Spectral Methods in Causal Dynamical Triangulations

Dipartimento di Fisica Enrico Fermi

Corso di Dottorato in Fisica

Spectral Methods in Causal Dynamical

Triangulations

Author

Giuseppe Clemente

Supervisor

Prof. Massimo D’Elia

PhD Thesis

Spectral Methods in Causal Dynamical Triangulations

Giuseppe Clemente

Abstract

Different aspects of the problem of formulating a consistent and predictive theory of Quantum Gravity are still open. Nevertheless, in the last few decades, many approaches have been proposed and the research on quantum gravity is now proceeding by exploring connections and obtaining new insights from the cross-fertilization between them. A fam-ily of approaches is drawing more and more interest recently: these are based on usual formulations of gravity as the Einstein-Hilbert theory or generalizations, but the mecha-nism which would make renormalization possible relies on the existence of a nonpertur-bative point in the space of couplings where the theory is well defined even at arbitrarily small scales (ultraviolet), regime which seems inaccessible to standard perturbative tech-niques. The occurrence of such non-trivial point in the space of parameters is called the asymptotic safety scenario, which can be investigated by means of different techniques, such as functional renormalization group approaches and numerical approaches based on Monte Carlo integration of the path-integral over geometries. In the latter class belongs the approach known as Causal Dynamical Triangulations (CDT), for which the geome-tries are approximated by triangulations (also called simplicial manifolds), i.e., manifolds composed of elementary flat building blocks of finite size (simplexes) and with a causal condition of global hyperbolicity that enforces a globally well-defined cosmological time. In the CDT approach, where the path-integral is formulated in terms of a Monte Carlo sampling of configurations from an appropriate statistical distribution, asymptotic safety can be tested by searching for a point in the phase diagram of the equivalent statistical system where a second-order transition with diverging correlation length occurs, and for which the continuum limit can be investigated. In literature, there is strong support for the existence of two second-order lines in the phase diagram, but whether one of these points corresponds to a continuum theory of quantum gravity consistent and has the expected semiclassical behavior has still to be established.

The first part of this thesis deals with the problem of defining useful observables for the study of critical properties in CDT, for which we present and investigate new tech-niques based on the analysis of eigenvalues and eigenvectors of the Laplace–Beltrami

(LB) operator on simplicial manifolds. Indeed, the solutions to the LB operator eigen-problem define what is the Fourier transform on the manifold under investigation, and allow us to classify hierarchically the characteristic length scales of the geometries, in particular their large scale (relevant) properties.

We start by considering the spectrum of dual graph representation of triangulations. This representation encodes information about the adjacency relations between simplexes, and, from these relations, one can build the Laplace matrix, which acts as an approxima-tion to the LB operator. From the spectrum of the Laplace matrix we can extract some useful quantities; in particular, the smallest non-zero eigenvalue λ1, also called spectral

gapgives us information about the largest characteristic length and connectivity proper-ties of the geometries, while the effective dimension dEFF tells us how the characteristic

lengths change when observed at different scales. By applying the spectral graph analysis to the spatial slices of CDT triangulations, we first identify the spectral characteristics of the four phases of the phase diagram and then we investigate the critical behavior of the transition line between the so-called Cb and CdS phases (a promising candidate for

a consistent continuum limit) by employing the spectral gap as order parameter for the transition, vanishing only in the CdS phase.

However, in the effort of generalizing the spectral graph analysis to full-fledged four-dimensional causal dynamical triangulations, we run into a couple of issues with the dual graph representation. We analyze the origin of these issues, and propose, as solution, an-other representation for the LB operator, coming from the framework of Finite Element Methods (FEM) and based on a weak formulation of the eigenproblem. Using FEM, which come backed up by theorems guaranteeing convergence, it is possible to set up a strategy, called refinement, that allows us to approximate with arbitrary accuracy the ex-act spectrum of the infinite-dimensional LB operator, while this strategy is not available for dual graphs (except in dimensions two). We compare the two methods first on test geometries and then on spatial slices of CDT configurations that had been already inves-tigated using dual graphs, and we find significant discrepancies in the estimates of both the effective dimension in the CdS phase and the critical index of the Cb-CdS transition.

The second part of this thesis, in the spirit of pushing toward a realistic connection with phenomenology, involves the minimal coupling of compact gauge fields of the Yang– Mills type with CDT, where link variables are placed in the edges of the dual graph. Our setup us is general, but we focus in particular to an exploratory numerical investigation of the system of two-dimensional CDT minimally coupled to gauge fields with either Up1q or S Up2q gauge groups, providing an explicit construction for the Markov chain moves. By

studying the effects of gauge fields on gravity using as observables the total volume and the correlation length of the distribution of spatial volumes along spatial slices (volume profiles) we find that the backreaction of fields on the geometry trivially amounts to a shift in the bare parameter associated to the cosmological constant. The effects of gravity on the gauge fields, investigated by using the correlation length of flux-related quantities called torelons, turn out to be less trivial. We also study the θ-dependence of the Up1q case by analyzing the topological charge, which behaves like a winding number for the gauge fields, and the topological susceptibility. Incidentally, we discover that the problem of critical slowing down, affecting different lattice field theories in the continuum limit, seems mitigated by orders of magnitude when one consider locally variable geometries, suggesting possible applications also in different contexts.

Ringraziamenti

Innanzitutto devo ringraziare Massimo, per tanti motivi: non solo come supervisor perch´e mi hai offerto tante opportunit`a per gestire il mio lavoro in modo libero, indicandomi sempre la via negli eventuali momenti di smarrimento, ma anche come amico per il tempo passato assieme, tra pizzate, giocate a calcetto, palline che fai smarrire sopra gli armadi dell’ufficio e altri momenti che ricordo sempre con piacere.

Se il clima del gruppo dei tuoi tesisti e dottorandi `e cos`ı spassoso `e anche grazie al tuo modo di essere. . . e per le prese in giro collettive a Marco.

Marcolino (cit. Loris), anche te devo ringraziare, insieme a tutto il gruppo di stanza 170 e compagni: Andrea, Antonio, Ciccio, Claudio-i, Claudio-o, Dadde, Gaia, Lorenzo e Umberto, e tutti i compagni di pranzo (scusate se la lista non `e completa). Abbiamo passato assieme solo pochi anni, ma di cose ne sono successe, e spero di avervi lasciato un segno positivo come quello che avete lasciato voi a me.

Ringrazio i miei amici sin dai primissimi anni a Pisa e che sono stati al mio fianco: Andrea P., Ivano, Marco I., Nicol`o. Spero che di rivedervi spesso, anche se le nostre strade si dividono per me siete come dei fratelli acquisiti.

Non posso fare a meno di ringraziare i miei amici di Palermo, che nonostante passassi la stragrande maggiorparte del tempo a Pisa non hanno mai smesso di accogliermi con calore e affetto quando scendevo. Scusatemi tanto, lo so, sono un pessimo amico. . .

Saluto con affetto anche i laureandi, ormai laureati, coi quali mi sono trovato bene a lavorare assieme, e tutti i membri del gruppo di gravit`a quantistica comparata. Chiunque, in questo momento di stanchezza, potrei avere dimenticato di menzionare, vi prego, per-donatemi.

Infine, ma non per ordine di importanza, ringrazio la mia famiglia, che mi ha sempre lasciato libero di portare avanti i miei interessi.

