A microscopic model of black hole evaporation in two dimensions

Abstract

We present a microscopic model of black hole (BH) ‘evaporation’ in asymptotically AdS₂ spacetimes dual to the low energy sector of the SYK model. To describe evaporation, the SYK model is coupled to a bath comprising of N_f free scalar fields Φ_i. We consider a linear combination of couplings of the form O_SY K(t)∑_iΦ_i(0, t), where O_{SY K} involves products of the Kourkoulou-Maldacena operator $iJ/N{\sum}_{k=1}^{N/2}{s}_k^{\prime }{\psi}_{2k-1}(t){\psi}_{2k}(t)$ specified by a spin vector s^′. We discuss the time evolution of a product of (i) a pure state of the SYK system, namely a BH microstate characterized by a spin vector s and an effective BH temperature T_BH, and (ii) a Calabrese-Cardy state of the bath characterized by an effective temperature T_bath. We take T_bath ≪ T_BH, and T_BH much lower than the characteristic UV scale J of the SYK model, allowing a description in terms of the time reparameterization mode. Tracing over the bath degrees of freedom leads to a Feynman-Vernon type effective action for the SYK model, which we study in the low energy limit. The leading large N behaviour of the time reparameterization mode is found, as well as the $O\left(1/\sqrt{N}\right)$ fluctuations. The latter are characterized by a non-Markovian non-linear stochastic differential equation with non-local Gaussian noise. In a restricted range of couplings, we find two classes of solutions which asymptotically approach (a) a BH at a lower temperature, and (b) a horizonless geometry. We identify these with partial and complete BH evaporation, respectively. Importantly, the asymptotic solution in both cases involves the scalar product of the spin vectors s.s^′, which carries some information about the initial state. By repeating the dynamical process O(N²) times with different choices of the spin vector s^′, one can in principle reconstruct the initial BH microstate.