[Paper Review] Pure states in the SYK model and nearly-$AdS_2$ gravity

Ioanna Kourkoulou, Juan Maldacena|arXiv (Cornell University)|Jul 7, 2017

Black Holes and Theoretical Physics27 references68 citations

TL;DR

The paper constructs pure states in the SYK model via Euclidean projection, shows diagonal correlators match thermal correlators at leading order, analyzes off-diagonal correlators, and discusses a gravity interpretation with behind-horizon regions and traversable-wormhole-like modifications.

ABSTRACT

We consider pure states in the SYK model. These are given by a simple local condition on the Majorana fermions, evolved over an interval in Euclidean time to project on to low energy states. We find that "diagonal" correlators are exactly the same as thermal correlators at leading orders in the large $N$ expansion. We also describe "off diagonal" correlators that decay in time, and are given simply in terms of thermal correlators. We also solved the model numerically for low values of $N$ and noticed that subsystems become typically entangled after an interaction time. In addition, we identified configurations in two dimensional nearly-$AdS_2$ gravity with similar symmetries. These gravity configurations correspond to states with regions behind horizons. The region behind the horizon can be made accessible by modifying the Hamiltonian of the boundary theory using the the knowledge of the particular microstate. The set of microstates in the SYK theory with these properties generates the full Hilbert space.

Motivation & Objective

• Define a class of pure states in the SYK model by simple local Majorana conditions and Euclidean time evolution to low energies.
• Show that diagonal two-point functions in these states reproduce thermal correlators at leading order in large N.
• Characterize off-diagonal correlators and their behavior under Lorentzian evolution.
• Provide numerical evidence from exact diagonalization that subsystems become entangled after a finite interaction time.
• Propose a gravity interpretation in nearly-AdS2 that mirrors SYK state symmetries and allows access to behind-horizon regions via modified boundary Hamiltonians.

Proposed method

• Use a Majorana fermion SYK model with random quartic couplings and an O(N) symmetry analysis.
• Define boundary-pure states |B_s> by eigenvalue conditions on pairs of Majorana fermions, forming a 2^(N/2)-dimensional basis.
• Evolve these states by Euclidean time ℓ to form low-energy states |B(ℓ)>, with β=2ℓ, and compute correlators in these states.
• Apply replica techniques and large-N Schwinger-Dyson equations to show diagonal correlators G_diag(τ,τ') equal Gβ(τ−τ') at leading order, and off-diagonal G_off(τ,τ') relate to products of thermal correlators.
• Perform low-energy (almost conformal) analysis using the conformal two-point function Gβ(τ) with Δ=1/4 for q=4 SYK.
• Use exact diagonalization for finite N (e.g., N=24,30) to study |B> state coefficients, correlator decay, and entanglement entropy growth.
• Discuss a gravity interpretation in nearly-AdS2 where a boundary shockwave yields a region behind the horizon, and relate to traversable-worldline-like boundary modifications.

Experimental results

Research questions

• RQ1 Do the diagonal two-point functions in the constructed pure states reproduce the finite-temperature thermal correlators at leading order in 1/N?
• RQ2 How do off-diagonal correlators behave in these pure states, and can they be expressed in terms of thermal correlators?
• RQ3 What is the time evolution of entanglement in subsystems of the pure states, and how quickly do they approach typical random-state entanglement?
• RQ4 What gravity configurations in nearly-AdS2 correspond to these SYK pure states, and can a boundary modification reveal behind-horizon regions?
• RQ5 How well do finite-N numerical results support the large-N analytic predictions for diagonal/off-diagonal correlators and overlaps (e.g., ⟨B_s|e^{-2ℓH}|B_s⟩)?

Key findings

• Diagonal correlators in the pure states are exactly the same as thermal correlators at leading order in large N.
• Off-diagonal correlators decay in time and can be expressed simply in terms of thermal correlators, up to 1/N corrections.
• A numerical diagonalization shows that subsystems become typically entangled after a finite interaction time, with entanglement saturation roughly independent of subsystem size.
• A gravity interpretation identifies nearly-AdS2 configurations with a shockwave and a behind-horizon region, suggesting a basis of microstates with smooth horizons accessible via boundary Hamiltonian modifications.
• Averaging over the sign choices in the initial |B_s> states reproduces the thermal ensemble, while individual |B_s> states yield correlators nearly thermal at leading order.
• The low-energy conformal limit yields explicit forms for G_off(t,t') that decay with time in a way consistent with thermalization.