Physics Letters B 743 (2015) 435–439
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Physics Letters B
www.elsevier.com/locate/physletb
A c =1 phase transition in two-dimensional
CDT/Horava–Lifshitz
gravity?
J. Ambjørn
a,b
, A. Görlich
a,c,∗
, J. Jurkiewicz
c
, H. Zhang
c
a
The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
b
Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radbaud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
c
Institute of Physics, Jagiellonian University, Lojasiewicza 11, 30-348 Krakow, Poland
a r t i c l e i n f o a b s t r a c t
Article history:
Received
30 December 2014
Received
in revised form 2 March 2015
Accepted
5 March 2015
Available
online 9 March 2015
Editor:
L. Alvarez-Gaumé
Keywords:
Quantum
gravity
Ho
ˇ
ra
va–Lifshitz gravity
Causal
dynamical triangulation
Monte
Carlo simulations
We study matter with central charge c > 1coupled 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 Ho
ˇ
rava–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 = 0theory 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 > 1in CDT
is very different from the corresponding phase observed for DT.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Two-dimensional models of quantum gravity are useful toy
models when it comes to study a number of conceptual prob-
lems
related to a theory of quantum gravity: how to define dif-
feomorphism
invariant observables, how to define distance when
we at the same time integrate over geometries, etc. Some two-
dimensional
models have the further advantage that they can be
solved analytically both as continuum quantum field theories and
as regularized “lattice” theories. Quantum Liouville gravity (2d Eu-
clidean
quantum gravity) can be solved as a conformal field the-
ory [1–3] and
also using dynamical triangulations (DT) [4–6]. Sim-
ilarly
2d (Euclidean) quantum Ho
ˇ
rava–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 =1barrier: the ge-
ometries
for c < 1 and c > 1 look completely different.
1
However,
it has not been easy to study the transition in either of the cases
*
Corresponding author at: The Niels Bohr Institute, Copenhagen University, Bleg-
damsvej 17,
DK-2100 Copenhagen Ø, Denmark.
E-mail
addresses: ambjorn@nbi.dk (J. Ambjørn), goerlich@nbi.dk (A. Görlich),
jerzy.jurkiewicz@uj.edu.pl (J. Jurkiewicz), zhang@th.if.uj.edu.pl (H. Zhang).
1
Whenever we talk about a field theory with central charge c coupled to quan-
tum
geometry, the central charge refers to the central charge of the field theory
defined in flat spacetime, before it is coupled to the fluctuating geometry.
since no analytic solutions exist for c > 1 and since it is difficult to
vary c continuously in numerical simulations.
There
exist many numerical studies (and a few analytic stud-
ies)
of DT coupled to matter in the c > 1region (for a partial list
see [9]), and a few numerical studies of CDT coupled to matter in
the same region [10,11]. In this paper we will study the transi-
tion
from c < 1to c > 1in a CDT model coupled to four Gaussian
matter fields. In order to be able to interpolate between the two
regimes we introduce a mass for the Gaussian fields. When the
mass is large (of the order of the inverse lattice spacing) we ex-
pect
the Gaussian fields to decouple from the geometry (when
we work with Euclidean signature geometries). The geometry will
then be that of pure 2d HLG, corresponding to c = 0. If the mass
is zero the Gaussian fields will represent a conformal field theory
with c = 4. We have already studied this system numerically [11]
and
we observed a change of the geometry compared to the c = 0
case.
Decreasing the mass will bring us from c = 0to c = 4. On the
way we will observe a phase transition between the two geometric
regimes.
Let
us briefly describe what has already been observed before
the present study. The numerical studies of (1 + 1)-dimensional
CDT
are conducted using an (Euclidean) spacetime with topol-
ogy
S
1
× S
1
. In the original formulation of the CDT model in
1 +1dimensions[12] the geometry is represented by a discretized
spacetime built of triangles. The vertices of triangles are located at
integer times, with two vertices at a time t and one at t ±1. Spatial
http://dx.doi.org/10.1016/j.physletb.2015.03.008
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.