGOING WORKS WITH NIRANJAN KAMATH , TANAY KIBE AND PRABHA MANDAYAM

INTRODUCTION: THE NEED FOR A MODEL OF A QUANTUM BLACK HOLE

COMPUTING THE PAGE CURVE A remarkable question has been answered: how to compute the Page curve via AdS/CFT! The semi-classical geometry itself reproduces the answer via the Quantum Extremal Surface! However, we do not yet understand the mechanism which leads to the resolution of the information paradox. S rad t t Page

CENTRAL QUESTIONS - 1 The AMPS Paradox: Strong subadditivity of entanglement entropy (considering three subsystems : early radiation + late radiation + interior) implies that semiclassical horizon and the equivalence principle cannot be valid simultaneously. Resolution 1: The physical Hilbert spaces are not factorizable (wormholes?). Yet such a factorization should work for simple observables in the semiclassical limit. How does this distinction between interior and exterior emerge in quantum gravity operationally? Resolution 2: Following Hayden and Harlow (2013) we can argue that mining information of the interior from the radiation system is exponentially complex in entropy. Given that Hawking evaporation time is polynomial in entropy, no practical paradox arises. However, this leads to its own set of questions. We need both most likely.

BLACK HOLE AS A QUANTUM ERASER Combining both that black hole is a fast scrambler and that enough Hawking quanta have been 𝒫 ( r s ln r s ) emitted, it follows that the infalling qubits should be rapidly encoded in time in the Hawking quanta. [Hayden and Preskill 2007] So the old black hole is an information mirror! It reveals infalling information fast and its own formation much later.

CENTRAL QUESTIONS - 2 1. How does the factorization of interior and exterior emerge? 2. How come there are observables that does information mirroring without revealing “interior”? 3. What makes the complex encoding of interior into the radiation system possible? 4. How do we fi nd the decoding algorithms for the mirroring and the interior from the dual CFT? 5. How can we ask all these and more questions precisely?

WE DO HAVE ONE PHYSICAL MODEL 1. Lots and lots of microstates with and without hair 2. Energy absorbing and relaxation property (quick transition between microstates post infalling perturbations with almost complete energy absorption) realized. So unless we resolve microstates it looks like a classical black hole. 3. The hair on the horizon provides the key to decode the infalling quanta (mirroring). There does exist observables blind to interior through which we can decode the infalling information after the fast relaxation. 4. The dynamics is pseudorandom (needs to be quanti fi ed). So the information of interior is indeed encoded in a complex way in the “decoupled” hair. 5. Many many questions can be posed precisely via QIT.

A SUCCESSFUL AND SIMPLE MODEL REF: KIBE, AM, SOLOVIEV, SWAIN; 2006.08644 (PHYS. REV. D 102 (2020) 8, 086008)

A CARTOON The near-extremal horizon can fragment into multiple throats via instantons [Brill (1992); Maldacena, Michelson, Strominger (1999)] This inspires a lattice model of fragmented throats with lattice dimensionality same as that of horizon. These “hard” degrees of freedom crystallize and interact via “soft” gravitational mobile charges

CONCRETELY INGREDIENTS OF A ONE-DIMENSIONAL HORIZON 1. SYK type quantum dots dual to NAdS2 throats — degrees of freedom t i ( u ) where is lattice site and i u is the experimentalist’s time. The reparametrized time t i ( u ) carries all information about the quantum dot’s state. 2. The mobile gravitational hair which carries mobile 𝗥 i gravitational charges which take values in SL(2,R). 3. Crucially the full system must have only one overall SL(2,R) symmetry corresponding to the original horizon geometry.

The quantum dots do carry SL(2,R) charges too! Conserved when decoupled. i = t i ( 3 ) − t ′ 2 2 i = t ′ 2 − t ′ ′ ′ t ′ ′ 2 − t ′ ′ ′ ′ i i i i i 𝓡 + 𝓡 0 3 t ′ t ′ t ′ t ′ t ′ i i i i i 3 ) − 2 t i ( i ( t i ) 2 t ′ ′ 2 − t ′ ′ ′ t ′ ′ − t ′ i i i i 𝓡 − i = t 2 t ′ t ′ t ′ i i i The mass of the dual black hole is simply the Casimir of these charges j + 1 j ) 2 ( 𝓡 + i 𝓡 − j + 𝓡 − 𝓡 i ⋅ 𝓡 j := − 𝓡 0 i 𝓡 0 i 𝓡 + SL(2,R) INVARIANT DOT PRODUCT M i = − 𝓡 i ⋅ 𝓡 i = − 2 Sch ( t i ( u ), u ) = − 2 ( 2 ) 2 t ′ ′ ′ t ′ ′ − 3 i i t ′ 2 t ′ i i

OUR EQUATIONS i = − λ ( 𝓡 i − 1 + 𝓡 i +1 − 2 𝓡 i ) ⋅ 𝗥′ M ′ i i = 1 σ 2 ( 𝗥 i − 1 + 𝗥 i +1 − 2 𝗥 i ) + 1 λ 2 ( 𝓡 i − 1 + 𝓡 i +1 − 2 𝓡 i ) 𝗥′ ′ The fi rst equation will turn out to be a di ff usion type equation responsible for relaxation and decoupling for λ > 0 The second equation will ensure information mirroring. Non-trivially we will need to know only ∑ ∑ 𝗥′ 𝗥′ i = 0 ′ which is conserved (since ) analogous to primordial information in early radiation i i i for the decoding of infalling qubits and not at all anything about the interior. It is most likely possible to add higher derivative terms in hair and also higher order interactions retaining phenomenological features. The hair must be quantized but we will consider coherent states admitting classical description.

