Abstract
To model the interior of a black hole, a study is made of a spin system with long-range random four-spin couplings that exhibits quantum chaos. The black hole limit corresponds to a system where the microstates are approximately degenerate and equally likely, corresponding to the high temperature limit of the spin system. At the leading level of approximation, reconstruction of bulk physics implies that local probes of the black hole should exhibit free propagation and unitary local evolution. We test the conjecture that a particular mean field Hamiltonian provides such a local bulk Hamiltonian by numerically solving the exact Schrodinger equation and comparing the time evolution to the approximate mean field time values. We find excellent agreement between the two time evolutions for timescales smaller than the scrambling time. In earlier work, it was shown bulk evolution along comparable timeslices is spoiled by the presence of the curvature singularity, thus the matching found in the present work provides evidence of the success of this approach to interior holography. The numerical solutions also provide a useful testing ground for various measures of quantum chaos and global scrambling. A number of different observables, such as entanglement entropy, out-of-time-order correlators, and trace distance are used to study these effects. This leads to a suitable definition of scrambling time, and evidence is presented showing a logarithmic variation with the system size.
Highlights
The anti–de Sitter/conformal field theory correspondence (AdS=CFT) [1] has been tremendously successful in providing a framework for addressing questions in quantum gravity, that goes far beyond the successes of perturbative string theory
The holographic mapping from conformal field theory operators to bulk spacetime operators is only well understood as a perturbative expansion around asymptotically AdS regions [2,3], and there is much current debate about how the holographic mapping can be extended deep into the bulk spacetime, where the presence of apparent horizons and global horizons make the application of the perturbative holographic mapping problematic
We have explored a four-spin interacting system that exhibits fast scrambling feature in the high temperature limit which is conjectured to be holographically dual to a black hole spacetime in the vicinity of the horizon
Summary
The anti–de Sitter/conformal field theory correspondence (AdS=CFT) [1] has been tremendously successful in providing a framework for addressing questions in quantum gravity, that goes far beyond the successes of perturbative string theory. In a series of papers, it has been argued a more general holographic mapping should take the form of a mean field theory approximation, where the bulk degrees of freedom are to be extracted after suitable averaging over the microscopic exact representation [4,5,6,7] In some sense, this is not a new idea, and similar proposals have been made in the context of loop quantum gravity, fuzzballs, etc. Since the systems have finite dimensional Hilbert spaces, by construction, the Hamiltonian may always be diagonalized, and questions of chaos boil down to whether the spectrum of energy eigenvalues is suitably dense, and whether the “local” basis of states one might be interested in have a simple representation in terms of energy eigenstates The latter condition is not satisfied if we restrict to interactions with a range comparable to the system extent. This in turn will lead to a definition of the thermalization time as we detail below, and we will see that even in this high temperature limit, the spin model exhibits a version of scrambling, where the thermalization time is logarithmic in the system size
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
