Large N matrices from a nonlocal spin system

TL;DR

This paper demonstrates how large N matrices, crucial for models of emergent spacetime, can arise from a simple nonlocal spin system, with analytical and numerical evidence supporting the emergence and behavior of the matrix saddle.

Contribution

It introduces a nonlocal spin model that exhibits emergent large N matrices, providing a new perspective on how spacetime models can arise from quantum spins.

Findings

Large N matrices emerge from a nonlocal spin model.

The matrix saddle dominates at high temperatures and disappears below a critical temperature.

Monte Carlo simulations confirm the analytical predictions.

Abstract

Large N matrices underpin the best understood models of emergent spacetime. We suggest that large N matrices can themselves be emergent from simple quantum mechanical spin models with finite dimensional Hilbert spaces. We exhibit the emergence of large N matrices in a nonlocal statistical physics model of order N^2 Ising spins. The spin partition function is shown to admit a large N saddle described by a matrix integral, which we solve. The matrix saddle is dominant at high temperatures, metastable at intermediate temperatures and ceases to exist below a critical order one temperature. The matrix saddle is disordered in a sense we make precise and competes with ordered low energy states. We verify our analytic results by Monte Carlo simulation of the spin system.