Spectral Analysis and the Phase Diagram of Causal Dynamical Triangulations

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Dipartimento di Fisica E. Fermi Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Spectral Analysis and The Phase

Diagram of Causal Dynamical

Triangulations

Candidate: Supervisor:

Alessandro Ferraro Prof. Massimo D’Elia

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Introduction

A quantum description of the gravitational field does not exist so far. One could try, as usual in the quantum field theory formalism, to quantize fluc-tuations around the solutions of the classical equations of motion. The re-sulting quantum theory is perturbatively non-renormalizable. The failure of standard methods demands the development of new ideas in theoretical physics.

A possible path is suggested by the so-called Asymptotic Safety conjec-ture. In the Renormalization Group framework, we can define a parameter space of the theory and follow RG flow lines generated by scale transforma-tions. Asymptotic Safety assumes the existence of a UV fixed point in the parameter space that is the limit of a whole class of theories. Indeed, it is possible to define the UV critical surface as the region of the parameter space in which RG flows arise from the UV fixed point. If the critical surface can be reached by fixing only a finite number of parameters, it will be possible to define a theory of gravity that is predictive at high energy-scale.

Causal Dynamical Triangulations is a non-perturbative approach to Quan-tum Gravity with the aim of proving this scenario. It is based on the Regge formalism, which defines a discretized version of the Einstein-Hilbert action by simplicial approximation of space-time configurations. In particular, in CDT one considers a specific regularization of the Euclidean path integral of the theory that is suitable to sample the configuration space by means of nu-merical methods. The specificity of this approach is that a causal structure is enforced on CDT configurations by considering a space-time foliation.

The resulting statistical system is investigated using the Monte Carlo approach. The study of the dependence of appropriate observables on the bare parameters of the lattice theory allows defining the CDT phases. The main goal is to find critical points in the phase diagram. Critical points of second-order phase transitions are characterized by the divergence of corre-lation lengths, measured in units of the lattice spacing. Approaching such points it is possible to define the continuum limit of the lattice theory. A possible non-trivial continuum theory obtained in this way could identify the non-perturbative UV fixed point of the Asymptotic Safety program.

Promising results have been found by the CDT community. Four different phases have been identified in the CDT phase diagram: A, B, C_dSand C_b. C_b

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and C_dS phases describe universes that have both spatial and time extension and are considered the more physically interesting phases. In particular, C_dS phase shows an average geometry that closely matches Euclidean de Sitter space. For the C_b|C_dS phase transition evidence of second-order behaviour has been provided. Still, more efforts are required to obtain a definitive proof. Furthermore, observables in standard CDT analysis are defined as global counting of elementary geometrical structures of a given type. This is not a suitable choice to describe local properties of CDT configurations. In our opinion, it may be important to have a more precise characterization of geometrical features of the different phases.

A new set of observables has been defined with the method known as Laplace-Beltrami spectral analysis. It is a generalization of Fourier analysis and has proved to be a very versatile tool. By calculating spectra of the LB operator, it is possible to study diffusive processes on very general structures. Eigenfunctions of the LB operator correspond to normal modes that bring with them information about different scales of the system.

It is possible to define the LB operator on a graph, that is defined by a set of equal points and an adjacency relation between them. Graphs can be used to model relations and processes of many types and therefore LB spectral analysis can provide useful information in a lot of different fields. In particular, spatial slices in CDT can actually be described as a graph and the Laplace-Beltrami analysis in CDT is performed on these structures. The case of full CDT configurations is more involved and spectral analysis has not been applied to them so far.

We will show how the analysis of spectral properties of spatial slices correctly reproduces the CDT phase diagram. For this purpose, the more interesting results are the presence of a gap in the Laplace spectra and the definition of an effective dimension that regulates diffusion processes. Study-ing their behaviour, we have found that it is possible to extract useful in-formation about geometrical properties of slices. Moreover, these quantities have proved useful in the characterization of the phase transitions. The pres-ence or abspres-ence of a gap in the spectrum of the Laplace-Beltrami operator allows distinguishing C_b and C_dS phases. The study of the critical behaviour of the spectral gap near the transition line gives information about the order of the transition. In particular, if the gap disappears in a continuous way, it will be evidence of a second-order phase transition. Indeed, the spectral gap could be interpreted as (the square of) a mass scale running as a function of the bare parameters of the lattice theory. Using the spectral gap, it is possible to define a typical length-scale of the system that will diverge ap-proaching the critical point if the Cb|CdS transition is second-order.

The thesis is organised as follows.

In chapter 1 we will present the foundations of CDT and analyze his role in the Asymptotic Safety program. We will shortly discuss the concept

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Introduction

of UV fixed point and how it is related to renormalization. We will define the CDT configuration space and the CDT action used to perform Monte Carlo sampling. Then, we will describe the CDT phase diagram, the differ-ences between its phases and the order parameters used to investigate phase transitions.

In chapter 2 we will discuss Laplace-Beltrami spectral analysis. To do so, we will define the Laplace-Beltrami operator and then we will show the form it assumes on a graph (the Laplace matrix). We will then discuss its general properties and the geometrical interpretation of some useful quantities. We will show typical results that can be obtained in the different phases of the CDT phase diagram. This chapter should provide all the elements needed to understand methods that will be used in the following chapters.

In chapter 3, we will make some preliminary analysis on simple cases. We will consider the A|B transition line that is expected to be first-order and discuss what kind of results can be obtained. Even if this case is not interesting for what concerns the continuum limit, it gives the possibility to show practical examples of the method discussed in previous chapters. Then, we will make a few first comments on the Cb|CdS phase transition.

In chapter 4, we will discuss in more details the C_b|C_dS transition. As stated, it is a possible second-order transition line. We will see in more details how the first eigenvalue of LB spectra can be defined as the order parameter for the transition. Spectra of LB operator depend on spatial volumes of slices and we will see that the correct order parameter has to be defined in the thermodynamical limit. We will see how the extrapolation is performed in several real cases, leading to the distinction between phases with or without a spectral gap that persists in this limit. The spectral gap allows us to define a typical length scale on spatial slices. We will show how the order parameter and the associated length scale (that can be interpreted as a correlation length) evolve approaching the transition line. As stated, if the spectral gap vanishes in a continuous way and (consequently) the related length scale diverges, it will be evidence of a second-order phase transition.

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Chapter 1

A lattice approach to Quantum

Gravity

1.1 The problem of Quantum Gravity

An interesting and unsolved problem in theoretical physics is how to reconcile General Relativity (GR) and Quantum Mechanics (QM).

We consider the Einstein-Hilbert (EH) action, that describes the (classi-cal) theory of gravity without matter fields:

SEH =

1 16πG

Z

d4xp|g|(R − 2Λ) (1.1)

where G is the Newton constant, g = det(g_µν) is the determinant of the metric tensor, R is the Ricci scalar (or scalar curvature) and Λ is the cosmo-logical constant.

