Fuzzy Scalar Field Theories: Numerical and Analytical Investigations
CCENTRO DE INVESTIGACION Y DEESTUDIOS AVANZADOS DELINSTITUTO POLITECNICO NACIONAL
DEPARTAMENTO DE FISICA
Teor´ıas de Campo EscalaresDifusas:
Investigaciones Anal´ıticas y Num´ericas
Tesis que presenta
Julieta Medina Garc´ıa
Para obtener el grado de:
DOCTORA EN CIENCIAS
En la especialidad de
F´ısica
Directores: Dr. Denjoe O’ConnorDr.Wolfgang Bietenholz
M´exico, D.F. April, 2006 a r X i v : . [ h e p - t h ] J a n ENTRO DE INVESTIGACION Y DEESTUDIOS AVANZADOS DELINSTITUTO POLITECNICO NACIONAL
PHYSICS DEPARTMENT
Fuzzy Scalar Field Theories:Numerical and AnalyticalInvestigations
Thesis submmited by
Julieta Medina Garc´ıa
In order to obtain the
Doctor in Science degree, speciality in
Physics
Advisors: Dr. Sc. Denjoe O’ConnorDr. Sc. Wolfgang Bietenholz
M´exico City April, 2006 ontents
I Simulations of the λφ Model on the Space S F × S λφ model on a fuzzy sphere 15 λφ model . . . . . . . . . . . . . . . . . . 153.2 The fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.1 Limits of the fuzzy sphere . . . . . . . . . . . . . . . . 173.2.2 The scalar action on the fuzzy sphere . . . . . . . . . . 183.3 Numerical results on the two dimensional model . . . . . . . . 19 λ T > ¯ λ > λ > ¯ λ T . . . . . . . . . . . . . . . 365.2.1 Thermalisation problems . . . . . . . . . . . . . . . . . 395.3 Estimating the maximal number of minima . . . . . . . . . . . 415.4 The equilibrium configurations . . . . . . . . . . . . . . . . . . 43 I − II . . . . . . . . . . 706.5 Collapse of observables . . . . . . . . . . . . . . . . . . . . . . 726.5.1 Collapse for ¯ λ < ¯ λ T . . . . . . . . . . . . . . . . . . . . 726.5.2 Collapse for ¯ λ (cid:29) ¯ λ T . . . . . . . . . . . . . . . . . . . 74 λφ on the fuzzy sphere . . . . . . . . . . . . . . . . . 887.1.2 Non-commutative lattice studies . . . . . . . . . . . . . 89 II Scalar Field Theory on S C P to a fuzzy S F S F in analogy to S F . . . . . . . . . . . . . . . . . . . . . . . 999.2 Review of the construction of C P . . . . . . . . . . . . . . . . 1019.2.1 C P as orbit under Spin (6). . . . . . . . . . . . . . . 104ONTENTS iii9.2.2 C P as orbit under Spin (5). . . . . . . . . . . . . . . . 108
10 Decoding the geometry of the squashed C P F ds . . . . . . . . . . . . . . . 119
11 Conclusions from part II 12312 General conclusions and perspectives 127A A small description of the Monte Carlo method 131
A.1 The Metropolis algorithm . . . . . . . . . . . . . . . . . . . . 132A.1.1 Modifying the Metropolis algorithm . . . . . . . . . . . 133A.1.2 An adaptive method for independent simulations . . . 134A.2 Methods to estimate the error . . . . . . . . . . . . . . . . . . 137A.2.1 Binning method . . . . . . . . . . . . . . . . . . . . . . 138A.2.2 Jackknife method . . . . . . . . . . . . . . . . . . . . . 138A.2.3 Sokal-Madras method . . . . . . . . . . . . . . . . . . . 138A.3 Technical notes . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B Polarisation tensors for SU (2) B.1 Explicit form of the generators of SU (2) IRR of dimension d L C Aside results 145
C.1 Criteria to determine the phase transition . . . . . . . . . . . 145C.2 Free field results . . . . . . . . . . . . . . . . . . . . . . . . . . 147
D Tables 149E Representations and Casimir operators 151
E.1 Explicit form of the generators of SO (6) in the 4 dimensionalIRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151v CONTENTSE.2 Gell-Mann matrices of SU (4) . . . . . . . . . . . . . . . . . . 152E.3 SO ( N ) Casimir operators . . . . . . . . . . . . . . . . . . . . 154E.4 Dimension of representations of SO ( N ) and SU ( N ) . . . . . . 156 F Calculation of the induced metric of C P as Spin (5) orbit 159 cknowledgement
It is with sincere appreciation that I wish here to acknowledge my debt toall those who contributed so much to making this study possible.I want to thank to CONACYT for supporting me with a Ph.D. scholar-ship.I am very indebted to the CINVESTAV which made available a grant-in-aid.To the
Dublin Institute for Advanced Studies where I did a predoctoralstay, I am indebted for many courtesies which facilitated my research includ-ing financial support.To the
