Euclidean wormhole in the SYK model

TL;DR

This work links ensemble averaging in a two-site SYK model with complex couplings to a concrete Euclidean wormhole in JT gravity with matter. The SYK system exhibits a low-temperature, gapped phase only after averaging, a feature mirrored by a double-trumpet wormhole in gravity sustained by imaginary sources for a marginal operator with $\Delta=1$. The gravity analysis, including the replica trick, shows a first-order transition from a disconnected black-hole phase to a connected wormhole, with the marginal operator expectation value acting as an order parameter. The qualitative agreement between the SYK thermodynamics and the gravitational saddle supports the interpretation that ensemble averaging in field theory can realize Euclidean wormholes, shedding light on factorization and the role of wormholes in holography and spectral statistics.

Abstract

We study a two-site Sachdev-Ye-Kitaev (SYK) model with complex couplings, and identify a low temperature transition to a gapped phase characterized by a constant in temperature free energy. This transition is observed without introducing a coupling between the two sites, and only appears after ensemble average over the complex couplings. We propose a gravity interpretation of these results by constructing an explicit solution of Jackiw-Teitelboim (JT) gravity with matter: a two-dimensional Euclidean wormhole whose geometry is the double trumpet. This solution is sustained by imaginary sources for a marginal operator, without the need of a coupling between the two boundaries. As the temperature is decreased, there is a transition from a disconnected phase with two black holes to the connected wormhole phase, in qualitative agreement with the SYK observation. The expectation value of the marginal operator is an order parameter for this transition. This illustrates in a concrete setup how a Euclidean wormhole can arise from an average over field theory couplings.

Figures (8)

• Figure 1: Top: Complex spectrum of the combined Hamiltonian (\ref{['hami']}) for $N = 30$ and, $\kappa = 1$ (left) and $\kappa = 0.1$ (right). Note the different scale of each figure. Bottom: Spectral density of the real (left) and imaginary (right) part of the eigenvalues for $\kappa = 1$, $N = 34$ after average over $45$ disorder realizations. The real part looks qualitatively similar to that of a single SYK model with real couplings. Indeed, it agrees well (solid line) with the analytical prediction, see $(24)$ of garcia2017, valid everywhere except in the tail of the spectrum. The best fitting is close to the analytical prediction for $2N$ Majoranas. Regarding the imaginary part, it is characterized by a peak at zero energy followed by a suppression for small energies whose origin at present we do not understand well.
• Figure 2: Left: Free energy after ensemble average of $300$ disorder realizations for $N = 30$ Majoranas and different strengths of the imaginary part $\kappa$. Right: Free energy for $\kappa = 1$ and different $N$'s.
• Figure 3: Dependence of free energy of the Hamiltonian (\ref{['hami']}) on the number of disorder realizations $N_{dis}$ for $N = 30$ and $\kappa = 1$. A flat free energy in the low temperature limit that signals a gapped spectrum and a wormhole phase is only observed after ensemble average with a large number of disorder realizations.
• Figure 4: Left: Double trumpet geometry corresponding to the wormhole phase. Right: Hyperbolic disks corresponding to the disconnected phase with two black holes. We observe a low temperature transition between these two phases.
• Figure 5: Plot of the free energy of the wormhole and the black holes at low temperature. The solid black line represents the free energy of the system. It corresponds with that of the wormhole, characterized by a constant negative free energy, at sufficiently low (high) temperatures. As temperature increases, we observe a first order transition separating the wormhole from the two black hole phase. This qualitatively matches the behavior seen in the SYK with complex couplings. Notice that there is also another solution at smaller $b$ whose free energy is always larger. We use $k=1$ and $S_0=10^3$.