The theory is non-renormalizable by power counting, because Newton’s constant has dimension −2 in natural units ~, c = 1. Indeed, redefining gµν = ηµν + κhµν (ηµν = diag(1, −1, −1, −1) is the flat metric) with κ =

16πG, the form of the action is given by ([1]): SEH =

Z

d4x(∂µ∂νhµν− hαα+ O(h2)) (1.2)

In eq. (1.2), we have considered the expansion of R for det(h_µν) 1/k and the approximationp|g| ' p|η| = 1. Terms of order hnin the expansion of R (n gravitons interaction) carries a coupling constant κn−2. Therefore, we will find that all n points correlation functions diverge if we take enough terms in their perturbative loop expansion. Actually, it has been shown that the theory is finite at one-loop order and diverges at two loops (see [2]) In order to cancel divergences, we need to fix an infinite number of counterterms. It is different from what happens for renormalizable theories, where only a finite number of correlation functions show divergent diagrams.

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In this case, we have to consider EH action as an effective theory that gives consistent results in the low-energy regime E p1/G. However, the theory is not predictive at higher energies, where quantum corrections be-came appreciable. Still, the generally accepted idea is that a fundamental quantum description of gravity must exist. The situation has strong analo-gies with the case of the Fermi theory of beta decay. It describes a four fermions interaction with a coupling constant G_F, where G_F has mass di-mension −2. Fermi theory, today, is known to be an approximation at low energies of the electroweak theory, which successfully predicted W± and Z particles.

There does not seem to be an obvious way to extend GR to a renormaliz-able theory by adding new degrees of freedom. Furthermore, one of the great difficulties in Quantum Gravity is that quantum corrections became impor-tant at very high energies. The energy scale MP l≡p1/G ∼ 1019 GeV that

separate classical and quantum regime is currently out of reach as regards ex-perimental techniques. Therefore, the appearance of new phenomena cannot be directly observed in a laboratory.

1.2 Asymptotic Safety

Another way to proceed is suggested by the Asymptotic Safety (AS) conjec-ture for Quantum Gravity, proposed by Steven Weinberg ([3]).

In the framework of Wilsonian Renormalization Group (RG), renormal-ized coupling constants are defined as a function of a given energy scale. A change in such a scale is called scale transformation. For example, it is possible to fix a cut-off Λ and to integrate out all the short-distance degrees of freedom (Fourier components of fields associated with large momentum k > Λ). We obtain an effective theory that provides a good description of the considered physical system with a given “resolution”. In this example, an RG transformation consists in changing the cut-off from Λ to bΛ (with 0 < b < 1). It is possible to follow the change of parameters in RG trans-formations. We define ~g = (g1, g2, ...) as the set of coupling constants of a

given theory. Each constant is associated with a possible interaction (i.e. that respects the symmetries of the theory) between the fields we are con-sidering. In general, if we change the energy scale, we will change the values of renormalized coupling constants. The path of ~g in the parameter space G is called Renormalization Group flow .

There are no inverse RG transformations because integrating out degrees of freedom is not an invertible process. However, a fundamental and renor-malizable quantum field theory has to be form invariant with respect to scale transformations, because it should be able to describe physical systems at all scales. This means no new field is necessary to describe higher energies behaviour. Therefore, it is possible to follow the RG flow backwards.

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Chapter 1. A lattice approach to Quantum Gravity

Figure 1.1: UV critical surface. Arrows indicate the direction of RG flows.

If a theory with a particular value of ~g is completely scale invariant (not only in its form) it denotes a fixed point in the parameter space. Hence, by definition of scale invariance, if we can define a typical length ξ of the system (that in general would depend on the parameter of the theory), in a fixed point the only possible values are ξ = 0, ∞. In statistical mechanics, diverging correlation lengths indicate the presence of a second-order phase transition in the phase diagram of a statistical system. This analogy allows relating the fundamental theory to a statistical system defined in an appro-priate way. Actually, this is the starting point of the methods that will be described in the following sections.

We can distinguish IR and UV fixed points, considering the different behaviour of RG flows in the nearby area. If a fixed point “attracts” flows of nearby points in the UV limit (this actually means that nearby points are repelled when an RG transformation is performed), it is said to be a UV fixed point. We can define the UV critical surface in parameter space as the area in which sets of parameters are attracted toward the UV fixed point (figure 1.1). UV fixed points have great importance in the theory of renormalization. All theories defined by a set of parameters inside the UV critical surfaces are equivalent in the UV limit. The UV behaviour of a whole class of theories could therefore be characterized by specifying only the parameters that identify the critical surface. These parameters are called relevant . As for counterterms in standard renormalization, if there is a finite number of relevant parameters to fix, we can define a theory that is predictive at high energies.

The perturbative expansion is well defined at high energies if coupling constants remain small in the UV limit. Then, perturbative methods provide meaningful results if the theory sits on the UV critical surface built around the Gaussian fixed point (that is actually the free theory). QCD, for example, is characterized by asymptotically vanishing interactions between particle as the energy scale increases: this property is called Asymptotic Freedom.

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Figure 1.2: Simplices appearing in 4-dimensional simplicial manifolds

Unfortunately, this does not happen in the case of the gravitational theory. However, there could exist a non-trivial fixed point around which a mean-ingful quantum theory of gravity could be defined. This is actually a formu-lation of the AS conjecture. In this sense, what we have discussed is a more general and non-perturbative renormalization method.

1.3 Simplicial manifolds and Regge action

Dynamical Triangulations (DT) ([4]) is an approach to Quantum Gravity with the aim of proving the AS scenario. It applies Monte Carlo methods to a discretized version of the EH action. The main goal is to find a second-order phase transition of the associated statistical system. If that is the case, it is possible to define the continuum limit of the lattice theory approaching the critical point. The result is a continuum theory that is well-defined also in the UV limit. Actually, the continuum theory may be trivial, in the sense that it is non-interacting. A standard regularization is provided by Regge calculus. In Regge’s formalism ([5]), manifolds are approximated with sim-plicial manifolds. A simplex is a d-dimensional object with d + 1 vertices (figure 1.2) that in our case represents an elementary block of (flat) space-time. Simplicial manifolds are defined as lattices where every point is one of the vertices of a simplex. A triangulation T = (S, h) is defined as a simplicial manifold S homeomorphic to a given manifold M, together with an home-omorphism h : S → M. Two triangulations associate to the same manifold differ by a reorganization of simplices that can be interpreted as a change of coordinates. Triangulations are sampled using a discretized version of the EH action to define the probability distribution1 P [τ ] ∝ e−SE[τ ]_{. To define}

Regge action, it is necessary to rewrite quantities appearing in the EH action as quantities depending on the way simplices are linked between them. For

1_{In path integral formulation, fields configuration are weighed with a factor e}iS_{. To}

interpret this factor as a probability distribution e−SE _{we need a transformation from}

Minkowsky space to Euclidean space (Wick rotation). In general, this is not a trivial task in Quantum Gravity, but for CDT triangulations we will see that is possible to perform this transformation in a very simple way.

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Figure 1.3: 2-dimensional triangulation, where a vertex (left) and triangles sharing it (right) are pointed out.

example, consider a 2-dimensional simplicial manifold. It is composed by triangles (2-dimensional simplices), links (1-dimensional simplices) and ver-tices (0-dimensional simplices). Two triangles are connected by a link (they have a link in common) and each triangle is connected with exactly three other triangles. However, there is no restriction in the number of triangles that can share the same vertex (figure 1.3). Assume that all simplices of the same dimension are equal, so that all links have the same length and all triangles are equilater (this is not true in a general simplicial manifold).

With this assumption, it can be defined the deficit angle around a vertex as

θ = 2π − nπ

6 (1.3)

where n is the number of triangles sharing that vertex. It is proved that all the information about local curvature in 2-dimension is contained in deficit angles around vertices (figure 1.4). In the general case d-simplices are linked with other d + 1 d-simplices by (d − 1)-simplices. The information about local curvature is contained in deficit angles around (d − 2)-simplices.