Institut f¨ur Physik, Humboldt Universit¨at zu Berlin for allowingme do a stay there.To my advisors Dr. Denjoe O’Connor and Dr. Wolfgang Bietenholz forall of their guidance, encouragement, and support, I am grateful for theircareful review of my thesis, and his valuable comments on my work.I wish to thank all my former colleagues, especially Idrish Huet, FernandoGarc´ıa, Pavel Castro, Rodrigo Delgadillo and Juan Aguilar.To people in the Physics Department, especially to Dr. Gabriel L´opezCastro for his help and support, to Dr. Jos´e Mendez. To my friends fromthe Physics Department, Xavier Amador, Carlos Soto, Alfredo L´opez, SendicEstrada, Sara Cruz and Luz del Carmen Cort´es.To people belonging to the fuzzy gang : Xavier Martin, Brian Dolan,Professor Balachandran, Marco Panero and Badis Ydri. A special acknowl-edgement goes to Frank Hofheinz and Jan Volkholz for introducing me toparallel computing and invaluable comments on lattice topics. v would like to thank all of those people who provided technical support,especially to Ronan Cunniffe, system administrator from
DIAS during mystay there.Additionally, I wish to thank all of secretariat staff of the Physics De-partment, especially to Patricia Villar, Ma. Eugenia L´opez and Flor Ib´a˜nez.All my friends deserve my special thanks for bringing joy into my life.My deepest thanks go to my parents, for their loving care and encour-agement and for giving me the foundation from which to build my life. I amalso grateful to my sister Gabriela.Finally, I owe a very special thanks to my husband Amilcar for his un-failing love and support through it all. His understanding attitude towardsmy work and confidence in me has been essential for the completion of thisthesis.vi esumen
Este trabajo est´a dedicado al estudio de Teorias de Campo (QFT) en espaciosdifusos .Los espacios difusos son aproximaciones al ´algebra de funciones de unespacio continuo por medio de un ´algebra matricial finita. En el l´ımite dematrices infinitamente grandes la aproximaci´on es exacta.Una caracteristica atractiva de esta aproximaci´on es que muestra de unamanera transparente como son preservadas las propiedades geom´etricas delespacio continuo.En el estudio del r´egimen no perturbativo de QFT los espacios difusosproveen una posible alternativa a la red como m´etodo de regularizaci´on. Estatesis est´a dividida en dos partes:1. Realizamos la simulaci´on Monte Carlo de la teor´ıa λφ en un espacioEuclideano de tres dimensiones. La regularizaci´on se compone por unaesfera difusa de dos dimensiones, S F , para las direcciones espacialesm´as una red convencional para la direcci´on temporal. Se identifica eldiagrama de phase de este modelo. Adem´as de las fases desordenaday ordenada uniforme usuales encontramos una tercera fase de orde-namiento no uniforme. Ello indica la existencia del fen´omeno conocidocomo mezclamiento UV-IR en el r´egimen de acoplamiento fuerte.2. Como segundo punto presentamos un an´alisis geom´etrico de una teor´ıaescalar general en una esfera difusa de cuatro dimensiones, S F . Unaaproximaci´on para S es de un interes especial dado que S es la subs-tituci´on natural de R en estudios de QFT Euclideana. viiin embargo la versi´on difusa de S no puede obtenerse mediante la cuan-tizaci´on del espacio cl´asico.El problema es rodeado definiendo una teor´ıa escalar en un espacio m´asgrande , que es C P el cu´al es de dimension seis. Incluye grados de libertadde S m´as otros que no lo son. Esos grados de libertad extras se eliminandin´amicamente mediante un m´etodo probabil´ıstico. El an´alisis de las estruc-turas geom´etricas nos permite interpretar a este procedimiento como unareducci´on de Kaluza-Klein de C P a S . bstract This work is devoted to the study of Quantum Field Theories (QFT) on fuzzyspaces .Fuzzy spaces are approximations to the algebra of functions of a continu-ous space by a finite matrix algebra. In the limit of infinitely large matricesthe formulation is exact.An attractive feature of this approach is that it transparently shows howthe geometrical properties of the continuous space are preserved.In the study of the non-perturbative regime of QFT, fuzzy spaces providea possible alternative to the lattice as a regularisation method. The thesis isdivided into two parts:1. We perform