Contents

1 Introduction 9

1.0.1 Structure of the thesis. . . 13

2 Overview of Causal Dynamical Triangulations 15 2.1 Path-integral . . . 15

2.2 Regge discretization. . . 17

2.2.1 Deficit angle as a measure of local curvature . . . 18

2.2.2 The Einstein-Hilbert action in Regge form. . . 20

2.3 Wick rotation . . . 24

2.3.1 Volume fixing. . . 26

2.4 Phase diagram of Causal Dynamical Triangulations . . . 27

3 Spectral Methods and Spectral Graphs Analysis 31 3.1 General concepts on spectral methods . . . 32

3.1.1 Interpretation of Laplace–Beltrami eigenvalues and eigenvectors . 33 3.1.2 Weyl’s law and effective dimension . . . 34

3.2 Spectral graph analysis . . . 35

3.2.1 An example: the spectrum of a 3-torus graph . . . 37

3.3 Numerical results of spectral graph analysis applied to spatial slices . . . 39

3.3.1 Spectral graph characterization of the phases in the CDT phase diagram . . . 40

3.3.2 Spectral features of the Cbphase and Cb-CdS transition . . . 44

3.4 Spectral visualization of dual graphs . . . 49

3.4.1 Laplace embedding . . . 50

3.4.2 Spring embedding . . . 52

3.5 Issues with the graph formulations . . . 54

3.5.1 Loss of metric information . . . 55 7

Contents

3.5.2 Overestimation of geodesic distance . . . 56

4 Spectral Analysis via Finite Element Methods 59 4.1 Weak formulation . . . 60

4.2 From weak form to finite elements . . . 62

4.2.1 Computation of the L and M matrix elements for generic simpli-cial manifolds . . . 64

4.2.2 FEM refinements . . . 67

4.3 Comparison between FEM and dual graph methods on test geometries . . 71

4.3.1 Issues of the dual graph method: discussion and a toy model . . . 75

4.4 Numerical results on spatial slices . . . 78

4.4.1 Large-scale spectral dimension of spatial slices in CdS phase . . . 80

4.4.2 Critical index of Cb-CdS transition . . . 84

5 Minimal Coupling to Compact Gauge Theory in 2D CDT 87 5.1 Minimal coupling in the dual graph. . . 88

5.1.1 Yang–Mills theories on a flat lattice . . . 88

5.1.2 Yang–Mills action coupled to CDT triangulation . . . 89

5.2 The algorithm . . . 92 5.2.1 Detailed balance . . . 93 5.3 Numerical results . . . 100 5.3.1 Phase diagram . . . 102 5.3.2 Gauge observables . . . 105 6 Conclusions 113

Chapter 1

Introduction

The quest for a self-consistent Quantum Theory of Gravity is still far from being settled. Standard Quantum Field Theory fails in providing solutions of the Einstein-Hilbert ac-tion which are renormalizable from a perturbative point of view, i.e. with ultraviolet (UV) divergences reabsorbed at all orders in the coupling expansion by adding a finite num-ber of counterterms [1]. On top of that, from the experimental point of view, quantum

gravitational phenomenology is difficult to test, since, by dimensional considerations, the full quantum regime is thought to take place near the scale of the Planck energy EPl ” mPlc2 ”

a

~c{Gc2 » 1.22 ˆ 1019GeV. Nevertheless, some quantum effects on phenomenology could still be observed in the next future, for example from cosmological (e.g., B-mode graviton polarization) or gravity-induced (e.g., decoherence and entangle-ment) observations, which would be useful to set constraints on theories of Quantum Gravity [2,3]. In the perturbative setting, it is still possible to renormalize some

general-ization of the Einstein-Hilbert action, for example by introducing higher-derivative metric terms [4–6]; the resulting theories, however, are either affected by non-unitarity induced by the presence of ghosts, or require a modification to the quantization prescription, which is still matter of debates.

Another possibility is to search for non-perturbative solutions to the renormalization of the Einstein-Hilbert theory or generalizations: a promising approach is represented by the so-called asymptotic safety program [7]. The program is rooted in the renormalization

group (RG) framework: the main idea is to find a non-perturbative ultraviolet (UV) fixed 9

Chapter 1

point in parameter space, with an RG-flow line stemming from it and reproducing the theory of gravity at lower energy scales.

Consistent progress in this direction has been achieved by analytic studies of the RG-flow [8–11]. A complementary approach is numerical: one considers a discretization of

the Euclidean path integral of the theory in configuration space, suitable to be studied by Monte-Carlo simulations, and looks for possible critical points, i.e. for values of the bare parameters where the correlation length, measured in units of the elementary discretiza-tion scale, diverges.

In this class we find Causal Dynamical Triangulations (CDT) [12–19], a numerical

Monte-Carlo approach to Quantum Gravity where the path-integral over (Lorentzian) manifolds is approximated by a sum over simplicial manifolds, also called triangulations, which are collections of elementary building blocks of flat space-time called simplexes, glued by their faces in such a way to form a piecewise-flat manifold. The action SCDTrT s

employed in the Monte-Carlo sampling is the Einstein-Hilbert action, appropriately dis-cretized for triangulations (T ) according to the Regge formalism [20]. Another relevant

feature of CDT is the causal condition of global hyperbolicity [21], which allows the

ex-istence of a cosmological time and enforces a foliated structure on the space-time, where spatial slices are characterized by a fixed (compact) topology. This causal structure allows to identify links (1-simplexes) in two well defined classes: timelike and spacelike; this, in turn, makes it possible to define unambiguously an analytic continuation from Lorentzian to Euclidean signature, which is essential for the path-integrals to be correctly computed by Markov processes using Monte-Carlo techniques, since triangulations can be sampled according to a distribution with weight 9e´SCDT_.

The physics in CDT has proved to be sensibly different from the one observed in the parallel approach called Dynamical Triangulations [22–26], similar in every aspect except

for the absence of any causal condition. Moreover, in two-dimensional gravity, where computations can be usually carried fully analytically, it has been shown that the causal condition present in the CDT case makes it differ also from other formulations of quantum gravity in agreement with each other, both in the discretized setting, like matrix models and Polyakov formulations [27], and in continuous one like Liouville gravity [28, 29],

Chapter 1

unless topology changes and the creation of the so-called baby universes are admitted in the CDT path-integral [30]. Another notable regularization proposed recently, is based on

a first-order formulation (Cartan gravity) augmented with a Yang–Mills term and treated with tensor network techniques [31]; even if apparently very different from the aforemen-tioned two-dimensional formulations, this approach seems in agreement with them (and therefore not with CDT), since it appears to be formally connected to both the Liouville and Polyakov form. From these two-dimensional considerations, we can expect that the causal condition present in CDT is quite relevant also in higher dimensions, and makes it somewhat distinct and unique from other related but physically dissimilar formalisms.

The ultimate goal of the CDT program is to show that in the phase diagram of the theory there exists a second order critical point where the some definition of autocorrela-tion length diverges, and that the critical region around such point describes a theory of quantum gravity consistent with our current understanding of the expected semiclassical behavior. An updated sketch of the phase diagram is shown in Figure 2.4 for simula-tions with slices with toroidal topology (T3_{), even if the qualitative features seem to be}

independent on the slice topology [32,33].

In order to characterize the phase diagram of the pure-gravity theory, in particular the critical properties of the system near second-order transition lines (candidate points for extracting a continuum theory), we would need, in principle, a complete set of physically descriptive observables which could probe every relevant feature of the geometries under investigation. The quest for meaningful observables has been overlooked for some time, and it has only recently gained more attention by the CDT community.

The main aim of this thesis is to describe our works in CDT, where we introduced new techniques based on spectral methods, i.e., the analysis of eigenvalues and eigenvectors of the Laplace-Beltrami operator on (simplicial) manifolds, which make it possible to define a hierarchy of new pure-gravity observables, and allow us to extract large-scale behaviors from any local observable. Indeed, the main idea behind spectral methods on (simplicial) manifolds is to identify and organize the characteristic length scales of geometries from the information contained in the spectrum, and the shapes of the wave (or diffusion) modes from the information contained in the corresponding eigenvector.