EH action can be written as SEH = 1 16πG Z dxd√−g R − Λ 8πG Z dxd√−g (1.4)

where the first integral is the total curvature and the second integral is the total volume (which in a triangulation is related in a simple way to the total number of maximal simplices). So, it is obvious that both terms can be expressed as quantities defined on triangulations, as assumed above.

In particular, Regge proved that the discretized version of EH action is SRegge[τ ] = 1 8πG( X σd−2_∈τ _σd−2V_σd−2− Λ X σd_∈τ V_σd) (1.5)

where τ is the triangulation, σk is an index running over k-simplices, _σk is

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Figure 1.4: Deficit angles with non-vanishing value are associated to curved space: elliptic geometry is obtained for positive values of deficit angles and hyperbolic geometry is obtained for negative deficit angles.

However, sampling the complete space of simplicial manifolds would be impossible in practice. In DT and similar approaches (like CDT, that we are going to introduce in the next section) the configuration space is restricted to triangulations where all lengths (volume of 1-dimensional simplices) are related in a simple way to a fixed value. This value is usually indicated with a and can be considered as the lattice spacing of the statistical system.

1.4 Causal Dynamical Triangulations

It is useful to restrict the configuration space of the discretized theory to triangulations that present a foliated structure. In Causal Dynamical Tri-angulations (CDT) ([6]) to each vertex is assigned an integer “time” label (figure 1.5). Sets of vertices with equal time label are called spatial slices. In a d-dimensional triangulation, spatial slices are (d−1)-dimensional simplicial manifolds and adjacent slices are connected by d-dimensional simplices. The foliated structure is required in order to satisfy global hyperbolicity2, but it also imposes that all slices have the same topology ([7]). Algorithms only use local transformations in sampling triangulations, then topology of slices cannot change in simulation time and it has to be fixed. In the following, slices with S3 topology will be considered. Then, periodic boundary

con-2

global hyperbolicity is a causality condition that can be imposed to a Lorentzian manifold and it is a sufficient condition to ensure that no closed causal curves exist. This is what the term “causal” stands for in CDT.

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Figure 1.5: Comparison between 2-dimensional triangulations with (right) and without (left) foliated structure.

ditions will be imposed on time labels, so that the triangulations will have S3× S1 _topology.

Because of the foliated structure, two kinds of links exist: links between two vertices on the same spatial slice are space-like links and links that connect vertices on adjacent spatial slices are time-like links. We impose that links of the same type must have the same length. We have that length-squared of space-like links is simply l2_s = a2 whereas length-squared of time-like links is l_t2 = −αa2 (α > 0). The parameter α can be considered as an index of an enforced space-time asymmetry in the lattice theory. Of course, dependence on α is expected to disappear in the possible continuum limit. k-simplices that connect two slices can be classified by a couple of number (m, k −m+1) (m is an integer with value between 1 e k) which indicates how many vertices are in each slice (figure 1.6). Considering the parameter α, we can perform the Wick rotation t → τ = it associating to each triangulation an Euclidean triangulation with l_t2 = ˜αa2 _{( ˜}_{α > 0)}3_{. Therefore, rotation to}

the Euclidean signature is obtained through the transformation α → ˜α = −α. With these prescriptions and with the fixed topology S3 _{× S}1_{, the}

Euclidean action in the 4-dimensional case takes the form

SCDT = −k0N0+ k4N4+ ∆(N4+ N41− 6N0) (1.6)

where N₀ is the total number of vertices, N₄ is the total number of 4-dimensional simplices (called pentachorons) and N₄₁ is the number of 4-simplices of kind (1, 4) or (4, 1). The coupling constants k0, k4 and ∆ are

dimensionless parameters linked to Λ, G and α. The derivation of CDT action has been discussed in appendix A.

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Euclidean simplices have some well known condition to be met. In d = 2, triangles inequality ls> 2ltmust be satisfied, so that ˜α > 1₄. The corresponding inequality in d = 4

is ˜α > q

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Figure 1.6: 2-dimensional spatial slices linked by different classes of 3-simplices.

Choices made about the properties of the lattice are intended to simplify calculations and implementation of algorithms. However, they are expected to become irrelevant in the limit a → 0 (continuum limit) and V → ∞.

The standard way of proceeding in the sampling of triangulation is to insert or to remove simplices in a small region (see appendix B for a detailed discussion of the standard CDT algorithm). Then, it is possible to compare values of the action for configurations before and after changes, accepting or rejecting new configurations with a certain probability (Metropolis-Hastings algorithm).

Unlike usual lattice simulations, the total volume of spacetime changes in simulation time. In fact, the number of pentachorons is one of the degrees of freedom of the discretized theory and it is proportional to the volume.

This is a problem for two main reasons:

• simulation time needed to obtain a suitable set of measures is expected to grow when the volume of the lattice becomes bigger. To let the vol-ume fluctuate in an uncontrolled way may be dangerous in the practical implementation of the algorithm;

• the possibility of fixing total volume makes available finite-size scaling methods where the behaviour of observables is studied and extrapo-lated in the infinite volume limit.

The standard procedure used in CDT is to dynamically tuning k4(with fixed

values of ∆ and k₀) in order to enforce hN₄i ' V for a given value of the target total volume V . Fluctuations δN₄ around the mean can be reduced by adding a term to the CDT action:

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Figure 1.7: Phase diagram of CDT as presented in [8]. A, B, C_b, C_dS phases are shown, with relative transition lines.

for a small enough value of . This must be done during the first part of the simulation, before making any measure.

Actually, in some cases, it can be useful to fix the value of N41 instead

of N₄ in a similar way. For each 4-simplex of type (4, 1) there is exactly one 4-simplex of type (1, 4) with a spatial tetrahedra in common. The sum of the spatial tetrahedra of a spatial slice defines the spatial volume VS of

that slice. Then, it is possible to define the total spatial volume of a full triangulations as V_S,tot= N41/2.

In the following, we will specify whether simulations have been performed fixing N₄ or V_S,tot.

1.5 CDT phase structure

Evaluating averages (in simulation time) of suitable observables for different values of k0 and ∆, it has been defined the CDT phase diagram (figure

(1.7)). Up to now, four different phases have been distinguished (see [8]). The phases have been found through analysis of observables related to the total number of different classes of simplices. Further analysis performed with spectral methods, which will be described in the following chapter, show the same phase structure. An evident difference between phases can be observed defining the volume profiles. A volume profile of a triangulation is the function N3,s(t) or, in other words, the spatial volume considered as

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Figure 1.8: Typical volume profile in A phase.

a function of the time label t. Here follows a brief description of the main features of the different phases:

• A phase: triangulations in this phase show a volume structure given by a seemingly random distribution of vanishing and non vanishing spatial volumes (figure (1.8)).

This phase is associated to multiple short-living universes.

• B phase: in this phase, the spatial volume is all concentrated in a single big spatial slice (figure (1.9)). Triangulations in this phase cor-respond to spatial extended4 _{universes with no temporal extension.}

• C_dS and Cb phases: both C phases have a similiar behaviour in

terms of volume profile. Triangulations in these phases show a stuc-ture where consecutive slices (which form the bulk of the triangulation) contain all the spatial volume (figure (1.10)).

