[Paper Review] Wormholes without averaging

Phil Saad, Stephen H. Shenker+2 more|arXiv (Cornell University)|Mar 31, 2021

Physics of Superconductivity and Magnetism48 references42 citations

TL;DR

The paper analyzes fixed-coupling SYK and shows that wormhole saddles persist while new “half-wormhole” saddles arise, enabling factorization without averaging; after averaging, half-wormholes vanish leaving non-factorizing wormholes.

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.

Motivation & Objective

• Investigate the fate of wormhole saddles in SYK when couplings are fixed rather than ensemble-averaged.
• Identify and characterize new saddles (half-wormholes) and their coupling sensitivity.
• Understand how multiple bulk descriptions coexist and how factorization emerges without averaging.
• Assess the role of fixed couplings in preserving or destroying factorization in decoupled systems.

Proposed method

• Use a simplified SYK reduction with one time point and fixed random couplings.
• Employ G, Sigma (collective field) techniques to represent z^2 and z^4 after averaging and without averaging.
• Compute moments such as <z^2> and <z^4> via saddle-point analysis and contour rotations.
• Introduce a sigma-tilde (sigma) and G_LR variable to represent wormhole-like correlations between replicas.
• Demonstrate the existence and interplay of wormhole saddles and non-self-averaging half-wormhole saddles.
• Show that with fixed couplings, wormholes persist with weak coupling dependence, while half-wormholes depend strongly on the couplings.

Experimental results

Research questions

• RQ1 Do wormhole saddles identified in the ensemble-averaged theory persist when couplings are fixed?
• RQ2 What new saddle points arise in the fixed-couplings setting, and how do they depend on the specific couplings?
• RQ3 Can wormhole and half-wormhole saddles together approximate fixed-coupling observables like Z_L and Z_LR?
• RQ4 How do multiple bulk descriptions coexist when not averaging, and how is factorization recovered?
• RQ5 What is the fate of half-wormholes after ensemble averaging?

Key findings

• Wormhole saddles persist under fixed couplings with weak dependence on the couplings (self-averaging).
• New saddles, termed half-wormholes, appear in fixed-coupling calculations and are strongly sensitive to the specific couplings (non-self-averaging).
• Wormholes and half-wormholes together provide an accurate semiclassical approximation for Z_L and Z_LR in the fixed-coupling theory.
• There exist multiple consistent bulk descriptions (with or without G_LR) that are equivalent in the averaged theory but differ in the fixed-coupling setting; factorization is restored by including half-wormholes in the manifestly factorized approach.
• After averaging over couplings, half-wormholes vanish, leaving non-factorizing wormholes as the residual description.
• The framework suggests these structures may extend to the full SYK model beyond the simple reductions.