A c = 1 phase transition in two-dimensional CDT/Horava–Lifshitz gravity?

J Ambjørn,A Görlich,J Jurkiewicz,H Zhang

doi:10.1016/j.physletb.2015.03.008

Abstract

We study matter with central charge c>1 coupled to two-dimensional (2d) quantum gravity, here represented as causal dynamical triangulations (CDT). The 2d CDT is known to provide a regularization of (Euclidean) 2d Hořava–Lifshitz quantum gravity. The matter fields are massive Gaussian fields, where the mass is used to monitor the central charge c. Decreasing the mass we observe a higher order phase transition between an effective c=0 theory and a theory where c>1. In this sense the situation is somewhat similar to that observed for “standard” dynamical triangulations (DT) which provide a regularization of 2d quantum Liouville gravity. However, the geometric phase observed for c>1 in CDT is very different from the corresponding phase observed for DT.

Highlights

Two-dimensional models of quantum gravity are useful toy models when it comes to study a number of conceptual problems related to a theory of quantum gravity: how to define diffeomorphism invariant observables, how to define distance when we at the same time integrate over geometries, etc
We analyzed spatial volume distributions n(t, m2) for causal dynamical triangulations (CDT) geometries interacting with 4 massive scalar fields
There seem to be two regimes: a small mass regime with a universal distribution identical to the distribution obtained for massless fields, i.e. for a conformal field theory with central charge c = 4, containing a blob and a stalk, and with the blob scaling with Hausdorff dimension D H = 3

Summary

Introduction

Two-dimensional models of quantum gravity are useful toy models when it comes to study a number of conceptual problems related to a theory of quantum gravity: how to define diffeomorphism invariant observables, how to define distance when we at the same time integrate over geometries, etc. 2d (Euclidean) quantum Horava–Lifshitz gravity (HLG) [7] can be solved both by using continuum methods and as a lattice theory [8] In both cases there seems to be a c = 1 barrier: the geometries for c < 1 and c > 1 look completely different.. In this paper we will study the transition from c < 1 to c > 1 in a CDT model coupled to four Gaussian matter fields. For each configuration we define t = 0 as the “center of volume” of the blob.2 In this way one can obtain the average spatial volume distribution of the blob with high accuracy: n(t) = 2 α N1−1/3 cos π t α N1/3.

The model

Small masses

Large masses

Study of the phase transition

Discussion and conclusion