Chapter 1

In our first attempts [34, 35] we investigated the characteristic of the different phases and some critical regions in the phase diagram by analyzing the spectrum of the Laplace matrix on graphs dual to CDT configurations (spatial slices in particular); this works led us to identify in the spectral gap λ1(the smallest non-zero eigenvalue) a parameter order for

the Cb-CdS transition (accompanied by higher-order eigenvalues), and to the introduction

of a new definition of running dimension, which we dubbed effective dimension, and which, as the name suggests, describes the effective behavior of geometries at different length-scales.

However, our efforts to properly extend the spectral analysis to full four-dimensional triangulations made us realize the limitations of the dual graph representation of simplicial manifolds. First of all, we recognized that undirected graphs, in the way they have been used for defining different observables in CDT, like the Hausdorff dimension [36, 37], the spectral dimension [38–40] and the quantum Ricci curvature [41,42], threat all nodes

and edges on equal footing, neglecting possible metric differences between simplexes (due to the different square-lengths between spacelike and timelike links in the triangu-lation). Second, but not by importance, we observed that the effects of approximating the Laplace-Beltrami operator on a simplicial manifold, acting on an infinite-dimensional Sobolev space, using the finite-dimensional Laplace matrix of the corresponding dual graph can have non-negligible effects even on the lowest part of the spectrum (the one related to the largest scales). The main reason of this misrepresentation has to do with the fact that geodesics between different points in the simplicial manifold are sistematically overestimated by path lengths in the dual graph, in particular in the presence of regions with high negative curvature. This geodesic overestimation does not involves only spec-tral observables, but in general any observable defined on dual graphs, since it affects their large and intermediate scale behavior, and calls for a reconsideration of earlier results in literature.

In order to manage these issues, we sought new ways to represent, with arbitrary ac-curacy at least on larger scales, the infinite-dimensional Sobolev space of scalar functions on simplicial manifolds; this led us to propose and investigate in [43] representations of

un-Chapter 1

like the ones obtained with dual graphs (except in two dimensions), are guaranteed to converge to the true solutions of full Sobolev space of functions.

Besides the investigations in the direction of pure-gravity observables, in [44] we

treated the problem of minimally coupling gauge theories to CDT. This is important for the search for a continuum and renormalized quantum theory of gravity, not only because quantum fields can change the critical properties of the phase diagram, but also because it makes possible to properly study the quantum cosmological phenomenology where the in-terplay between fields and geometry are relevant. This problem, in two dimensions, was considered also in earlier works both analytically, by locating link variables in links of the triangulation [45], and numerically, using non-compact representations of the gauge

group [46], Our setting is somewhat different: as a pilot study, we numerically inves-tigated the minimal coupling of compact Yang-Mills gauge groups, with link variables associated to edges of graphs dual to triangulations, which produces the correct counting of degrees of freedoms in dimensions higher than two, making our algorithmic strategy more general. In particular, we numerically studied the coupling of gauge groups Up1q and S Up2q in two dimensions.

1.0.1

Structure of the thesis

In Chapter 2 we will give a general overview of the Causal Dynamical Triangulations approach by introducing some concepts useful for the next chapters.

The theory of spectral methods and spectral graph methods will be discussed in Chapter3, where we will also show some numerical results obtained from the analysis of eigenvalues and eigenvectors of the Laplace matrix of graphs dual to slices of four-dimensional CDT configurations (Section3.3). In Section3.5 we will discuss the issues that we identified in the usage of the dual graph representation.

As anticipated above, these issues persuaded us to search for another representation, which resulted in the introduction of the Finite Element Method (FEM), which is the topic of Chapter4. After having given an overview of the FEM formalism, we will com-pare dual graph and FEM spectra for some test geometries (Section 4.3), stressing again how the dual graph issues can affect observables even at large scales; a comparison of

Chapter 1

the new FEM results with earlier dual graph results on spatial slices of four-dimensional CDT configuration is shown in Section4.4.

In Chapter5we will discuss the minimal coupling of gauge fields of the Yang-Mills type with CDT in the dual graph representation, firstly by introducing the simulation algorithm (Section5.2), and then by showing some numerical results of its application for the Up1q and S Up2q gauge groups on a two-dimensional CDT (Section5.3).

Chapter 2

Overview of Causal Dynamical

Triangulations

As introduced above, Causal Dynamical Triangulations (CDT) is a path-integral approach to Quantum Gravity where renormalizability of the theory is realized in a non-perturbative fashion according to the asymptotic safety scenario, suggested by Weinberg in the seven-ties [7].

In this chapter, we will give a broad outline of the main developmental steps of the CDT approach. We will start from reviewing its path-integral formulation as a quantum field theory, and then we will proceed by introducing the Regge discretization of the configuration space and the causal condition of global hyperbolicity, which allows to perform an analytic continuation from the Lorentzian to the Euclidean signature (Wick rotation). Finally, we briefly discuss about the phase diagram of CDT and some features of the different phases.

2.1

Path-integral

Non-perturbative tools are essential to study any approach in the asymptotic safety sce-nario, since the behaviour of the theory around a non-perturbative fixed point is, by def-inition, poorly represented by perturbative expansions with small couplings. Some of the tools which can be applied to undertake this challenging task are based on numerical techniques, in particular Markov Chain Monte Carlo (MCMC) algorithms, which have

2.1. Path-integral Chapter 2

already been applied with success in the investigation of non-perturbative features of sta-tistical and quantum field systems. For quantum field theories, MCMC methods typically apply within the path-integral formalism, which therefore appears to be suitable as the natural setting for the CDT approach. In the following, we will describe some informa-tion about the path-integral, which will come useful later.

Denoting, in complete generality, all the fields involved by using a single symbolΦ, and the space of field configurations by C, we can write the path-integral evaluation of the vacuum expectation value for any observable O as:

xOy “ 1 Z

ż

ż

In this form, however, the expressions in Equation (2.1) are of no use for numerical im-plementations. First of all, memory constraints forbid the management of the uncount-ably infinite dimensionality of the configuration space C, making therefore necessary to discretizethe dynamical variables in play into a finite-dimensional space, which however should be guaranteed to represent the relevant features of the original infinite-dimensional space to arbitrary accuracy (in the thermodynamical and continuum limit). The particu-lar discretization employed in CDT is described more in depth in the next section (Sec-tion2.2).

In the second instance, the term exp p_~iS q is complex-valued, and the fact that it is usually wildly oscillating through configurations makes the integrals in Equation (2.1) unfeasible to be integrated numerically with sufficient accuracy. Nevertheless, it is possi-ble, in some cases, to perform an analytic continuation of the action functional S pαq, with respect to some parameter α Ñ rα, such that the argument of the exponential becomes real1 _S

Eprαq ” ´iS prαq P R. There are many cases for which it is not simple to obtain a real and positive weight exp p´SEq which could be treated as a probability distribution;

this situation is called the sign problem2, and it is a non-deterministic polynomial-time (NP) hard problem [48]. Fortunately, as pointed out in Section2.3, the CDT action

(with-1_{The E in S}

E stands for Euclidean, as will be clear below.

2_{Many (more or less satisfying) solutions to the sign problem have been proposed over the years; one}

promising solution, which however relies on further improvements in the current technologies, is the use of quantum algorithms [47].

Chapter 2 2.2. Regge discretization

out matter fields) is free from the sign problem. This procedure, named Wick rotation when the parameter for the analytical continuations is the time coordinate, allows us to interpret the path-integral measure DrΦse´SErΦs{~ _{as a probability measure, and Z as the}

partition functionof an associated statistical system3, such that we can define a probabil-ity densprobabil-ity functional value PrΦs ” 1

Ze

´SErΦs{~ _{to any field configuration} Φ P C. After

Wick-rotation, the vacuum expectation value of any observable O, can be computed as follows: xOy “

ż

whereΦ is now interpreted as a random variable with values in C, Ppφq as the probability of the eventΦ “ φ, and Op¨q is the function on Cdiscr.associated to the observable O.