These phases are the most physically interesting because universes de-scribed are extended both in space and time. In particular, considering the CdS phase, it can be observed that the average volume distribution

of slices matches that of Euclidean de Sitter space ([9]). Indeed, this is the meaning of the subscript dS. Instead, the b in C_b stands for bifurcation. The name is due to the observation of two different types of spatial slices in C_b triangulations, as will be discussed below. Several tests have been done to determine the order of the transitions be-tween the different phases ([6, 8, 10]). The A|B line transition does not

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the spatial extension must be understood as a non-vanishing number of 3-simplices on the given slice. As will be explained later, this does not coincide, in this case, with a real spatial extension. Studying local properties, it has been seen that distances between points on the slice remain small even for large volumes.

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Figure 1.9: Typical volume profile in B phase.

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need particular attention, because it is a transition between phases which cannot reproduce our universe. Anyway, it can be seen that A|B transition is first-order (see chapter 3). The A|CdS and the B|Cb transition lines occur

for almost constant values of k0 and ∆, respectively (figure 1.7). Then, as

usual in the analysis of critical systems, conjugated quantities in the action conj(k0) ≡ N0/N41(in A|CdS transition) and conj(∆) ≡ (N4+N41−6N0)/N4

(in B|C_b transition) are analyzed as order parameters5. While A|C_dS seems to be another first order phase transition, B|C_b appears to be of second-order. There is no simple order parameter for Cb|CdS phase transition. It is

observed that C_b triangulations consist of two different types of alternating slices, one of which is characterized by the presence of vertices shared by a large number of links. Considering this property, it is possible to define an appropriate order parameter. More specifically, the order parameters studied in C_b|C_dS phase transition is defined as

OP2 =

1

2(|Omax(t0) − Omax(t0+ 1)| + |Omax(t0) − Omax(t0− 1)|) (1.8) where Omax(t) is the highest coordination number6 in the slice t and

Omax(t0) = max

t (Omax(t)) (1.9)

The analysis provides also in this case hints for a second-order phase transi-tion.

In the next chapter, a new class of observables will be defined, in order to give a more precise description of the geometrical structure of slices. As we will see, quantities related to these observables reproduce correctly the differences between phases. Moreover, such observables allow a more natural definition of the order parameter in Cb|CdS phase transition. As we will see,

the new order parameter is related in a simple way to a typical length-scale defined on spatial slices.

5_{Actually, order parameters are usually related to transitions between phases with}

different symmetry. It is not clear if all CDT phase transitions are associated with a symmetry breaking. In any case, in the literature these quantities are referred to as order parameters of the phase transitions in CDT.

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Chapter 2

Spectral methods in CDT

2.1 Laplace-Beltrami spectral analisys

One of the main difficulties encountered in the study of Quantum Gravity is the paucity of useful observables.

In CDT, observables related to the total number of different kind of simplices are used to perform the standard analysis. Still, they are not a suitable choice to describe, for example, the geometric structure of slices. It is useful to define observables that describe their local features.

Studying eigenvectors and eigenvalues of the Laplace-Beltrami (LB) op-erator, it is possible to define a new set of observables. This set is not com-plete, nevertheless, it allows us to describe slices at different length-scales. This method is called Laplace-Beltrami spectral analysis. The LB operator is a generalization of the Laplace operator (both will be indicated in formu-las with −∆): given a Riemannian manifold (M, gµν), LB operator acts on

functions f ∈ C∞(M) as

−∆f = − 1

p|g|∂µ(p|g|g

µν_∂

νf ) = −gµν(∂µ∂ν − Γαµν∂α)f (2.1)

where gµν is the inverse of the metric gµν and Γαµν are Christoffel symbols.

For M = RN and gµν = δµν (flat Euclidean manifold), we have Γαµν = 0 for

any choice of α, µ, ν, so −∆f = − N X i=1 ∂_i2 (2.2)

that is the standard Laplace operator.

Functions u ∈ C∞(M) that satisfy −∆e = λe form an orthogonal basis of the algebra C∞(M). The basis can obviously be chosen orthonormal and this particular choice is called spectral basis.

A non-trivial example is given by spherical harmonics. Defined on a 2-sphere, spherical harmonics Y_l,m(θ, φ) are labeled by two integers l, m:

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l identifies an eigenspace associated to the eigenvalue l(l + 1)r−21, while m = −l, −l + 1, ..., l − 1, l is the index that catalogues the eigenfunctions on that eigenspace. Therefore, the eigenspace identified by l has dimension µl = 2l + 1.

Spectral bases are related to the description of local properties of a man-ifold. To show that, we will consider a familiar example. The diffusion process on a generic manifold M is described by a generalized version of the heat equation

∂tf (x, t) − k∆f (x, t) = 0 (2.3)

where −∆ is, in our case, the LB operator instead of the Laplace operator and k is an arbitrary constant. Solutions of the equation can be expanded in terms of eigenfunctions en(x) of the LB operator. Eigenfunctions can be

ordered for increasing values of the associated eigenvalue. Then, the spectral basis can be written as BM = {en|λn ∈ σM, λn+1 ≥ λn}, where σM is the

spectrum of the eigenvalues. Then,

f (x, t) =

|σM|−1

X

n=0

fn(t)en(x) (2.4)

Heat equation can then be rewrited as a set of decoupled equations

∂tfn(t) = −λnkfn(t) ∀n (2.5)

and the solution is given by

f (x, t) = |σM|−1 X n=0 e−λnkt_f n(0)en(x) (2.6)

Thus, {λ_n} are related to diffusion rates and bigger values of n correspond to faster diffusion modes.

With little modification, it is possible to study the generalized version of the wave equation:

∂_t2f − ∆f = 0 (2.7) with solutions f (x, t) = |σM|−1 X n=0 (fn(0)cos( p λnt) + f 0 n(0)sin( p λnt))en(x) (2.8)

In the last equation, we can interpret λnas the frequency squared of

propa-gating modes in the manifold. Normal modes en(x) bring with them

infor-mation about the manifold for different length-scales.

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Chapter 2. Spectral methods in CDT

A spectrum σM does not uniquely denote a manifold M (examples of

isospectral manifolds are present in the literature). Nevertheless, eigenvalues and eigenvectors can be used to extract geometric features of the manifold at different scales (see [11]).

2.2 Laplace matrix on a graph

The LB operator can be defined on more general structures, for example, the algebra of functions on a graph. A graph is a pair (V, E) where V is a set of elements v called vertices and E is a set of adjacency relations (v_i, vj)

between vertices (E ⊂ V × V ). The degree of the vertex v_i is the number of elements (vi, vj) ∈ E. We will consider only undirected graphs, which means

that if (v_i, vj) ∈ E then (vj, vi) ∈ E, too. A basis of the set of functions

f : V → R is fi(vj) = δij with i = 1, .., N , where N = |V | is the number

of vertices of the graph. Therefore, the algebra of the functions on a graph can be represented as the vector space RN. In this representation the LB operator is a N × N matrix which can be written as

L = D − A (2.9)

D is a diagonal matrix and the i-th element Diiis the degree of the vertex vi.

Instead, A is the adjacency matrix, defined so that Aij = 1 if (vi, vj) ∈ E,

Aij = 0 otherwise.

