Abstract
In this paper, we develop a general effective theory for two copies of the Sachdev-Ye-Kitaev (SYK) model with a time-dependent bilinear coupling. For a quantum quench problem with an initial state of the thermofield double state, we show how the evolution of the system is described by a complex reparametrization field with a classical Hamiltonian. We study correlation functions in this system and compare the large-q theory with the bulk low energy effective theory. In particular, we study the special case of a “rescued black hole”, which describes how a time-evolved thermofield double state can evolve to the ground state of a coupled SYK model by a carefully tuned time-dependent coupling. In the low energy region, there is a holographic dual interpretation, which is a geometry that crosses over from an eternal black hole to a global AdS2 vacuum. This family of geometries allow us to access the bulk region that would be the black hole interior without the rescue process. By comparing the large-q and low energy theory, we find that even in the low energy region the deviation from the low energy theory cannot be neglected if the rescue process starts late. This provides evidence that the low energy effective theory of the bulk fails near the inner horizon of the black hole. We note the possibility of a connection to a two-dimensional analog of the higher-dimensional black hole singularity.
Highlights
Still many open questions about the black hole interior
We show that the large-N large-q dynamics of the SYK model with a generic timedependent coupling and a thermofield double initial state can be described by a complex reparametrization field, which is a generalization of the boundary location in AdS2
We study the rescued black hole geometry, and show that even if the initial state is a low temperature thermofield double state, and the couplings are kept small, there are certain four-point functions for which the low energy description in terms of free matter propagating on AdS2 is wrong by an order-1 fraction if the rescue process is turned on at a late time ur−
Summary
We describe the boundary model we study and the full solution for the two-point function. We will defer the majority of discussion of the bulk interpretation to the sections. The key building block is the SYK ensemble of Hamiltonians on a system of N Majorana fermions, the set of which we just denote by χ and elements χj ∈ χ with j ∈ 1, . N ; the elements obey the algebra {χj, χk} = δjk. The ensemble is defined by an even number q > 2, a Gaussian random anti-symmetric tensor Jj1···jq , and Hamiltonian HSYK[J ] with iq/2 N.
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