The particular discretization used in CDT is discussed in the next section.

2.2

Regge discretization

The action describing the classical theory of General Relativity with a cosmological con-stant term is the so called Einstein-Hilbert action

SEH “

1 16πG

ż

ddx ?´g pR ´ 2Λq, (2.3)

where the two parameters G andΛ are respectively the Newton constant and the cosmo-logical constant, while R is the Ricci curvature scalar.

The Einstein-Hilbert action is also the one usually adopted in CDT, but with an appropri-ate discretization.

Different discretization schemes seem possible; for example, in Numerical Relativity [49] one usually localizes the relevant degrees of freedom (spatial metric and extrinsic curva-ture) on the sites of a regular lattice, and use Cauchy methods to evolve them in physical time. However, in Monte-Carlo simulations of Field Theories, the whole time evolution constitutes a single path of the path-integral, so a discretization which represent the full spacetime as a single configuration is in order.

3_{If the time direction is compactified, Z can also be interpreted as the partition function for a}

thermody-namical system at finite temperature.

2.2. Regge discretization Chapter 2

A more appealing discretization scheme for Monte-Carlo simulations in Gravity, where the configurations are approximations to manifolds, comes from Regge calculus [20]: in

this formalism, a manifold is approximated by a set of elementary building blocks of flat spacetime called simplexes (or simplices), glued together in such a way to form a so called simplicial manifold, or more commonly, triangulation, since triangles are the highest-order simplexes in 2-dimensional manifolds (surfaces). In a general dimension d, a simplicial manifold is represented by a hierarchy of k-simplexes (k “ 0, 1, . . . , d), where the 0-simplexes are the vertices, the 1-simplexes are the links, which connects 2 vertices, the 2-simplexes are the triangles, containing 3 links and 3 vertices, and so on as depicted in Figure2.1.

Figure 2.1: All types of k-simplices up to k “ 4

The way in which all these simplexes are glued and their link-lengths encode the whole geometric information about the simplicial manifold to which they belong; in par-ticular, the distance between two physical points on a triangulation, for example localized in the interior of different d-simplexes, can be easily computed since geodesics are rep-resented by straight lines in the interior of each triangle, and the transition maps between two adjacent d-simplexes ensures a trivial parallel transport at the simplex hypersurfaces. In the following section we will discuss about how the information on the curvature is encoded in the triangulation.

2.2.1

Deficit angle as a measure of local curvature

While the flat interior of any d-simplex cannot contain any information about the cur-vature of a d-dimensional triangulation, Regge showed that the local curcur-vature can be completely encoded in the so called deficit angle around the pd ´ 2q-simplexes; this quan-tity is computed as 2π minus the angular amount by which a vector rotates when parallel

Chapter 2 2.2. Regge discretization

Figure 2.2: Relation between deficit angle and local curvature: flat (A), positive (B), negative (C).

transported in a closed loop around a given pd ´ 2q-simplex. For illustration purposes, let us consider a 2-dimensional triangulation made up of equilateral triangles (π{3 radiants per angle), where vertices are the carriers of information about the local curvature; as depicted in Figure 2.2, we can observe three types of situations depending on the num-ber nv of (equilateral) triangles around a vertex, since the deficit angle is computed as

εv “ 2π ´ nvπ{3, which is associated to a positive (nv ă 6), negative (nv ą 6) or flat

(nv “ 6) local curvature.

Since, in the following chapters, we will often have to deal with equilateral simplexes, it is convenient to define the idea of coordination number nσ for a k-simplex σ in a

d-dimensional triangulation, as the number of d-simplexes which contain σ (in the discus-sion above the coordination number for the vertex v was nv).

The sum of all the local curvature contributions from each pd ´ 2q-simplex is the total curvature of a triangulation, which will contribute to the Regge discretization of the Einstein-Hilbert action, as discussed in the following.

2.2. Regge discretization Chapter 2

2.2.2

The Einstein-Hilbert action in Regge form

The action in Equation (2.3) is written in terms of metric variables; in order recast it in the formalism of triangulations, it is convenient to separately consider its two constituents, the total curvature (first integral) and the total volume (second integral):

S rgµνs “ 1 16πG

ż

ż

The total volume term, whose coupling is the ratioΛ{G, can be simply represented, in the Regge formalism, by the sum of the volumes of all the d-simplexes:

ż

where, in general, Tpkqdenotes the whole set of k-simplexes of the triangulation T , while

Vσk denotes the k-dimensional volume of a specific k-simplex σk.

From the discussion above, the Regge form of the total curvature term can be written as the sum of all the local curvature contributions, represented as the deficit angles around pd ´ 2q-simplexes. As shown by Regge, the actual local contribution for a single simplex σd´2_{is computed as 2ε}

σd´2V_σd´2, so that the action term becomes:

ż

Therefore, replacing the expressions (2.5) and (2.6) into Equation (2.4) we obtain the so called Regge action:

SReggerT s “ 1 8πG ´ _ÿ σd´2_PT pd´2q εσd´2V_σd´2´Λ ÿ σd_PT pdq Vσd ¯ , (2.7)

or, in dimensionless form:

SReggerT s “ κ ÿ σd´2_PT pd´∈q εσd´2V_σd´2´ λ ÿ σd_PT pdq V_σd, (2.8)

Chapter 2 2.2. Regge discretization

where we have introduced the dimensionless quantities:

V_σk ” Vσk ak , κ “ ad´2 16πG, λ “ 2Λad 16πG, (2.9)

where a is an arbitrary length scale for the moment, and κ, λ replace the Newton constant and the cosmological constant as new dimensionless parameters in the action.

Apart from dimension 2, where the total curvature term in Equation (2.8) is a topologi-cal constant (proportional to the Euler characteristic χ of the surface, by the Gauss-Bonnet theorem), we have to compute the action contributions given by the deficit angles εσd´2,

which, for a generic triangulation, would involve the evaluation of dihedral angles and volumes for each (d ´ 2)-simplex. In CDT however, one can simplify this computation by choosing a specific discretization for triangulations. Indeed, triangulations in CDT are foliated4, which means that vertices lie only on spatial slices at various slice times, and d-simplexes fill the spacetime between adjacent slices (usually called slab), so that they have m vertices in a slice and d ` 1 ´ m vertices in another contiguous slice (with 1 ď m ď d).

This causal structure admits only two kinds of 1-simplices: spacelike links, connecting two vertices of the same slice, and timelike links, connecting two vertices of adjacent slices. We can then fix the links square-lengths ∆s2 so that each spacelike link would have ∆s2 “ a2, while each timelike link would have∆s2 “ ´αa2 (with α ą 0), without loss of generality5_.

With this constraints, there are only d types of d-simplices, since they are completely determined by the pair pm, d ´ m ` 1q indicating the number of vertices in two consecutive slices. In a (Lorentzian) 2-dimensional triangulation, for example, there will be only two types of 2-simplices, related by time-reversal: p1, 2q and p2, 1q, containing one spacelike link and two timelike links as shown in figure 2.3, together with the different types of simplices in three and four dimensions. This extremely simplifies computations, since, in

4_{A preferred foliation such as the one usually employed in CDT is certainly convenient, but actually not}

strictly necessary. Indeed, in Ref. [50] it is shown that also non-foliated triangulations can be used, but are more complicated and the results seem compatible with the ones using foliated triangulations.

5_{For example, any Euclidean manifold can be approximated, with arbitrary precision, using equilateral}

simplexes with small enough side length a. There is no loss of generality, but there is some loss in accuracy at a fixed number of simplices.

2.2. Regge discretization Chapter 2

Figure 2.3: From the top to the bottom, all the types of maximal simplices in 2D, 3D and 4D, modulo time reversal.

order to write the Regge action for CDT triangulations, only a limited number of distinct volumes and dihedral angles are required.