The total conserved energy of the system is simply the sum of the black hole masses, the kinetic energy and gradient energy of the hair. ℰ = ℰ 𝒭 + ℰ 𝖱 ℰ 𝒭 = ∑ M i = M i ℰ 𝖱 = λ 3 ′ + λ 3 ( 𝗥 i +1 − 𝗥 i ) ⋅ ( 𝗥 i +1 − 𝗥 i ) 2 σ 2 ∑ 2 ∑ ′ ⋅ 𝗥 i 𝗥 i i i For a nice split between interior and exterior, crucially there is NO interaction energy.

MICROSTATES Microstates are just the static/stationary solutions of our model. One can readily obtain all such solutions. We choose a global SL(2,R) frame. THE QUANTUM DOTS (NADS2 THROATS) 𝓡 ± 𝓡 0 i random and constant giving di ff erent M i i = Q , THE HAIR i = − σ 2 i + 𝓛 ± = 𝗥 ± loc 𝗥 ± λ 2 𝓡 ± i Locked with the interior

mon + 𝗥 0 rad 𝗥 0 i = 𝗥 0 i i α = ∑ ′ Q 0 Monopole term remains i mon = α u 𝗥 0 i una ff ected by perturbations i q i = ∑ rad = q i ( u ), ∑ 𝗥 0 q ′ i = 0 i Radiation that can propagate on top of any microstate i i without a ff ecting it i − 1 q ′ ′ σ 2 ( q i +1 + q i − 1 − 2 q i ) = 0

Recall that generally the total energy can be split into two parts ℰ 𝒭 = ∑ ℰ = ℰ 𝒭 + ℰ 𝖱 M i = M i ℰ 𝖱 = λ 3 ′ + λ 3 ( 𝗥 i +1 − 𝗥 i ) ⋅ ( 𝗥 i +1 − 𝗥 i ) 2 σ 2 ∑ 2 ∑ ′ ⋅ 𝗥 i 𝗥 i i i But in the hairy microstates we have non-trivially (as a consequence of dynamics) neat split between interior and exterior components of energy. ℰ 𝖱 = ℰ pot 𝖱 + ℰ mon + ℰ rad 𝖱 𝖱 𝖱 = − σ 2 ( 𝓡 i +1 − 𝓡 i ) ⋅ ( 𝓡 i +1 − 𝓡 i ) 2 λ ∑ ℰ pot INTERIOR (LOCKED) i = − 1 ℰ mon 2 λ 3 α 2 EXTERIOR (PRIMODIAL) 𝖱 𝖱 = λ 3 2 + λ 3 2 2 σ 2 ∑ 2 ∑ ( q i +1 − q i ) ℰ rad q i ′ EXTERIOR (DECOUPLED RADIATION) i i

Also, t i , t ′ i , t ′ ′ need to be continuous for −∞ < u < ∞ and therefore i 𝒭 ± i + 𝒭 − 𝒭 + We choose this. Q ≤ − M i , i ≤ 0, i ≥ 2 Q or 𝒭 ± i + 𝒭 − 𝒭 + Q ≥ − M i , i ≥ 0, i ≤ 2 Q Remarkably these inequalities imply that t ′ i ≥ 0 or t ′ i ≤ 0 for all . i The uniform arrow of time emerges from our model. We choose the future direction.

Ensemble of microstate solutions: Fix total mass M , Q and . α M i , 𝒭 ± Allocate subject to inequalities discussed. i Adding hair on top: 𝗥 rad Each microstate solution supports hair oscillations that can propagate freely over i the lattice without a ff ecting it

SHOCK AND VERIFY Do our model reproduce energy absorption and relaxation properties of classical black hole? If yes, when we perturb a random initial microstate with one or more sequence of shocks, then it will QUICKLY relax to another microstate absorbing all energy in the continuum limit. In presence of shocks, the equations of motion are: i + ∑ i = − λ ( 𝓡 i − 1 + 𝓡 i +1 − 2 𝓡 i ) ⋅ 𝗥′ M ′ e i , A δ ( u − u i , A ) A i = 1 σ 2 ( 𝗥 i − 1 + 𝗥 i +1 − 2 𝗥 i ) + 1 λ 2 ( 𝓡 i − 1 + 𝓡 i +1 − 2 𝓡 i ) 𝗥′ ′

Results for a single shock We shock a 5-site periodic chain at site 1. The initial microstate is always randomly chosen. Note that the shock energy is almost totally absorbed in the black hole mass

𝒭 0 Crucially the relaxes to a new i homogeneous value in the microstate. The conservation of the monopole charge implies that the fi nal homogenous component should also be in same direction

𝒭 ± Also relax to constants as they i should in a ( fi nal) microstate.

The dynamics is pseudorandom (necessary for Harlow-Hayden scenario) The fi nal microstate is hairy with decoupled oscillations q i