A CDT slice is a 3-dimensional triangulation where all tetrahedra are equal (this is not true for the whole 4-dimensional triangulation due to the difference between timelike and spacelike links). It is possible to define an adjacency relation between tetrahedra that share a common face. All tetra-hedra are adjacent in this sense to exactly 4 other tetratetra-hedra. Therefore, a triangulation can be represented as an undirected graph where each vertex has degree 4. The graph associated to a given triangulation is called dual triangulation (figure 2.1). The Laplacian matrix on a dual triangulation assumes a very simple form:

L = 4 · 1 − A (2.10)

Then, the eigenvalue problem only consists of diagonalizing the adjacency matrix A. This is a relatively simple task considering that A is a sparse and symmetric matrix with only real coefficients. Several algorithms have been developed to efficiently calculate eigenvalues and eigenvectors of this particular category of matrices.

Some general properties of spectra of the Laplace matrix have to be dis-cussed in order to extract geometric information. First, Laplace matrix is positive semi-definite. Then, we can define an ordering of the eigenvalues {λi | i ∈ N, λi ≤ λi+1} and we have (if there are no fixed boundary

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Figure 2.1: Graph associated to a 2-dimensional triangulation.

of Laplace matrix with λ₀ = 0 as eigenvalue. The dimension of the λ0 = 0

eigenspace is equal to the number of connected components of the graph. Indeed, if we have more than one connected component, functions e0,i(x)

defined so that

e0,i(x) ∝

(

1 on the i−th component

0 elsewhere (2.11)

form a basis of the λ0 eigenspace. Anyway, we will consider only connected

graphs, so that the multiplicity of λ0 is µ0 = 1.

The first non-trivial (and most interesting in the following analysis) eigen-value is λ1. We will refer to it as spectral gap.

The spectral gap is related to the idea of connectivity: the higher the value of λ₁, the more the graph is connected. Let us put it in a more formal way. Given a compact Riemannian manifold M, the Cheeger isoperimetric constant h(M ) is defined as the minimal area of a hypersurface ∂A dividing M into two disjoint pieces A and M \ A.

h(M) ≡ inf vol(∂A)

vol(A)vol(M \ A) (2.12)

The same idea can be implemented on a graph G = (V, E) by the definition h(G) ≡ min{∂A

A : A ⊂ V, |A| ≤ V

2} (2.13)

where ∂A ∈ E is a set of links connecting vertices of A with vertices of V \ A. This is related to the concept of “bottleneckness”. Indeed, h(G) can be interpreted as a measure of the minimal number of edges that is necessary

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Figure 2.2: Typical CDT dual slices for λ1 6= 0 (left) and λ1 ' 0 (right)

visualized in a 3-dimensional space through standard techniques in graph theory for the analysis of complex network (see [12, 13, 14]).

to cut the graph and to obtain two separated components. Given a k-regular graph, the spectral gap (that in this context is sometimes called algebraic connectivity) satisfies the Cheeger’s inequalities

1

2λ1≤ h(G) ≤ p

2kλ1 (2.14)

In a flat manifold, it is known that oscillating modes with arbitrarily small frequencies are permitted for arbitrarily large values of the volume. This is not true in general and some CDT slices present a different behaviour, with a spectral gap approaching a finite non-zero value as V → ∞.

We can consider A as an n-dimensional ball in Rn. Its volume is pro-portional to rn, where r is the radius, while the area of its boundary (the n-sphere with radius r) is proportional to rn−1. So, vol(∂A)/vol(A) can be arbitrarily small and h(Rn) = 0. A manifold with infinite total volume (or a graph with an arbitrarily large number of vertices) that have a non-zero value of the spectral gap presents a higher degree of connectivity (consider figure 2.2). A peculiar feature of this kind of graph (that are known in the literature as expander graphs ([15])) is that, if λ₁ 6= 0 in the limit V → ∞, then diffusion time through the whole universe stays finite. Therefore, its size (intended as maximal distance between vertices in the graph) does not actually grow significantly.

Other interesting quantities are the density of eigenvalues ρ(λ), defined so that ρ(λ)dλ is the number of eigenvalues with value in the range [λ, λ + dλ], and its integral

n(λ) = Z λ

0

ρ(λ0)dλ0 (2.15)

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Figure 2.3: Plot of λn in function of n/V , both represented in logarithmic

scale, for an hypercubic lattice with L_x = 3, Ly = 15, Lz = 1800. Dotted

lines are plots of Weyl law for different values of d.

behaviour for n(λ) (on a generic Riemannian manifold) is given by the Weyl law ([16, 17]). For large λ, we have

n(λ) = ωd (2πd₎V λ

d/2 _(2.16)

where V is the volume of the manifold, d is its dimensionality and ω_d is the volume of the d-dimensional ball of unit radius.

It is observed that Weyl law can be taken seriously also in the discretized case. Moreover, it is a good approximation also for the lower eigenvalues.

Consider, for example, a 3-dimensional hypercubic lattice with periodic boundary conditions. We call Lx, Ly, Lz the lengths (number of sites) in the

three different directions. This is a very simple case and a solution for the eigenvalue problem can be found analytically. The graph is the discretized version of a 3-torus and eigenvectors are linear combinations of normal modes of oscillation on a (discretized) circle. Therefore, eigenvectors are labeled by a vector of wave numbers ~k = (kx, ky, kz) with ki = 2πm_L_ii and mi integers so

that −Li/2 ≤ mi≤ Li/2. With these definitions, eigenvalues are completely

characterized by the relation

λm~ = 4π2( m2 x L2 x +m 2 y L2 y +m 2 z L2 z ) (2.17)

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Each eigenspace is at least 2-dimensional, due to the symmetry under inversion with respect to a given axis.

With the assumption Lx ≤ Ly ≤ Lz, the first eigenvectors2 are normal

modes that oscillate only in z direction. When m_z ' L_z/Ly, oscillation

modes in y direction start to be excited and then, for larger values of λ, also modes in x direction are present. It is possible to observe three different regimes in which eigenvalues follow the Weyl law with d = 1, d = 2 or d = 3 (figure 2.3). What emerges is a mechanism of dimensional reduction at higher length scales, with d that can be interpreted as a scale depending effective dimension d_{EF F} of the system.

Operatively, dEF F is defined by the relation

2 dEF F

= d log λ

d log n/V (2.18)

dEF F, by definition, can in general take any real positive value. The

emer-gence of a non-integer dimension in simplicial manifolds with fixed dimension is discussed in appendix C.

Phase diagram of CDT can be analyzed in terms of the new quantities defined above. We have seen that in some cases, a finite value of the spectral gap persists in the infinite volume limit. This feature can be related to a peculiar behaviour of n(λ). If we take Weyl law seriously even for small values of λ, then we must conclude that λ → 0 in the limit V → ∞ for every finite value of the effective dimension. In fact, n(λ) is a increasing function of λ and it can be inverted:

λ ∝ (n/V )2/dEF F _(2.19)

By definition, when λ1 6= 0, successive eigenvalues can not be arbitrarily

small. It must be concluded that d_{EF F} diverges in this case. Intuitively, one can think that the high degree of connectivity of the graph makes available a large number of directions for the propagation of normal modes.