In general, angles and volumes in a Lorentzian manifold are complex valued, but the action, fortunately, turns out to be real even after analytical continuation to the Eu-clidean (more details can be found in [14]). As a concrete example, let us consider the

2-dimensional Regge action in the form of Equation (2.8). The total curvature in 2 dimen-sions is a topological invariant 2πχ, therefore, denoting by V2the common dimensionless

(Lorentzian) volume for all triangles, the action simply reads

SRegge,2DrT s “ κ 2πχ ´ λV2N2rT s, (2.10)

where N2rT s denotes the total number of triangles in the triangulation T , which

corre-sponds to the sum of p2, 1q and p1, 2q types of triangles: N2rT s “ N p2,1q

2 rT s ` N p1,2q 2 rT s.

Chapter 2 2.2. Regge discretization

The dimensionful spacetime simplex volume V2 is the same

for both p2, 1q and its time-reversed counterpart p1, 2q, and can be easily computed by using the Lorentzian form of the Pythagoras theorem, as depicted in the figure on the right

V2 “ ? 4α ` 1 4 a 2 ùñ V<sub>2</sub> “ V2 a2 “ ? 4α ` 1 4 , (2.11)

so that the dynamical part of equation (2.10) becomes SRegge,2DrT s “ ´λ

? 4α ` 1

4 N2rT s. (2.12)

Similar considerations apply in d dimensions, where it is sufficient to compute the ex-pressions of dihedral angles and volumes for all the types of simplices pm, d ´ m ` 1q from m “ 1 to m “ tpd ` 1q{2u, since the types related by time-inversion, pm, d ´ m ` 1q and pd ´ m ` 1, mq are congruent, and therefore share the same volume and dihedral angles. The sums in Equation (2.8) become then a counting over all the simplex types involved. In general, we will follow the conventions of Ref. [14], where Nk denotes the

total number of k-simplexes, N_kpm,k´m`1q denotes only the number of k-simplexes of the pm, k ´ m ` 1q type, and N_kS L denotes the number of k-simplexes with all vertices in the same slice (i.e., spacelike). These numbers are not independent in a triangulation, but are related by geometric identities. For example, one identity comes from the Euler charac-teristic χ “řd_k“0p´1qkNk, which, as already mentioned, is a topological invariant6.

Other identities can be computed by combinations of local properties of simplicial manifolds and boundary conditions; for example, in a 3-dimensional foliated triangulation with time-periodic conditions each spatial triangle is shared by two tetrahedra of type p1, 3q and p3, 1q, so that we obtain the relation N₂S L“ N

p3,1q 3

2 . In general, the d-dimensional

Regge action takes the form

SReggerT s “ÿ

σ

fσpαqNσrT s, (2.13)

6_{For 2-dimensional orientable surfaces χ “ 2p1 ´ gq, where g is called the genus of the surface, which,}

informally, corresponds to the number of its holes.

2.3. Wick rotation Chapter 2

where the sum extends over all simplex types tσu involved and the information about dihedral angles and volumes for the σ simplex type are contained in the functions fσpαq, which still have to be analytically continued from the Lorentzian (α ą 0) to the Euclidean signature (α ă 0) as we will briefly discuss in the next section. Explicit expressions for these functions can be found in Ref. [14], but they are not essential for the purposes of

this thesis.

A feature stemming from the properties of CDT triangulations, is that the topology of all spatial slices must be the same independently of their slice time, since the space-time between two spatial slices with different topologies cannot possibly be filled with d-simplexes without involving singular points having a degenerate light-cone structure. Indeed, the interior points of any d-simplex is flat, and embeds a Minkowskian light-cone, so that the transfer maps between adjacent d-simplexes forces the path of causal geodesics to be always well defined. Therefore, there are only a finite number of full spacetime topologies with compact slices and time-periodic conditions, homeomorphic to S1ˆΣ, whereΣ can be any of S3_{, S}2_{ˆ S}1_{and T}3

 S1ˆ S1ˆ S1. In principle, the path-integral over geometries should involve the sum over all contributions from distinct topologies. However, the Monte-Carlo updates in actual CDT simulations are currently implemented as local homeomorphisms, which do not provide for transitions between topologies, and make possible only simulations with fixed topology. In this thesis, our results come from simulations performed using spatial slices with spherical topology (Σ  S3), but recent works suggest that the phase diagram structure of other choices of slice topology is very similar to the spherical case [16,32,33].

2.3

Wick rotation

As mentioned above, Monte-Carlo techniques applied on quantum field theories require an analytic continuation from the Lorentzian signature to the Euclidean signature, , such that the path-integral exponent iS with integration in the real time t becomes ´SE, where

now the integration takes place in the imaginary time direction τ “ it7. This

transfor-7_{Actually, two solutions are possible, since τ “} ?

´1t “ ˘it. In quantum field theory, the choice of the sign is a matter of convention, provided that it is consistent with the pole prescription for the Feynman

Chapter 2 2.3. Wick rotation

mation should be performed continuously in the complex plane, and for this reasons it is usually called Wick rotation. Same reasoning can be applied to fσpαq functions of the Regge action in the form of Equation (2.13). Indeed, the α parameter determines the square-lengths of triangulation links∆s2 “ ´αa2(α ą 0), so that, in order to obtain the Euclidean form of the Regge action and of the triangulation, we can use it to perform the analytical continuation α ÞÑ ´γ where γ ą 0 makes the Euclidean lenght-square of timelike links positive: ∆s2 “ γa2. Notice that the length-square of spacelike links a2 is kept constant during the process.

For example, the analytic continuation applied to the 2-dimensional Regge action in (Equation (2.12)) acts as follows:

iSRegge,2Dpαq Ñ iSRegge,2Dp´γq “ ´λ a 4γ ´ 1 4 N2” ´S Regge,2D E pγq. (2.14)

It is possible now to reabsorb the γ dependence by a redefinition of the bare parameter λ Ð λ

?

4γ´1

4 (provided that γ ą 1{4), so that the Wick-rotated Regge action, used for

two-dimensional CDT simulations, takes the very simple form:

SRegge,2D_E “ λN2, (2.15)

where, from now on, the T dependence is understood. The probability weight of a generic triangulation T can then be computed as a simple count of triangles PrT s9e´λN2rT s_.

This apparent simplicity, however, hides the entropic nature of counting all the possible simplicial manifolds generated by gluing a certain number N2 of triangles. It is also

worth noticing that, for small values of λ, triangulations with arbitrary large volumes are not suppressed as they are for larger values of λ, resulting in an “explosion” of the lattice volume during two-dimensional simulations. Exact analytical results, involving the transfer matrix formulation of CDT in two dimensions, show that for simulations with λ below the critical value λc “ logp2q » 0.693, the average spacetime volume xN2y

diverges, while it is finite for λ ą λc (see Ref. [51, 52] for details). This fact will be

useful in Chapter 5, where we will show the shift in the critical value λc induced by the

propagator.

2.3. Wick rotation Chapter 2

introduction of gauge fields.

A strategy analogous to the one shown here can be done for the 4-dimensional case, which is the one we will investigate for most of the thesis. At the end of the day, after computing dihedral angles and volumes for all simplexes as functions of α, performing the Wick rotation as described above, and redefining the bare parameters in a simpler form (see Ref. [53] for details), one can write the 4-dimensional CDT action in the following

conventional form

SE “ ´k0N0` k4N4`∆pN4` N p4,1q

4 ´ 6N0q, (2.16)

where the original dependence on the γ parameter, the Newton constant and the cosmo-logical constant, has been absorbed into the new three parameters of the action∆, k0 and

k4. The parameter∆ encodes the amount of asymmetry between square-lengths of the

Eu-clidean timelike and spacelike links (∆ vanishes at γ “ 1), and represents the freedom of performing different, but equivalently possible, analytic continuations. In Equation (2.16), Notice that k4, as factor of the total volume N4, acts as the λ parameter in two-dimensions,

suppressing arbitrarily large triangulations; indeed, also in four dimensions it is observed a critical value k4 “ k4cpk0, ∆q below which volume averages diverge. It is then possible,

for fixed values of k0 and∆, to use k4 as a parameter that we can tune in order to make

the average volume xN4y of simulations fluctuate around a target volume V as explained

in the next section.