Monte Carlo simulations of a λφ theory on a 3-dimensionalEuclidean space. The regularisation consist of replacing space by afuzzy 2-dimensional sphere, namely S F , and Euclidean time by a con-ventional lattice. We identify the phase diagram of this model. Inaddition to the usual disordered and uniform ordered phases we finda third phase of non-uniform ordering. This indicates the existence ofthe phenomenon called UV-IR mixing in the strong coupling regime.2. Second we present a geometrical analysis of the scalar field theory on a4-dimensional fuzzy sphere, S F . An approximation to S is of specialinterest since S is the natural replacement of R in studies of Eu-clidean QFT. Nevertheless a fuzzy version of S cannot be achieved byquantisation of the classical space. ixhe problem is circumvented by defining a scalar theory on a larger space, C P which is 6-dimensional. It includes degrees of freedom related to S plusothers beyond S . Those extra degrees of freedom are dynamically suppressedthrough a probabilistic method. The analysis of the geometrical structuresallows us to interpret this procedure as a Kaluza-Klein reduction of C P to S . hapter 1Introduction Physics works best when there is a good interaction between experiment andtheory. Unfortunately, for many of the interesting questions that arise, eitherit is impossible to perform the appropriate experiments or they are too costly.This is where the power of computer simulations plays an important role. Insome sense, the computer simulation plays the role of the experiment. Inmodern particle physics, the experimental tools are large accelerators andthe theories are typically quantum field theories. One of these is that of thestrong interactions known as Quantum Chromodynamics. It is very difficultto extract some predictions from this theory as they fall in a non-perturbativeregime and many physicists have resorted to computer simulations to extractthe physical predictions.Similarly, many of the more speculative ideas emerging in physics involvestrongly interacting field theories, some of these have novel features suchas non-commutativity of the space-time coordinates. This type of structureis also suggested by string theory. The work of this thesis is dedicated todeveloping non-perturbative techniques adequate to these non-commutativetheories and hopefully to string theory.Fuzzy spaces are included in the wider framework of non-commutativegeometry.The idea of involving non-commutativity into Physics dated from themiddle of the last century, nevertheless the substantial development has taken1place in the last few years. There are several reasons why studying non-commutative (NC) spaces has become so popular in the physics community.Although our interest in the study of fuzzy spaces is related to the study ofQuantum Field Theories as we will discuss later, we would like to mentionsome other motivations for the study of NC geometry in Physics.Many interesting phenomena in Physics have been discovered by ex-tensions, therefore generalising commutative spaces into non-commutativespaces seems a natural extension. In this spirit, generalising commutativespaces to non-commutative spaces seems motivated. Non-commutativitycan be incorporated into many branches of Physics like Gravitational The-ories, Condensed Matter Physics and Quantum Field Theories. The firstattempts to involve NC theories in Quantum Gravity date from the lastdecade [1]. Fuzzy spaces can be found in String Theories (with D-Branes)under certain conditions — see Ref.[2]-[4]. In Condensed Matter Physics itwas found that the Quantum Hall Effect can be formulated in terms of non-commutative coordinates where a magnetic background field B is related tothe non-commutative parameter [5]-[7].One of the open problems in Field Theory is the existence of non welldefined finite quantities: the divergences . The regularisation procedure mod-ifies the Field Theory to remove those divergences. The three well establishmethods to the date are the dimensional regularisation, Pauli-Villars regu-larisation and the lattice regularisation [8]. The first two methods are forexclusive application at small coupling regimes. Regarding our motivation,we plan to test the feasibility of fuzzy