The Ising model with a boundary magnetic field on a random surface

TL;DR

This paper analyzes the critical Ising model on a randomly triangulated disk with a boundary magnetic field, revealing how boundary effects influence bulk magnetization and boundary behavior in a quantum gravity context.

Contribution

It provides an exact analytic expression for boundary magnetization as a function of boundary field h, connecting boundary effects to geometric fluctuations in a quantum gravity setting.

Findings

Bulk magnetization decreases with increasing boundary field h.

The disk partition function can be expressed as a function of an effective boundary length and bulk area.

The boundary magnetization is linear near h=0, indicating finite magnetic susceptibility.

Abstract

The bulk and boundary magnetizations are calculated for the critical Ising model on a randomly triangulated disk in the presence of a boundary magnetic field h. In the continuum limit this model corresponds to a c = 1/2 conformal field theory coupled to 2D quantum gravity, with a boundary term breaking conformal invariance. It is found that as h increases, the average magnetization of a bulk spin decreases, an effect that is explained in terms of fluctuations of the geometry. By introducing an $h$-dependent rescaling factor, the disk partition function and bulk magnetization can be expressed as functions of an effective boundary length and bulk area with no further dependence on h, except that the bulk magnetization is discontinuous and vanishes at h = 0. These results suggest that just as in flat space, the boundary field generates a renormalization group flow towards h = \infty. An exact analytic expression for the boundary magnetization as a function of $h$ is linear near h = 0, leading to a finite nonzero magnetic susceptibility at the critical temperature.