The Ising Model on a Dynamically Triangulated Disk with a Boundary Magnetic Field

TL;DR

This paper uses Monte Carlo simulations to explore phase transitions and boundary behaviors in a dynamically triangulated disk with Ising spins, revealing critical exponents and phase boundaries related to 2D quantum gravity.

Contribution

It introduces a detailed numerical study of a boundary magnetic field effect on a triangulated disk with Ising spins, connecting results to continuum quantum gravity predictions.

Findings

Identified three phases with distinct boundary growth behaviors.

Determined critical exponents for the magnetic phase transition.

Mapped phase boundaries including a tricritical point.

Abstract

We use Monte Carlo simulations to study a dynamically triangulated disk with Ising spins on the vertices and a boundary magnetic field. For the case of zero magnetic field we show that the model possesses three phases. For one of these the boundary length grows linearly with disk area, while the other two phases are characterized by a boundary whose size is on the order of the cut-off. A line of continuous magnetic transitions separates the two small boundary phases. We determine the critical exponents of the continuous magnetic phase transition and relate them to predictions from continuum 2-d quantum gravity. This line of continuous transitions appears to terminate on a line of discontinuous phase transitions dividing the small boundary phases from the large boundary phase. We examine the scaling of bulk magnetization and boundary magnetization as a function of boundary magnetic field in the vicinity of this tricritical point.