2.3.1

Volume fixing

In Monte-Carlo simulations of quantum field theories, as long as the total volume is finite, finite-volume effects have to be taken into account. Usually, this is done by making a finite size scalinganalysis, which consists in studying the dependence of observables from the total volume V and then extrapolate the results to the thermodynamical limit. However, unlike typical Monte-Carlo simulations, the size of the CDT lattice can change during the simulation. In order to apply the finite size scaling technique also in CDT, we can make the total volume of configurations fluctuate around a certain target volume V by

Chapter 2 2.4. Phase diagram of Causal Dynamical Triangulations

fine-tuningthe parameter k4 at some particolar value above the critical one k4cpk0, ∆q for

which xN4y “ V, and then study the thermodynamical limit as V Ñ 8. This procedure

makes us treat the target volume V, in practice, as a parameter of the system, besides k0,

∆, while k4is no more a free parameter of the Lagrangian, but must be tuned according to

the chosen value of V, as a Legendre transformation from the triple pk4, k0, ∆q to the triple

of new parameters pV, k0, ∆q.

It is useful, at this point, to impose a weak constraint on the space of configuration sampled, by multiplying the probability weight by a “volume fixing” factor such as8

e´∆S “ e´pN4´Vq2, _(2.17)

which suppresses the contributions from configurations with volumes far away the target one V, where  is a positive and sufficiently small constant. In alternative, it is possible to fix the average of the total spatial volume NS L

3 “ pN p4,1q

4 ` N

p1,4q

4 q{2, by using the same

form as Equation (2.17) but with N₃S Linstead of N4and such that xN₃S Ly “ VS,tot:

e´∆S

“ e´pN3S L´VS,totq2, (2.18)

The latter expression will be our preferred choice for the discussions in Chapters3and4, where we will often use the following notation: N41 “ N

p4,1q

4 ` N

p1,4q 4 .

2.4

Phase diagram of Causal Dynamical Triangulations

As pointed out above, of the three parameters k4, k0and∆ appearing in the four-dimensional

CDT action (2.16), k4 is traded for the total (spatial or spacetime) volume V, so that the

phase diagram of the system in the thermodynamical limit can be described by just the two parameters left k0and∆.

The phase structure of CDT, thoroughly discussed in the literature [13, 14, 17, 18,

32, 33], is characterized by four distinct phases, named A, B, CdS (also dubbed de Sitter

phase) and Cb(also dubbed bifurcation phase), as sketched in Figure2.4.

8_{Other choices can be employed, for example∆}

S “ |N4´ V|.

2.4. Phase diagram of Causal Dynamical Triangulations Chapter 2

Figure 2.4: Sketch of the phase diagram of CDT in 4D with T3 topology of the slices. The transition lines A-CdS and B-CdS are first order, where the transition lines B-Cb and

Cb-CdS have been identified as second or higher order. [Figure taken from Ref. [33]]

In order to get a first rudimentary look on the differences between the phases, we can use as observable the volume profile of triangulations, which represents the distribution of slice spatial volumes as a function of the (Euclidean) slice time t, and is denoted by VSptq

(T dependence implied).

In the B phase, the total spatial volume VS,totbecomes almost entirely localized in a single

spatial slice at slice time t0, so that the volume profile takes the form VSptq » VS,totδpt ´

t0q. Actually, the spatial volume of the other slices cannot possibly assume the value 0

(since that would introduce a singularity in the triangulation), but it must at least reach the configuration with minimal number of equilateral tetrahedra compatible with the slice topology Σ (i.e., VSptq “ 5 for Σ  S3); however, in practice, this region of minimal

volumes, usually referred as the stalk, becomes irrelevant in the thermodynamical limit, and one can safely ignore its contribution at finite volumes. Also configurations in the CdS and Cb phase have a stalk structure, which this time surrounds an extended region,

contiguous in slice time, with non-minimal spatial volume correlated in slice time, the so-called blob. The A phase configurations, instead, is characterized by multiple and uncorrelated peaks in the distribution of spatial volumes per slices VSptq.

Chapter 2 2.4. Phase diagram of Causal Dynamical Triangulations

possess the qualitative properties that one would expect from a spacetime representing a proper universe, namely a non-trivial extension in slice time and a (positive) correlation between slices at close times; for this reason, the A and B phase are considered unphysical. For what concerns the distinction between configurations in the CdS and Cbphases, the

distribution of volume profiles VSptq does not suffice in identifying their differences, and

we must recur to another observable. Indeed, the bifurcation phase Cb, differs from the

CdS phase by the presence of two different classes of slices which alternate each other in

slice time; these classes can be characterized by the presence (or absence) of vertices with high coordination number (i.e., the number of d-simplices to which they belong, usually denoted by Opvq), so that the Cbphase could be distinguished from the CdS by studying the

distribution of maximum coordination number for each slice (usually denoted by Omaxptq).

In order to investigate quantitatively each transition line, one has to employ an order parameterfor each specific line. Many definitions have been used in literature both in the cases of the spherical and toroidal spatial topologies [17,19,32,33,54–56], but, despite

the choices, the structure of the phase diagram turns out as in Figure2.4, with the B-CdS

and CdS-A transitions appearing to be first-order, and the B-Cband Cb-CdS transitions both

appearing as higher-order to the present state of research.

The higher-order lines with diverging correlation lengths are the ones that should be considered when exploring the continuum limit, because, in that case, the details of the discretization become irrelevant. The Cb-CdS transition, in particular, seems to be the

most promising candidate to search for a continuum theory of quantum gravity, because of the qualitative properties of both phases, being extended in slice time, and also be-cause of their volume profile distribution which fits surprising well with the one of a (Wick-rotated) de Sitter spacetime [42,57,58]. For this reasons, in the following, besides

a characterization of each different phase, we focus on the investigation of the critical properties of the Cb-CdS transition line.

Chapter 3

Spectral Methods and Spectral Graphs

Analysis

Nowadays, spectral analysis of gauge covariant derivatives find application in a huge va-riety of different fields. In Quantum Chromodynamics (QCD), for example, spectra and eigenvectors of the Dirac operator (i.e., the minimally-coupled gauge covariant deriva-tive of S Up3q) are studied and employed to extract useful information about topological properties of systems [59, 60]. In this chapter we will discuss in particular the spectral

analysis on graphs associated to triangulations. To mention just a few of recent appli-cations of spectral graph analysis of the graph analogue of the covariant derivative ∇ (or its square ∆ ” ∇2 _{more frequently), has been employed in shape analysis for computer}

aided design and medical physics [61, 62], in order to study dimensionality reduction

and spectral clustering for use in feature selection/extraction in some contexts of machine learning [63], in optimal ordering as for the PageRank algorithm of the Google Search

engine [64], in the analysis of connectivity and robustness for random networks [65] and

in many other fields. This makes the application of spectral analysis to CDT is just one of a well known analysis tool. On the other hand, some well known results which have been established in other fields will turn out to be useful in our investigation of CDT.

