Recent results in Euclidean dynamical triangulations

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by  J. Laiho

∗
Recent results in Euclidean dynamical triangulations
J. Laiho, S. Bassler, D. Du, J. T. Neelakanta
Department of Physics, Syracuse University, Syracuse, NY, USA
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1
0 D. Coumbe †
2

n The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100
a
Copenhagen, Denmark
J

4
2

]
h WestudyaformulationoflatticegravitydefinedviaEuclideandynami-
t caltriangulations(EDT).Afterfine-tuninganon-triviallocalmeasureterm
-
p wefindevidencethatfour-dimensional,semi-classicalgeometriesarerecov-
e ered at long distance scales in the continuum limit. Furthermore, we find
h that the spectral dimension at short distance scales is consistent with 3/2,
[ a value that is also observed in the causal dynamical triangulation (CDT)

approach to quantum gravity.
1
v
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2 1. Introduction
8
6 The perturbative nonrenormalizability of gravity motivates the search
0 for nonperturbative formulations, one example of which is the asymptotic
.
1 safety scenario [1]. This scenario requires a non-trivial fixed point that the
0 renormalization group flow of gravitational couplings are attracted to at
7
high energies, making gravity effectively renormalizable when formulated
1
: nonperturbatively. Although there is currently no rigorous proof for the
v
existenceofsuchafixedpoint, theaccumulationofevidenceisnowstrongly
i
X suggestive [2–10].
r In a lattice formulation of an asymptotically safe field theory, the fixed
a
point would appear as a second-order critical point, the approach to which
would define a continuum limit. The divergent correlation length character-
istic of a second-order phase transition would allow one to take the lattice
spacing to zero while keeping observable quantities fixed in physical units.
∗ Presented at the 3rd conference of the Polish society on relativity
† Speaker
(1)
2 proceedings˙psor printed on January 25, 2017
Euclidean dynamical triangulations (EDT) is a particularly simple for-
mulation of lattice gravity, and one that has already proved successful in
two dimensions [11]. However, early EDT simulations in four dimensions
encountered a number of problems. In particular, the parameter space was
foundtoconsistofonlytwophases,neitherofwhichresembledsemi-classical
general relativity in four dimensions [12–16]. Moreover, the phase transi-
tion separating the two phases was found to be first order, making the
existence of a continuum limit unlikely [17, 18]. The problems encoun-
tered in EDT caused Ambjorn and Loll to introduce a causality constraint,
whichdistinguishesbetweenspace-likeandtime-likelinksonthelatticeand
thereby permits an explicit foliation of the lattice into space-like hypersur-
faces of fixed topology, a formulation known as causal dynamical triangu-
lations (CDT). It has been shown that CDT possesses a 4-dimensional de
Sitter-like phase [8]. Here, we revisit the 4-dimensional EDT approach in-
cluding a non-trivial measure term, showing that for a specific fine-tuning
of the non-trivial measure we obtain results similar to those found in the
CDT approach.
EDTdefinesaspacetimeoflocallyflatd-dimensionaltriangles,eachwith
a fixed edge length. The model described in this work uses the partition
function
 
