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

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.