In the next Section we will define the Laplace–Beltrami operator in a general setting, describing some of the most relevant properties of its spectrum and eigenvectors. Then, in Section3.2we will introduce the concepts of graphs, their corresponding Laplace

3.1. General concepts on spectral methods Chapter 3

trix, and the association between triangulations and dual-graphs, which will be used in Section3.3 in the analysis of spatial slices of CDT configurations, while in Section 3.4

we introduce some techniques for visualizing dual-graphs and their Laplace eigenvectors. Finally, in Section3.5, we will describe some issues of the graph representations, which, despite the latter being definitely a source of useful information, made us give up on it in search for better and reliable methods, as the one which we will describe and investigate in the next chapter.

3.1

General concepts on spectral methods

The Laplace–Beltrami (LB) operator can be defined for space of many different types. For smooth Riemannian manifolds pM, gµ,νq is action on the algebra of smooth scalar functions f P C8 pM, Rq is described by1: ´∆ f ” ´a1 |g| Bµp b |g|gµνBνf q “ ´BµBµf ´Γµ_µνBνf, (3.1)

where gµν is the metric tensor and Γαµν are the Christoffel symbols. Since in the

follow-ing we will always deal with boundaryless manifolds, we will not bother to distfollow-inguish between the different types of eigenvalues which arise from using different boundary con-ditions.

The LB operator is both symmetric, real and positive semi-definite, and, for the spec-tral theorem, it is possible to find a set of mutually orthogonal eigenvectors BM solving

the eigenvalue problem ´∆ f “ λ f and making a basis for the algebra C8

pM, Rq. Fur-thermore, it is straightforward to show that the LB operator is linear, and invariant with respect to isometries of the manifold, as are its eigenvalues and eigenspaces2

1_{The minus sign is a convention mostly employed in mathematics; we chose this convention since it}

makes the eigenvalues non-negative.

2_{Eigenspaces are not affected by the application of isometries, while eigenvectors in general can be}

(except when they are associated to simple eigenvalues); this is associated to the freedom in the choice of the basis of orthogonal eigenvectors BM, while the eigenspaces are uniquely determined by the operator.

Chapter 3 3.1. General concepts on spectral methods

3.1.1

Interpretation of Laplace–Beltrami eigenvalues and

eigenvec-tors

In order to understand the physical content carried by eigenvalues and eigenvectors of the Laplace–Beltrami operator on a generic manifold M, we will now consider a heat di ffu-sion process starting from a point x0 P M, described by the following partial differential

equation (PDE): $ ’ ’ & ’ ’ % Btupx, x0; tq ´∆upx, x0; tq “ 0, upx, x0; 0q “ ?1 |g|δpx ´ x0q , (3.2)

where is the starting point of the diffusion. Given the LB spectrum σM ” λn|λnď λn`1

(₃

and an orthogonal basis of eigenvectors BM ” enpxq|λn P σM

(

, we can write the solution to Equation (3.2) as a linear combination of eigenvectors:

upx, x0; tq “ |σM|´1

ÿ

n“0

unpx0; tqenpxq , (3.3)

which allows us to solve Equation (3.2) as a set of decoupled first order equations:

Btunpx0; tq “ ´λnunpx0; tq @ n . (3.4)

Solving Equation (3.4) for each n, we can write the general solution to Equation (3.2) as follows: upx, x0; tq “ |σM|´1 ÿ n“0 e´λnt_e npx0qenpxq , (3.5)

where we used the fact that the initial condition completely determines the evolution, since, by orthogonality of eigenvectors, we have unpx0; 0q “

ş

a

|g| enpxqupx, x0; 0q “

enpx0q.

From the exponential decays in Equation (3.5), where diffusion modes corresponding to different eigenspaces are decoupled from each other, the physical content of eigen-values and eigenvectors becomes apparent: larger eigeneigen-values are associated to faster

3_{In the following discussions, eigenvalues will be always labeled in non-decreasing order, and they will}

be considered distinct even when they are not simple (multiplicity greater than 1).

3.1. General concepts on spectral methods Chapter 3

diffusion modes, whose form is dictated by their corresponding eigenvectors, while the converse is true for the smaller eigenvalues. The smallest mode is always a zero-mode4 λ0 “ 0, which just corresponds to the (uniform) solution reached after thermalization

e0pxq “ 1{

?

V and is not interesting.

The smallest non-zero eigenvalue λ1, called spectral gap, corresponds to the mode with

slowest diffusion rate, and therefore sets a lower bound on the overall rate of a general diffusion process.

Using the thermodynamical terminology, a relatively small value of the spectral gap for a manifold means that some regions in the manifold require relatively long times to ap-proach thermal equilibrium. From the geometrical point of view, long diffusion times, and therefore small spectral gaps, are associated to large volumes in extended geometries. Instead, a large spectral gap is typically connected to a small diameter5_{, since it} corre-sponds to fast thermalization rates between all the manifold regions. We will come back to these concepts many times in the next sections.

3.1.2

Weyl’s law and e

ffective dimension

Besides the spectral gap, we can define two quantities of interest: the spectral density:

ρpλq “ÿ

i

δpλ ´ λiq, (3.6)

and the integrated spectral density:

npλq “ÿ

i

θpλ ´ λiq, (3.7)

where tλiu are the LB eigenvalues and θ is the Heaviside step function6. As can be seen

from its definition (3.7), npλq counts the number of LB eigenvalues with values below λ,

4_{It is straightforward to prove that for a generic compact manifold with n connected components, there}

are exactly n zero-modes of the Laplace–Beltrami operator. The manifolds which we will consider in the following are always made of a single connected component, so λ0“ 0 is the only zero-mode, and λną 0

for n ě 1.

5_{The diameter of a manifold is the maximum of the minima of path lengths between any two points in}

the manifold.

6_{The spectral density and the integrated spectral density are actually functions of the complex parameter}

Chapter 3 3.2. Spectral graph analysis

and its asymptotic behavior at large spectral radius λ is called Weyl’s law [66–68]:

npλq „ ωd p2πqd

Vλd{2, (3.8)

where d and V are respectively the chart (UV) dimension of the manifold and its volume, while ωd is the volume of a d-dimensional ball of unit radius.

Even if Equation (3.8) actually holds only on the higher end of the spectrum (i.e., the part associated to the small scales), we can nonetheless make use of its physical content to extend its range of validity also up to large scales; this led us to define of new type of measure of dimension in [34], which we called effective dimension, and which runs depending on the length-scale considered (dictated by λ):

dEFFpλq ” 2

dlogpn{Vq

dlog λ . (3.9)

This definition of dimension is representation independent, and will be used both in the rest of this chapter, where we discuss the dual graph representation of the Laplace– Beltrami eigenvalue problem, and in the next one (Chapter 4), where we will consider instead the Finite Element Method representation.

3.2

Spectral graph analysis

In order to compute spectra and eigenvectors of the Laplace–Beltrami operator on trian-gulations, we need first to define it, by writing an appropriate discretization the space of functions on which it acts.

A possibility, the one considered in this chapter, is to represent triangulations by their associated dual-graph, and use the tools of graph theory and spectral graph analysis.

A graph G is formally a pair of (finite) sets pN, Eq: the set N contains the so called nodes, whereas E Ă N ˆ N is a binary relation on N encoding the connectivity between nodes in the form of ordered pairs of nodes tpna, nbqu. An element pna, nbq of E is

com-monly called edge since represent a connection from a node na to another one nb; in

particular, the kind of graphs which we are interested in are undirected graphs, that is

3.2. Spectral graph analysis Chapter 3

graphs G “ pN, Eq where the relation E is symmetric: pna, nbq P E ùñ pnb, naq P E

(each edge comes in both directions).

Graphs are abstract entities, and, in general, should not be considered as embedded into any space, or as encoding metric properties. Nevertheless, for d-dimensional equilat-eraltriangulations, it is useful to build an associated graph whose nodes can be imagined to be located at the centers of d-simplexes, while the edges would encode the adjacency relations between d-simplexes. In the following discussions, this graph will be called dual graph, to distinguish it from another possible graph construction which associates nodes to triangulation vertices and edges with triangulation links. An example of dual graph, “embedded” into the underlying triangulation, is shown in Figure3.1.

