Wormholes without averaging

TL;DR

The paper tackles the factorization puzzle in the SYK model by analyzing fixed couplings without ensemble averaging, revealing that wormhole saddles persist while new non-self-averaging half-wormhole saddles arise. The authors show that the combined wormhole and half-wormhole contributions reproduce factorization in a semiclassical approximation and explain why half-wormholes vanish under averaging. They propose a general framework to extend these ideas to full SYK via a half-saddle synthesis of $\langle \mathscr{Z}^2\rangle$ saddles and discuss implications for holography and bulk descriptions of factorization. Overall, the work clarifies how non-self-averaging structures can restore factorization in fixed-coupling theories and lays groundwork for understanding their bulk duals in more realistic models.

Abstract

After averaging over fermion couplings, SYK has a collective field description that sometimes has "wormhole" solutions. We study the fate of these wormholes when the couplings are fixed. Working mainly in a simple model, we find that the wormhole saddles persist, but that new saddles also appear elsewhere in the integration space -- "half-wormholes." The wormhole contributions depend only weakly on the specific choice of couplings, while the half-wormhole contributions are strongly sensitive. The half-wormholes are crucial for factorization of decoupled systems with fixed couplings, but they vanish after averaging, leaving the non-factorizing wormhole behind.

Figures (3)

• Figure 1: We plot the complex $\sigma$ plane. Outside the blue scalloped curve, the trivial saddle point dominates, and $\Phi(\sigma)$ is self-averaging. The black dots are the locations of the "wormhole" saddle points. They are in the self-averaging region: wormholes persist.
• Figure 2: We plot six different samples of the integrand for $z^2$, namely $\Phi(\sigma) \Psi(\sigma)$. Solid lines with different colors denote different samples. The black dashed line is the exact RMS value of the integrand using (\ref{['exactphi2']}), and the red dashed line is the exact averaged value, using (\ref{['averagedPhi']}). Here $N=40$, $q=2$. The wormhole saddle for $\langle z^2\rangle$ is at a value $\sigma = 1$, which is not within the self-averaging region.
• Figure 3: The solid blue curve is the logarithm of the RMS value of the integrand along the ray $\sigma = r e^{\frac{\mathrm{i} \pi}{4}}$, namely $\frac{1}{N}\log(\Phi_{\text{rms}}(e^{\frac{\mathrm{i}\pi}{4}}r)^2) - \frac{3}{4}r^{\frac{4}{3}}$. The dashed red curve is the logarithm of the averaged value of this integrand, namely $\frac{1}{N}\log(\Phi_{\text{mean}}(e^{\frac{\mathrm{i}\pi}{4}}r)^2) - \frac{3}{4}r^{\frac{4}{3}}$. The wormhole saddle point is at $r = 1$, and the half-wormhole is at $r = 0$. For the plot we took $N = 100$ and $q = 4$. (We don't know how to make a plot with samples for $q > 2$ because $\Phi(\sigma)$ seems to be intractable.)