2.3 Spectral gap and scaling properties of CDT slices

Spectral analysis in CDT ([18]) is defined on spatial slices. Therefore, the volume in (2.16) is the spatial volume VS. What emerges is that spatial

slices of the different phases are characterized by different behaviours in terms of λ1 (figure 2.4) and dEF F (figure 2.5). We will say that a phase

has no spectral gap if its slices have a behaviour consistent with the limit λ1 → 0 when VS → ∞. Neither A phase nor CdS phase shows a spectral

gap. In this sense, the two phases are quite similar, but they have different

2

as usual, we have considered standard ordering of eigenvalues (and related eigenvec-tors) {λn: λn≤ λn0 if n < n0})

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Figure 2.4: Density distribution ρ(λ) of the first 100 eigenvalues for A, C_dS, B phase slices in different range of volumes. C_dS and A cases (figures in the upper left and upper right respectively) show an accumulation of eigenval-ues towards λ = 0, while the B case (bottom figure) seems to suggest the presence of a minimum value λ 6= 0.

Figure 2.5: Effective dimension of A, B, C_dS slices plotted as function of n/VS.

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behaviour in terms of the effective dimension. At small scales (large values of n/V_S), slices of both phases are 3-dimensional. Going to larger scales, dEF F decreases and tends to a value ∼ 1.5. Differences between A phase

and C_dS phase emerge at intermediate scales: in A-slices d_{EF F} is a monotone decreasing function of V_S, while in C_dS there is a minimum and then d_{EF F} grows again approaching higher scales.

B phase instead has a finite value of the spectral gap and the spectral dimension has a completely different behaviour, too. At small scales, d_{EF F} ' 2.5, then it starts growing and diverges at large scales.

Cb phase has peculiar properties. As said, in a typical Cb triangulation

two different classes of slices can be distinguished. They are called B-type slices and dS-type slices (for similarity with slices of the phases already analyzed) and alternate each other in slice-time. B-type slices have a finite value of the spectral gap, while dS slices have no gap: given the distribution of λ1 for different slices in Cb phase, it is possible to distinguish two peaks,

which remain separate in the thermodynamic limit. However, approaching the C_b|C_dS line, the gap on the B-type slices progressively reduces (figure 2.6).

The presence of slices with non-zero values of λ1 allows us to distinguish

the two phases. Therefore, λ₁ can be used as the order parameter of the transition. The critical behaviour of λ1 suggests the possibility of a

continu-ous change approaching the transition line. Moreover, the dimension of LB eigenvalues is that of a mass-squared. A spectral gap that persists in the infinite volume limit can be used to define a typical length of the system. Such a length, namely l1 ∝ 1/

√

λ1, diverges in approaching the critical point

if λ₁ vanishes in a continuous way.

The difference of λ₁ values for the two classes of slices is clearly related to differences in scaling profiles (figure 2.7). One can see that higher eigenvalues of both types of slices collapse on the same curve, so no difference in the small scale behaviour is expected. At larger scales, however, the spectra are separated. The value of n/V at which the division takes place is called bifurcation volume Vb and the Cb phase is also called, for this particular

feature, bifurcation phase. Bifurcation volume plays an important role in the practical implementation of spectral analysis. Indeed, to distinguish dS and B slices, we need spatial volumes V_S > Vb. Parameters of simulations

have to be fixed taking account of this constraint.

It is reasonable to expect a closure of the bifurcation figure in approaching the C_b|C_dS phase transition. This is consistent with the decreasing of λ1 in

B-type slices. We will show later that the closure occurs with a shift of the bifurcation volume that is, therefore, dependent on ∆, k0. The hypothesis

of λ1 approaching zero in a continuous way is equivalent to the statement of

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Figure 2.6: Spectral gap plotted as a function of ∆ and with k0 = 2.2,

VS,tot= 40K.

Figure 2.7: Average of λ in function of n/V for slices in the center of the bulk of a typical Cb triangulations (with k0 = 2.2, ∆ = 0.10, Vtot = 40K).

Time label are defined so that t_slice = 0 is the label of largest B-type slice in each triangulation.

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Chapter 3

Preliminary tests

In this chapter, we will shortly present some simple results that can be obtained when LB spectral analysis is applied to the study of transition lines.

First, we will consider the case of simulations across the A|B phase tran-sition. As said, this transition occurs between phases that are not expected to reproduce our universe. Our interest in this transition is therefore purely academic because the transition is expected to be first-order. In our opinion, it could be useful to understand what observations can be made in such a case, before considering a second-order phase transition.

After that, we will focus on the study of the C_b|C_dS. In this case, we fix the value k0 = 0 and then we approach the transition line by changing ∆.

We are interested in observing whether or not the transition line disappear at a certain point1, to be able to better characterize the CDT phase diagram. The code employed for this study is an implementation in C++ (dis-cussed and tested in previous works ([18])) of the standard CDT algorithm. Eigenvalues and eigenvectors of the LB operator on dual triangulations have been obtained using the “Armadillo” ([19]) C++ library with “Lapack”, “Arpack” and “SuperLU” support for sparse matrix computation. Simula-tions and analysis presented in this chapter and in the next one have been performed at the Scientific Computing Center at INFN-Pisa, at Pisa Data Center in San Piero a Grado and at supercomputer Marconi, CINECA.

3.1 A|B phase transition

A phase can be observed for sufficiently large k0. B phase occurs for

suffi-ciently small k0and small ∆. The two phases present very marked differences

between them. Consider volume profiles: in the B phase, a large portion of the total spatial volume is concentrated on a big single slice. Instead, the

1

It has been observed, instead, that the first-order transition line B|Cb vanishes at

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Figure 3.1: Scatter plot of (λk, k/V ) obtained from spectra of slices in A

phase triangulations (top) with k0 = 5.0 and ∆ = −0.3 and B phase

trian-gulations (bottom) with k₀ = 5.0 and ∆ = −0.2. In both cases, the total spatial volume is VS,tot = 5K. Colours indicate the spatial volumes of the

slice on which a given point has been calculated, accordingly to the colour map on the right side of each figure.

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Chapter 3. Preliminary tests

Figure 3.2: Distribution of λ1 for different values of ∆, with k = 5.0 and

VS,tot= 5K.

spatial volume is randomly distributed in the A phase and slices never show a very large value of VS. Also in terms of spectral properties, slices of the

two phases have completely different behaviour. B phase is characterized by a finite value of the spectral gap and infinite value of the effective dimension. Instead, in A phase we observe a finite value of the effective dimension and a vanishing value of the spectral gap. These different behaviours are clearly visible in the scatter plot of (λ_n, n/V ) as can be seen in figure (3.1). In the B phase, the curve of eigenvalues is approximately flat for smaller values of n/V , while it has a non-vanishing slope in the A phase.

There is no observation of how A|C_dS, C_dS|C_b and C_b|C_dS lines merge in the A|B transition line: in the literature, it is supposed that all the transition lines merge in a quadruple point, but a more complicated picture could be effectively realized. However, we will study A|B transition in a region of the phase diagram where we do not take the risk of coming across other phase transitions.