Figure 3.1: Section of a two dimensional triangulations and its associated dual graph.

For a general undirected graph G “ pN, Eq, where N “ tniu |N|´1

i“0 is an (ordered) set of

nodes and E is the set of unordered pairs of connected nodes, we can define the associated Laplace matrix by the expression

L ” D ´ A, (3.10)

where D is the (diagonal) degree matrix such that the element Dii” |te P E|ni P eu|

counts the number of nodes connected to the node vi, while A is the symmetric

adja-cency matrixsuch that the element Ai j “ χErpni, njqs is 1 only if the nodes ni and nj are

connected (i.e. pni, njq P E) and zero otherwise.

Since each simplex in a boundaryless triangulation has exactly pd ` 1q adjacent d-simplexes, each node has the same degree pd ` 1q, and the graph is said to be regular.

Chapter 3 3.2. Spectral graph analysis

The degree matrix, therefore, takes the trivial form D “ pd ` 1q1, and all the interesting information comes from the adjacency matrix A.

3.2.1

An example: the spectrum of a 3-torus graph

Before applying spectral graph analysis in specific CDT contexts, we will show now some of the concepts discussed previously, but in a more familiar setting.

Let us consider a d-dimensional hypercube with sides of integer length Li P N, i “

1, . . . , d, and with the opposite faces identified (i.e., with the topology of a d-dimensional torus Td

” Śd_i“1S1). The space inside this hypercube can straightforwardly be dis-cretized by partitioning it into unit hypercubes, and constructing the so called torus grid graph, which consists in the graph where nodes correspond to the unit hypercubes, and graph edges correspond to pairs of adjacent unit hypercubes. The graph is abstract, but, just for visualization aid, one can imagine the nodes to be placed at each integer tuple ~n inside the hypercube (or cube for d “ 3 if it can help to visualize it), where ni P Z{LiZ

and i “ 1, . . . , d.

A straighforward computation shows that the eigenvalues of the Laplace matrix for a d-dimensional torus grid graph can be written as:

λ_~k “ 2d ´ 2 d ÿ i“1 cos pπmi{Liq “ 4 d ÿ i“1 sin2pki{2q, (3.11) where ´Li{2 ă mi ď Li{2 and ki ” 2πmi{Li(i “ 1, . . . , d).

Notice that, Taylor expanding Equation (3.11) for ~k around the origin ~0, we obtain the same form of the exact LB spectrum for a continuous torus Td, i.e., λ_~kpcont.q “ řd_i“1k2_i; for sufficiently small eigenvalues (large scales) the discretization effects are negligible, but become more and more pronounced above around λ _{& 1. Furthermore, the spectral} gap vanishes in the thermodynamical limit Li Ñ 8, as expected since the geometry is

extended.

The labeling in Equation (3.11) is made in term of a discrete version of the wave number ki, but we want to relabel the eigenvalues in a non-decreasing order λ~k Ñ λn

(λn ď λn`1), because the second form is the one yielded by numerical routines for the Giuseppe Clemente Spectral Methods in Causal Dynamical Triangulations 37

3.2. Spectral graph analysis Chapter 3

1e-05 1e-04 1e-03 1e-02 1e-01 1e+00

n / V

λ

Figure 3.2: Plot of λn against its volume-normalized order n{V, for a torus grid graph

different combinations of (integer) sizes Liin each direction. The straight continuous line

is the exact Weyl scaling, i.e. Equation (3.8), predicted for d “ 3; the dashed straight lines correspond to effective Weyl scalings or effective dimensions d “ 2 and 3. [Figure taken from Ref. [34]]

extraction of eigenvalues, and also the most useful for the following discussions.

Indeed, in view of the considerations done in Section3.1.2, by studying the eigenvalues λn in terms of their volume normalized order n{V, we can extract the running effective

dimension using Equation (3.9) by (twice of the inverse of) the slope in λnvs n{V plots.

Figure3.2shows three examples of spectra of 3-dimensional torus grid graphs with di ffer-ent number of nodes in each direction; as expected, the spectrum for the lattice with equal sides, apart from some discrepancy due to the eigenvalues degeneracies (sixfold, since we have left and right modes for each direction), lies, for all the large and intermediate scale range (i.e. λ ! 1), in a straight line corresponding to effective dimension dEFF » 3, while

the spectra of the other asymmetric combinations shows a “knee” at some intermediate value λ1_{, where the slopes change, so as the e}ffective dimensions. It is interesting to

no-tice that, the two cases with same largest size (Lz “ 600 " Lx, Ly) have exactly the same

low spectrum, and indeed, using the very interpretation of effective dimension, these ge-ometries can be considered as effectively 1-dimensional at the largest scales, as also the slope of the plot suggests. The case with Lx “ Ly ! Lzshows a single knee at the scale

Chapter 3 3.3. Numerical results of spectral graph analysis applied to spatial slices

λ1

» p2π{Lxq2, after which the spectrum is basically indistinguishable from the one with

equal sizes, showing that below length-scales l ă Lx the geometries are the same, while

the case with Lx ! Ly ! Lz, shows two very separate knees one at λ1 » p2π{Lyq2 and the

other almost at the scale of the lattice spacing at λ2

» p2π{Lxq2, with the presence of an

effective dimension dEFF » 2 in the intermediate region between λ1 and λ2.

Since the Laplace–Beltrami operator has the dimensions of the inverse of a length squared, we can associate to each eigenvalue a characteristic length scale ln „ 1{?λn.

Moreover, since the eigenvectors of the LB operator solve the wave equation as stationary waves, the lengths tlnu can be interpreted as the wavelengths of these modes: they are

related to the frequencies at which the manifold resonates [69].

3.3

Numerical results of spectral graph analysis applied

to spatial slices

In order to test the methods of spectral graph analysis, we considered configurations from each one of the four phase discovered, and using spherical topology. Furthermore, we will discuss only the analysis on CDT spatial slices, which are boundaryless 3-dimensional submanifolds of 4-dimensional CDT configurations: the reason of this choice is due to the observation that spatial slices are made of equilateral simplexes (tetrahedra), which allows to represent them faithfully by dual graphs; we will discuss more about this in Section3.5.

First of all, in the rest of this Section, we show some spectral properties of slices from simulation points all well inside the four phases [34], and then we discuss the critical

behavior of these spectra around the Cb-CdS critical line [35], which is a good candidate

for the continuum limit.

3.3. Numerical results of spectral graph analysis applied to spatial slices Chapter 3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1

B

C

b

C

dS

Figure 3.3: Distribution of λ1 at k0 “ 2.2 and variable ∆ for configurations with total

spatial volume VS,tot “ N41

2 “ 40k and for slices with spatial volume VS ą 2k. [Figure

taken from Ref. [34]]

3.3.1

Spectral graph characterization of the phases in the CDT phase

diagram

The coarsest spectral characterization of slices in different points of the phase diagram is given by their spectral gap λ1, introduced in Section3.1.

Figure 3.3 shows the general behavior of the distribution of spectral gaps for di ffer-ent simulation points across the B-Cb line and the Cb-CdS line (k0 “ 2.2 and varying ∆),

where darker dots correspond to more frequent values of λ1. The spectral gap distribution

of Cb slices shows two classes of slices, with either vanishing or non-vanishing spectral

gap, and due to this peculiar property we will discuss it separately in Section3.3.2. Nev-ertheless, from Figure3.3we can already anticipate that across the Cband the CdS phase,

for increasing values of∆, the gap closes: since the LB eigenvalues derive from scaled quantities (λ „ a´2_{), this observation suggests to use the λ}

1as an order parameter for the