Correlation Functions in the Multiple Ising Model Coupled to Gravity

TL;DR

This paper investigates the correlation functions in a coupled Ising spins and 2D gravity model, revealing diverging correlation lengths and critical exponents at phase transitions, with both toy and full models analyzed.

Contribution

It introduces a detailed analysis of correlation functions in the Ising model coupled to 2D gravity, including a toy model and the full model, and predicts critical exponents at phase transitions.

Findings

Two-point functions exhibit diverging length scales.

Correlation functions decay exponentially in the full model.

Critical exponents are predicted at phase transitions.

Abstract

The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar phi-cubed graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model.