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

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.

32 pages, latex

References in corpus (9)

• From Laurent Series to Exact Meromorphic Solutions: the Kawahara equation
• Quantum geometry of 2d gravity coupled to unitary matter
• Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian
• Simulations of Dynamically Triangulated Gravity -- an Algorithm for Arbitrary Dimension
• Two Ising Models Coupled to 2-Dimensional Gravity
• Dynamical Triangulations, a Gateway to Quantum Gravity ?
• A geometric approach to free variable loop equations in discretized theories of 2D gravity
• The Ising model with a boundary magnetic field on a random surface
• Boundary fields and renormalization group flow in the two-matrix model