Spectrum of the Laplace-Beltrami Operator and the Phase Structure of Causal Dynamical Triangulation

TL;DR

This paper introduces a spectral analysis method using the Laplace-Beltrami operator to distinguish phases in Causal Dynamical Triangulation quantum gravity, providing new geometric insights and potential tools for continuum limit investigation.

Contribution

It develops a novel spectral approach to characterize phases in CDT quantum gravity, extending previous diffusive process analyses with detailed geometric and dimensional information.

Findings

Different phases identified by spectral gap analysis.

Effective dimensionality varies across phases.

Spectral quantities can monitor continuum limit proximity.

Abstract

We propose a new method to characterize the different phases observed in the non-perturbative numerical approach to quantum gravity known as Causal Dynamical Triangulation. The method is based on the analysis of the eigenvalues and the eigenvectors of the Laplace-Beltrami operator computed on the triangulations: it generalizes previous works based on the analysis of diffusive processes and proves capable of providing more detailed information on the geometric properties of the triangulations. In particular, we apply the method to the analysis of spatial slices, showing that the different phases can be characterized by a new order parameter related to the presence or absence of a gap in the spectrum of the Laplace-Beltrami operator, and deriving an effective dimensionality of the slices at the different scales. We also propose quantities derived from the spectrum that could be used to monitor the running to the continuum limit around a suitable critical point in the phase diagram, if any is found.