(cid:88) 1 (cid:89)N2
ZE =  O(tj)βe−SE, (1)
C
T
T j=1
wheretheproductisoveralltriangles, andO(t )istheorderofthetriangle
j
j, i.e. the number of 4-simplices to which the triangle belongs. The term
in square brackets defines our non-trivial measure term, where β is a free
parameter. The Einstein-Regge action is given by
S = −κ N +κ N , (2)
E 2 2 4 4
where κ and κ are related to the bare Newton’s constant and the cosmo-
2 4
logical constant, respectively. κ is tuned to its critical value such that an
4
infinite volume limit can be taken [13], leaving a 2-dimensional parameter
space spanned by κ and β.
2
The parameter space of our model is depicted schematically in Fig 1.
Numerical simulations have established that the transition line AB is first
order and the line CD is a higher-order transition or analytic crossover. In
this work we determine the Hausdorff and spectral dimension close to the
transition line AB.
proceedings˙psor printed on January 25, 2017 3
β
A
κ
2
Branched
Polymer
C
Phase
B
Collapsed Crinkled
Phase Region
D
Fig.1: A schematic of the EDT phase diagram as a function of κ and β.
2
2. Global Hausdorff dimension
The Hausdorff dimension generalises the definition of dimension to non-
integer values and can be used to characterise a fractal geometry. We deter-
mine the Hausdorff dimension of our ensembles by studying the finite-size
scaling of the three-volume correlator C (δ) introduced in Ref. [19], where
N4
(cid:88)t (cid:10)Nshell(τ)Nshell(τ +δ)(cid:11)
C (δ) = 4 4 . (3)
N4 N2
τ=1 4
Nshell(τ) is the number of 4-simplices within a spherical shell one 4-simplex
4
thick at a geodesic distance τ from a randomly chosen origin, and t is the
maximum number of shells in the triangulation. N is the total number
4
of 4-simplices and the normalization of the correlator is chosen such that
(cid:80)t−1 C (δ) = 1. We rescale δ via x = δ/N1/DH, which allows us to deter-
δ=0 N4 4 (cid:16) (cid:17)
mine D as the value for which c (x) = N1/DHC δ/N1/DH becomes
H N4 4 N4 4
independent of N .
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Figure 2 shows the correlator c (x) for lattice volumes of 4K, 8K and
N4
16K four-simplices on the transition line AB for β = 0. We find that the
overlap between the curves is maximised for D = 4.1 ± 0.3, providing
H
strong evidence that the Hausdorff dimension close to the transition line
AB is consistent with four.
4 proceedings˙psor printed on January 25, 2017
Fig.2: Scaling of the volume-volume distribution as a function of the rescaled
variable x=δ/N1/DH using lattice volumes of 4K, 8K and 16K four-simplices.
4
3. Spectral dimension
The spectral dimension D defines the effective dimension of a fractal
S
geometry via a diffusion process. D is related to the probability of return
S
P for a random walk over an ensemble of triangulations after σ diffusion
r
steps, and is defined via
dlog(cid:104)P (σ)(cid:105)
r
D (σ) = −2 . (4)
S
dlogσ
Assuming the fit function D (σ) = a− b we obtain a large distance
S c+σ
spectral dimension in the range D = 2.7−3.3 [20], which is inconsistent
S
with 4-dimensional semi-classical general relativity. However, this discrep-
ancy may be due to finite volume or discretisation effects associated with
the lattice simulations. In order to investigate whether this is the case we
consider an additional extrapolation of D (∞) of the form
S
1
D (∞) = c +c +c a2, (5)
S 0 1 2
V
where c is a fit parameter, V is the volume and a the lattice spacing. This
i
particular ansatz is motivated by the fact that the data points are linear
in 1/V and a2. Extrapolation to the continuum and infinite volume limit
gives D (∞) = 3.94±0.16 and D (0) = 1.44±0.19, as shown in Fig. 3a
S S
REFERENCES 5
and Fig. 3b, respectively. A value of D (0) consistent with 3/2 may have
S
important implications for the asymptotic safety scenario [21].
(a) (b)
Fig.3: The large (a) and small (b) distance scale spectral dimension D as
S
a function of inverse lattice volume for 3 different β values, including an
extrapolation to the infinite volume and continuum limit.
4. Discussion and conclusions
In this work we determine the Hausdorff and spectral dimension for a
specific fine-tuning of the bare coupling constants in Euclidean dynamical
triangulations (EDT). Using a finite-size scaling analysis we determine the
Hausdorff dimension to be D = 4.1 ± 0.3 on the transition line AB for
H
β = 0, which is consistent with 4-dimensional general relativity and CDT
results [8]. Furthermore, by applying an additional extrapolation to the
continuumandinfinitevolumelimitswefindalargescalespectraldimension
of D (σ) = 3.94±0.16 and a small distance value of D (σ) = 1.44±0.19,
S S
results that are also similar to those reported in CDT [22, 23].
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