Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
Micha Berkooz, Prithvi Narayan, Joan Simon
TL;DR
This work computes exact two-point functions for random operators in a family of SYK-like almost Gaussian spin glass models by mapping spin-glass moments to chord diagrams and introducing an auxiliary bulk Hilbert space with a transfer operator T. The authors develop a novel, markable chord-diagram method and a hopscotch recursion that yields a symmetric transfer matrix, whose spectrum reproduces Erdős’ eigenvalue distribution and connects to a Liouville/Schwarzian bulk action in the infrared. They derive a closed integral formula for the exact two-point function of a random operator M, including important limits such as q→1 and the Schwarzian regime, and demonstrate how bulk reconstruction is encoded in the auxiliary T-operator with a clear bulk/boundary correspondence. The results illuminate factorization properties, operator dimensions in the conformal window, and provide a concrete bridge between microscopic spin-glass dynamics and bulk gravitational descriptions in AdS2-like contexts. Overall, the paper advances exact non-perturbative tools for out-of-equilibrium correlators in disordered quantum systems and strengthens the link between chord-diagram combinatorics, bulk reconstruction, and Schwarzian gravity.
Abstract
The exact 2-point function of certain physically motivated operators in SYK-like spin glass models is computed, bypassing the Schwinger-Dyson equations. The models possess an IR low energy conformal window, but our results are exact at all time scales. The main tool developed is a new approach to the combinatorics of chord diagrams, allowing to rewrite the spin glass system using an auxiliary Hilbert space, and Hamiltonian, built on the space of open chord diagrams. We argue the latter can be interpreted as the bulk description and that it reduces to the Schwarzian action in the low energy limit.