We have fixed as parameters of the simulations the total spatial volume VS,tot = 5K and k0 = 5.0. We recall that VS,tot is not really fixed and

changes at each step. At the beginning of the simulation, the value of k4

has been fine-tuned so that V_S,tot fluctuates around the considered value. Several values of ∆ have been chosen to go through the phase transition line: in this case, we consider a set of 11 equally spaced values of ∆ in the

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Figure 3.3: Evolution in simulation time of spatial volumes calculated on slices of three different simulations. The simulation indicated with sim3 (green) shows a very different trend with respects to the other two cases.

range [−0.3, −0.2] (figure 3.2). To compare values of λ1 on different slices,

we require that differences should be due only to the geometric structure and not to the volumes. To neglect volume dependence and perform a correct comparison one could try to consider only slices with volume V_S on a strict interval. Still, we recall that configurations in A phase and B phase have really different volume profiles. It is impossible to have a very populated statistical sampled of slices of both phases in a strict interval of values of VS. Still, we can consider that, for large values of VS, the curve of λ1(1/VS)

is expected to be flat (this is a direct consequence of the diverging effective dimension dEF F) in B phase. We can fix a lower bound on VS, that is

intended to select only A-slices with higher volumes, and consider λ₁ on B slices to be approximatively not dependent on VS. This should be sufficient

to make a meaningful comparison. In particular, we have considered only slices with volume V_S > 400. What emerges is a picture (figure (3.2)) not too different from the one seen in figure (2.6). The value of the spectral gap in B phase decreases approaching the phase transition, while it has an approximatively constant value in A phase. It is difficult to say how these different behaviours are connected at the critical point.

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Figure 3.4: Evolution in simulation time of λ1 calculated on slices of three

different simulations. In λ1case, no clear difference seems to emerge between

different simulations.

transition takes place. To get more hints about the order of the transition, we have to make further analysis in that point.

In particular, we have observed that for ∆ = −0.25 a peculiar situation is realized: different configurations extracted by simulations with the same parameters k0 = 5.0, ∆ = −0.25, VS,tot= 5K seem to be in different phases.

It could be seen that volume profiles are different: one can consider how the maximal spatial volume evolves in simulation time, obtaining figure (3.3). The coexistence of two phases at the same point of the phase diagram proves that A|B actually is a first-order phase transition.

We could ask whether or not these differences can be observed in terms of spectral properties of spatial slices. As a first attempt, one can try to compare the distribution of the spectral gap in the different cases (figure (3.4)), but no clear distinction seems to emerge.

Still, it is possible that the typical values of the spectral gap separate approaching higher values of the total volume. Indeed, as already explained, the two phases are characterized by different scaling profiles. We can expect that λ₁ does not have large variations in B slices when V_S becomes larger, while λ1 is supposed to decrease in A slices. The compatibility of λ1 in the

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Figure 3.5: Evolution in simulation time of the effective dimension calcu-lated on slices of three different simulations. Fluctuations of the effective dimension seem to qualitatively follow spatial volume fluctuations in figure (3.3).

Scaling properties of eigenvalues are regulated by the Weyl law, as already mentioned in chapter 2. Then, to validate this hypothesis is necessary to include the effective dimension in our discussion. Considering this quantity, it can be observed a difference in the behaviours of the different simulations (figure (3.5)). Therefore, we can conclude that the coexistence between A and B phases in points of the phase diagram near the transition line can be observed also considering their spectral properties.

3.2 Cb|CdS phase transition

The C phases share a very similar typical volume profile2_{. But when it comes}

to analyzing spectral properties of slices, conspicuous differences come out. If we take a look at the scatter plots (λ_n, n/V ) (fig. 3.6), we will see that two different behaviours are realized. In particular, eigenvalues in C_b phase are disposed on different curves: this is due to the distinction between

B-2

Typically, a Cb volume profiles present a tighter bulk and, coherently, slices on the

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Figure 3.6: Scatter plot of (λk, k/V ) obtained from spectra of slices in CdS

phase triangulations (top) with k0 = 0 and ∆ = 1.0 and Cb phase

triangu-lations (bottom) with k₀ = 0 and ∆ = 0.5. In both cases, the total volume is VS,tot= 20K. Colours indicate the spatial volumes of the slice on which a

given point has been calculated, accordingly to the colour map on the right side of each figure.

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type slices and dS-type slices. B-type slices are characterized by a non-zero spectral gap and a diverging effective dimensions and then are similar to slices in B phase. dS-type slices are characterized by a zero-value of the spectral gap and a finite effective dimension and are similar to slices in C_dS phase.

For the characteristics of the C phases, Cb|CdS is the more promising

transition to our purposes. Analysis performed until now seems to indicate that it could really be a continuous transition. However, the real order of the transition has not been definitively determined.

As in the case of the A|B transition, we will go through the transition line taking a set of equally spaced value of ∆, fixing the values k₀ = 0 and VS,tot= 20K. Once again, we have slices with different spatial volumes VS,

but the comparison between typical values of λ1 in different cases should be

done considering only the “geometrical nature” of slices. In C_b|C_dS case, it is generally possible to find a not too large interval of values of VS

popu-lated both by C_b-slices and by C_dS-slices. This restriction, however, greatly restrict the sample we can use for the analysis.

Another procedure that can be considered consists of removing the de-pendence on the volume from the value of λ₁estimated. We have, from Weyl law, that λn∝ (n/VS) 2 dEF F (3.1) For n = 1, we obtain λ1 ∝ (1/VS) 2 dEF F _(3.2) and then log(λ1) = − 2 dEF F log(VS) + const. (3.3)

The constant in equation (3.3) is with good approximation not dependent on the volume3 and we can define it as log(˜λ1). We assume that in ˜λ1 the

volume dependence has correctly been removed and we can use it to compare different phases using slices with different volumes.

Distributions of log(˜λ1) for different values of ∆ has been collected in

figure 3.2 and it is clearly visible that distribution of log ˜λ1 is a single

dis-tribution for ∆& 0.7 and it is splitted in two part for ∆ . 0.7. Then, the transition line still exists at k0 = 0 and it is not interrupted at some higher

values of k₀.

In the next chapter, we will analyze in more detail the Cb|CdS transition,

in order to establish whether or not the change we have observed in slices

3_{If we consider the Weyl law, the constant is dependent by the dimension. In the case}

of CDT slices, we have already stated that eigenvalues are correctly fitted by Weyl law with a value of the dimension dEF F, defined in chapter 1. It is observed that dEF F is a

function of the volume, then the independence of the constant in equation (3.3) from the volume is just an approximation that we expected to be valid at high values of VS.

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Figure 3.7: Scatter plot of log ˜λ as a function of ∆ for k0 = 0 and VS,tot =

20K.

properties occurs in a continuous way. In the following, we will approach the problem in a different way. Instead of artificially removing volume de-pendence, we will extrapolate the spectral gap in the limit VS → ∞. In this

limit, λ1 → 0 in CdS phase, while in B-like slices a finite value λ∞1 6= 0

per-sists. The question is whether the function λ∞₁ → 0 for some values k_c, ∆c

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Chapter 4

Running scales and a possible

second-order phase transition

In the previous chapter, we have asserted that the A|B phase transition is a first-order phase transition. Our main argument was the observation of the coexistence of both phases on the same point of the phase diagram. Now, we will focus on a more interesting case.

Differences in terms of spectral properties of C_b phase and C_dS phase have been discussed in chapter 2 and again at the end of chapter 3. Then, we only recall an important feature that we will use in the following discus-sion: C_dS slices are characterized by a vanishing value of the spectral gap while C_b triangulations consist in an alternating structure of dS-type slices (where λ1 ' 0) and B-type slices (with a finite non-zero value of the

spec-tral gap). In other words, looking at the values of λ1 for different slices of

different triangulations in C_bphase, it is possible to distinguish two different distributions. The value of λ1calculated in B-type slices defines a mass scale

of the system (λ1dimension is actually that of a mass-squared), that instead

is absent in C_dS slices. The main topic of this chapter is the behaviour of λ1 near the transition, for the purpose to determine how this change occurs.

