Gravitational collapse in SYK models and Choptuik-like phenomenon
Journal of High Energy Physics
SYK model is a quantum mechanical model of fermions which is solvable at strong coupling and plays an important role as perhaps the simplest holographic model of quantum gravity and black holes. The present work considers a deformed SYK model and a sudden quantum quench in the deformation parameter. The system, as in the undeformed case, permits a low energy description in terms of pseudo Nambu Goldstone modes. The bulk dual of such a system represents a gravitational collapse, which is
... e, which is characterized by a bulk matter stress tensor whose value near the boundary shows a sudden jump at the time of the quench. The resulting gravitational collapse forms a black hole only if the deformation parameter ∆ exceeds a certain critical value ∆ c and forms a horizonless geometry otherwise. In case a black hole does form, the resulting Hawking temperature is given by a fractional power T bh ∝ (∆ − ∆ c ) 1/2 , which is reminiscent of the 'Choptuik phenomenon' of critical gravitational collapse. (thermal) state; furthermore, the information about the initial pure state appears to be lost. How does one understand this puzzle within a unitary quantum mechanical framework? With the AdS/CFT correspondence, such a unitary description appears possible in terms of the dual CFT where gravitational collapse can be modelled by a quantum quench     and under a sudden perturbation a given pure state can evolve to a pure state with thermal properties. 1 Such models are not easy to construct in strongly coupled field theories in three and higher dimensions. In lower dimensions, however, there are powerful techniques to deal with the dynamics of strongly coupled conformal field theories. In one dimension, the relevant strongly coupled model        which has a holographic dual     is the SYK model. 2 In the present paper, we will discuss gravitational collapse in such a holographic dual. Besides the above issue of 'information loss', gravitational collapse is associated with another interesting phenomenon, namely that of Choptuik scaling. In his classic work  Choptuik analyzed a family of initial states characterized by a parameter p (which roughly corresponds to the amount of self-gravitation of the infalling matter) and evolved them numerically (see, e.g.  for a review). He found that while no black holes are formed for p < p c , they are formed for p > p c , with the mass of the resulting black hole given by M bh ∝ (p−p c ) γ . Here, γ is found to be a universal critical exponent, which depends only on the type of infalling matter and not on the details of the initial configuration. The results of  were in asymptotically flat space (for a review see, e.g.  ). This was generalized to asymptotically AdS 3 spaces (a) for scalar field collapse in  and (b) for formation of BTZ black holes in  from point particle collisions (the critical exponents are different in the two cases). In higher AdS D spaces, with D ≥ 4, the Choptuik phenomenon gets richer, because bounces from the boundary play a dominant role  . 3 It is important to obtain a field theory understanding of Choptuik criticality. SYK model allows us to explicitly perform a boundary calculation dual to Choptuik transition, as we will describe in the paper. 4 The basic framework for our study was laid out by Kourkoulou and Maldacena in , where they described an explicit construction of a complete set of thermal microstates for the SYK model. The dual spacetime for such a state, evolving under the SYK Hamiltonian H SYK , is given by a black hole in the Poincare wedge of AdS 2 with a special boundary condition at the corner. It was shown in  that, if one starts with a given such state and turns on a certain perturbation H (s) M with a suitably fine-tuned spin a and coefficient 1 We will henceforth call such pure states with thermal properties, thermal microstates. 2 The SYK model for charged fermions has been discussed in  and the holographic dual to such a model has been presented in [17, 18] . 3 The critical value pc in this case refers to critical value of p for black hole formation in the first pass for the imploding matter. For p < pc there are a new series of critical values pc,n so that for p > pc,n a black hole just about forms after n bounces; the mass of the resulting black hole after n bounces has the same critical behaviour Mn ∝ (p − pc,n) γ with the same critical exponent  . We thank M. Rangamani for important discussions on this issue. 4 Strictly speaking, we should call the phenomenon Choptuik-like since the precise details of a critical gravitational collapse are not yet possible to work out (see section 5.3 for the extent to which details of the gravitational collapse are possible to work out). In the following, phrases like Choptuik phenomenon and Choptuik transition will be used with this caveat in mind. 5 This perturbation is an example of a state-dependent operator which has been used in the context of understanding microstates in the black hole interior in ; see also [27-30]. 6 If the coefficient is lower than the threshold we get a smaller black hole than the black hole that would have formed with = 0, see . 7 In two dimensions, in the context of two-dimensional dilaton-gravity black holes [32-34], gravitational collapse has been discussed in . 8 Preliminary results were presented at https://indico.cern.ch/event/691363/timetable/#14-gravitationalcollapse-in-t. 9 Quantum quenches in SYK model have been discussed, in somewhat different contexts, in  and .