Spectrum of the Dirac operator coupled to two-dimensional quantum gravity

TL;DR

This paper investigates the spectrum of the Dirac-Wilson operator for Majorana fermions coupled to two-dimensional quantum gravity, revealing phase-dependent spectral discretization and critical scaling behaviors.

Contribution

It numerically analyzes the Dirac operator spectrum on dynamical triangulations, linking spectral phenomena to phase transitions in the coupled quantum gravity-fermion system.

Findings

Discretization of eigenvalues near a specific hopping parameter value

Connection between spectral behavior and antiferromagnetic phase

Determination of critical exponents and Hausdorff dimension

Abstract

We implement fermions on dynamical random triangulation and determine numerically the spectrum of the Dirac-Wilson operator D for the system of Majorana fermions coupled to two-dimensional Euclidean quantum gravity. We study the dependence of the spectrum of the operator (epsilon D) on the hopping parameter. We find that the distributions of the lowest eigenvalues become discrete when the hopping parameter approaches the value 1/sqrt{3}. We show that this phenomenon is related to the behavior of the system in the 'antiferromagnetic' phase of the corresponding Ising model. Using finite size analysis we determine critical exponents controlling the scaling of the lowest eigenvalue of the spectrum including the Hausdorff dimension d_H and the exponent kappa which tells us how fast the pseudo-critical value of the hopping parameter approaches its infinite volume limit.