If λ1 reaches the zero value in a continuous way, it will define a vanishing

typical mass-scale and then a diverging typical length-scale of the system. This is the case of the continuous transition that we are looking for.

A length-scale defined on Cb slices is the bifurcation volume (Vb), defined

qualitatively as the value of n/V_S at which the spectra of B-slices and dS-slices start being different.

In figure (4.1) we have considered triangulations with k₀ = 0.75, VS,tot=

40K and different values of ∆ (in particular ∆ = 0.2, 0.4, 0.6). As can be seen, in all cases the bifurcation is evident (and then all configurations are in

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1e-04 1e-03 1e-02 1e-01 1e+00 λ n ∆=0.2 1e-04 1e-03 1e-02 1e-01 1e+00 λ n ∆=0.4

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

n / V S 1e-04 1e-03 1e-02 1e-01 1e+00 λ n ∆=0.6

Figure 4.1: Scatter plot of λn as a function of n/VS for slices of single

configurations sampled at k = 0.75 and different values of ∆. The total slice volume VS for each configurations has been fixed to VS,tot= 40K.

Cb phase) and the value of Vb1grows approaching the transition line. So, the

separation between the two classes of slices can be observed only at higher and higher scales, while it disappears in C_dS phase. If the transition we are interested in turns out to be a continuous phase transitions, we could expect Vb to diverge approaching the transition line.

It is known that continuous phase transitions can be characterized by critical exponents that describe power law behaviours of specific observables. In our case, we want to show that, approaching the C_b|C_dS transition line, the spectral gap of B-type slices is consistent with these typical trends for a certain value of the critical index.

4.1 Spectral gap in the infinite volume limit

In this section, we will define the method used to estimate the spectral gap in a given point of the phase diagram. As well as k₀ and ∆, we also consider a fixed value of the total spatial volume VS,tot.

1

Actually, an accurate way to determine the value of Vbhas not been defined yet. Still,

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Chapter 4. Running scales and a possible second-order phase transition

Figure 4.2: Scatter plot of λ1 as a function of 1/V for a typical CdS

con-figurations. The parameters of the simulation are k₀ = 0.75, ∆ = 0.7, VS,tot= 40K.

The two types of slices in Cb phase have both to be analyzed to prove

that actually only one of them shows a spectral gap that does not vanish in the infinite volume limit. Because of the strong similarities between slices in CdS phase and dS-type slices on the one hand and between slices in B phase

and B-type slices on the other, we will shortly discuss results obtained in these simpler cases. This preliminary analysis can be considered as a prepa-ration for the study of the Cb phase.

4.1.1 Vanishing spectral gap: CdS phase

Consider the scatter plot in figure (4.2). We have plotted the inverse of the slice volume 1/V_S on the x axis and the value of λ₁ on the y axis. The slices have been extrapolated from triangulations of different indipendent simulations (∼ 400 configurations in total) running at the same point of the phase diagram (k₀ = 0.75, ∆ = 0.7 and VS,tot= 40K).

As explained in chapter 2, the function λn(n/VS) can be properly

de-scribed by using the Weyl law (3.1). The effective dimension d_{EF F} is a function of the volume. Still, in the C_dS case it can actually be fixed to a finite constant value for volumes that are large enough. We want to show that data in figure (4.2) (in which we have considered data only for n = 1)

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Figure 4.3: Best fit curves in the form (4.1) for hλ1i , hλ3i , hλ5i as a function

of h1/V_Si.

follow a similar trend (at least for small values of 1/VS).

To extract the trend of these points, we have to build a binning of the values of V_S: for each bin, we have considered the average value of λ₁(namely hλ1i) and 1/VS (namely h1/VSi) calculated on slices with volumes in that

range.

For technical reasons, volume bins have been chosen to be equally spaced in terms of VS (not in 1/VS, that is the variable represented in the scatter

plot). This ensures that volume bins are more fairly populated. However, this is only a detail of the practical implementation: no effect due to the volume bins definition is expected to appear for a large enough statistical sample. In this way, we can define a new set of data {hλ₁i ± d hλ₁i}2 _{as a}

function of h1/VSi that can be used to implement fit procedures. Actually,

the need to use volume bins rather than calculating averages at given values of 1/V_S, introduce a further source of uncertainty. We need to associate an error also to h1/V_Si. The standard error of the mean does not seem a correct choice. Indeed, the uncertainty we have to consider does not vanish until the width of bins is zero, even if the statistical sample was infinitely

2_{In order to give a correct estimation of the errors, we must take into account}

autocor-relations between data, for example using blocking methods. This consideration highlights a major problem that is particularly important in the analysis of Cb phase, where

auto-correlation time becomes very long approaching the phase transition line so that more and more computing resources are required to perform meaningful analysis.

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Chapter 4. Running scales and a possible second-order phase transition

A dEF F χ2/d.o.f.

λ1(1/VS) 28 ± 3 1.61 ± 0.03 44/46

λ3(1/VS) 33 ± 3 1.72 ± 0.03 31/46

λ5(1/VS) 70 ± 4 1.69 ± 0.03 38/46

Table 4.1: Fit parameters referred to the fit function (4.1) estimated for λ₁, λ3 and λ5 and evaluation of the reduced chi-squared.

populated. Then, the width of the bins or also the spread of data within them seem better alternatives. But even so, we found that (in all cases under examination) the errors associated with the presence of volume bins does not entail any appreciable contribution to fit results. So from now on, we will neglect them. The fit function we have considered is

_¯

λ1 = A h1/VSi2/dEF F (4.1)

and they have been considered data corresponding to values of the slice volume V_S > 500. Function (4.1) seems to correctly reproduce the trend of the data, with a value of the reduced chi-squared χ2/d.o.f. = 44/46. Fit parameters estimated are A = 18 ± 3 and dEF F = 1.61 ± 0.03. This confirms

that the spectral gap closes in the V_S → ∞ limit.

It can be argued that higher eigenvalues (λ_nwith n > 1) are expected to behave in an analogous way. We have considered several eigenvalues, with n up to a few tens. The best fit curves obtained in the cases n = 1, 3, 5 are plotted in figure (4.3), while estimated parameters and values of the reduced chi-squared have been collected in table (4.1). Similar results can be obtained for different values of n. Then, all the eigenvalues (at least in the range we have considered) seem to vanish in the thermodynamical limit. As further evidence, in all cases the scaling profiles are compatible with the effective dimension dEF F ' 1.6 measured in previous works ([18]).

Different total spatial volumes

We have shown that the spectral gap vanishes in C_dS phase. In order to do that, we have considered values of hλni on slices extracted from simulations

with fixed values of k₀, ∆ and VS,tot. Then, we have found that their trend

for 1/V_S→ 0 is compatible with a power law behaviour as expressed in (4.1). Changing the parameters k0 and ∆ (i.e. moving on the phase diagram), we

find different results for the spectral gap in the different phases. While re-maining in the same phase, a non-zero spectral gap can have different values changing ∆ and k0, as already explained (we will show that in more details

in section (4.2)).

Still, we need to fix also the value of VS,totand one might wonder whether

Spectral Analysis and the Phase Diagram of Causal Dynamical Triangulations