Pure states in the SYK model and nearly-$AdS_2$ gravity

TL;DR

The authors construct pure states in the SYK model via simple local Majorana boundary conditions and Euclidean evolution to low energy, revealing that diagonal two‑point functions match thermal correlators at leading order while off‑diagonal correlators are determined by thermal data and decay over time. They support these results with exact diagonalization showing rapid entanglement growth and a typical‑state behavior consistent with Page’s entropy expectations. A gravity interpretation in nearly‑AdS2 identifies shockwave configurations behind horizons and a boundary‑driven protocol to access interior regions, akin to traversable wormholes. The findings suggest these simple boundary states form a complete basis of low‑energy SYK states and offer a concrete link between pure‑state dynamics and horizon interior structure in a holographic context.

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.

Figures (8)

• Figure 1: (a) The Euclidean computation pictured as an interval of size $\beta = 2 \ell$. (b) The same Euclidean computation pictured as a circle, with a special point where we project on to the state $|B\rangle$. (c) Evolving by Euclidean time $\ell$ we get the state $|B(\ell)\rangle$, which we can then evolve using Lorentzian evolution. (d) By inserting simple operators $O_1$, $O_2$, at intermediate Euclidean times we can get states containing some small excitations.
• Figure 2: We set $N=30$. In the left we plot the square of the coefficients of the non-zero expansion of $|B\rangle$ in terms of the energy of the energy eigenstates. (Half of the coefficients are automatically are zero due to the $(-1)^F$ symmetry). They are random looking. The average value of the square of the non-zero coefficients is $2^{-N/2 +1}$, which is about $0.6 \times 10^{-4}$. Note that the density of eigenstates is not uniform along the horizontal axis. On the right see the phases of the coefficients. More precisely, since the phases of the energy eigenstates can be chosen independently for each eigenstate, we really plot the difference in phases for two different states $|B_s\rangle$, $|B_{s'}\rangle$.
• Figure 3: We set $N=24$. We plot the Lorenzian time expectation value of the operator $S_1(t)/2 = i G_{\rm off}(t,t)$ on the state $|B\rangle$ (\ref{['Bs']}) (with $\ell=0$). We see that it decays over at time of the order of the interaction time $1/J$. We also plot twice the square of the thermal correlator at $\beta =0$. We see that (\ref{['Twoptoff']}) holds pretty closely despite the low value of $N$.
• Figure 4: Here $N=24$. Ratio of the entanglement entropy of a subfactor of $N_A$ spins (or $2 N_A$ Majorana fermions), to the entanglement entropy for a typical random state in the Hilbert space Page:1993df. Something to note is that the time it takes to saturate is independent of the size of the subsystem. This is different from a local spin chain and it reflects the all to all nature of the SYK Hamiltonian.
• Figure 5: On the left we have $N=24$ and we plot the Euclidean time thermal answer and also the ratio (\ref{['RatioAvg']}) averaged over 150 choices of the couplings. We also ploted the ratio (\ref{['RatioAvg']}) for one particular value of the couplings to display how it differs from the average. On the right we took $N=30$ and we show the euclidean correlator (\ref{['RatioNat']}) and the thermal one for a single realization of the couplings. We see that as we increase $N$ we are approaching the result (\ref{['Twoptdi']}). Comparing with the error for a single realization of the couplings for $N=24$ (left) and $N=30$ (right) we see that it decreases as we increase $N$.