Discrete JT gravity as an Ising model

TL;DR

This paper demonstrates that discrete JT gravity on a hyperbolic tiling corresponds to an Ising model with a topological constraint, providing insights into holographic dualities and discretized gravity models.

Contribution

It establishes a novel correspondence between discrete JT gravity and an Ising model with a topological constraint, exploring the classical limit and finite truncations.

Findings

The Ising model encodes JT gravity via a topological domain wall.

Classical JT gravity corresponds to the low-temperature limit of the Ising model.

Finite hyperbolic lattice truncations show a regime where the map between theories is valid.

Abstract

Inspired by the program of discrete holography, we show that Jackiw-Teitelboim (JT) gravity on a hyperbolic tiling of Euclidean AdS$_2$ gives rise to an Ising model on the dual lattice, subject to a topological constraint. The Ising model involves an asymptotic boundary condition with spins pointing opposite to the magnetic field. The topological constraint enforces a single domain wall between the spins of opposite direction, with the topology of a circle. The resolvent of JT gravity is related to the free energy of this Ising model, and the classical limit of JT gravity corresponds to the Ising low-temperature limit. We study this Ising model through a Monte Carlo approach and a mean-field approximation. For finite truncations of the infinite hyperbolic lattice, the map between both theories is only valid in a regime in which the domain wall has a finite size. For the extremal cases of large positive or negative coupling, the domain wall either shrinks to zero or touches the boundary of the lattice. This behavior is confirmed by the mean-field analysis. We expect that our results may be used as a starting point for establishing a holographic matrix model duality for discretized gravity.