spaces as a regularisation scheme inQuantum Field Theories. It should work, as the lattice procedure, at anyregime.Suppose we want to study QFT through the path integral formalism on agiven space. If we want to access the non-perturbative regime it is necessaryto discretise the space in order to get a finite number of degrees of freedom.The standard method is to approximate the space by discrete points — alattice — representing the space and then calculate the observables over thatset of points. This simple idea has generated some of the most successful Introduction M if we have thealgebra A of functions over M , a Hilbert space, H , and a differential operatorable to specify the geometry (in Ref. [10] that operator is the Dirac operator, D , although for scalar theories as those studied in the present thesis it wasconjectured in [11] that the Laplace-Beltrami operator ∆ is enough to specifythe geometry). Then, instead of discretising directly M by means of pointswe can discretise the triplet ( A , H , D ) and here the fuzzy spaces enter: theyare essentially discretisations at algebraic level. If we want to obtain a finitenumber of degrees of freedom — namely, the coefficients in the expansion ofa function in the algebra basis — the algebra has to be finite dimensional,i.e. a matrix algebra of dimension N , M at N , and as a consequence we havea non-commutative algebra. The elements in the algebra act on a finitedimensional version of the Hilbert space, H N , and an appropriate version of D is needed. In the limit N −→ ∞ (called the commutative limit) we haveto recover M .Summarising, the fuzzy discretisation consist in the replacement:( A , H , D ) −→ ( M at N , H N , D N ) . The term “fuzzy” originates from the following observation: since in thefuzzy space the coordinates will be matrices and they do not commute, this Spaces are included in this more general notion of manifold . For our purposes we workonly with spaces. We denote it as fuzzification . will mean in the Quantum Mechanics spirit that the notion of points doesnot exist, i.e. the space turns fuzzy.The seminal work on fuzzy spaces is due to Madore; in his work [12] afuzzy approximation of a two-dimensional sphere is constructed. Since thenthere exists a large compendium of fuzzy literature, e.g. Refs. [13]-[18]. Mostof fuzzy spaces have been constructed based on the following observation: Ifwe quantise a classical phase space we obtain a finite dimensional Hilbertspace. This implies that the candidates to be quantised are manifolds offinite volume which have a symplectic structure. Co-adjoint orbits of Liegroups fall into this class. A didactic example of them are the complexprojective spaces, C P n , they are 2 n -dimensional spaces that can be definedas SU ( n ) orbits. A discussion of its fuzzy version is given in Ref. [18]. Thefamily of C P nF is especially interesting since C P F ∼ = S F ; S F and S F can beobtained form C P F (see Ref. [16]).Once we count with a fuzzy version of a space, the next step in ourprogram is to define a Field Theory on it, e.g. Refs. [19]-[25]. Then we needto construct fuzzy versions of Laplacians, Dirac operators, etc. The solidmathematical background of the fuzzy approach makes it easy to identifysuch fuzzy versions. Of course one has to check whether the proposed theoryreproduces the continuum theory.Field Theories on fuzzy spaces share a generic property of Field Theorieson general non-commutative spaces called the UV-IR mixing [26]. The UV-IR mixing was originally discovered in perturbative calculations Refs. [27]-[29]. In non-commutative spaces we have two kinds of diagrams, those thatreduce to commutative diagrams and those diagrams without a commutativecounterpart, these are divergent at low momenta.Simulations on fuzzy spaces are a relatively recent topic, see Refs. [30]-[34].In the present thesis we concentrate on the study of Scalar Field Theorieson fuzzy spaces. We cover two important remarks in the fuzzy program:1. Test the feasibility as a discretisation method through a numerical sim- Introduction λφ model on the 3dimensional Euclidean space S ⊗ T by means of the Metropolis algorithm.Our regularisation consists of • the fuzzy sphere S F for the spatial coordinates • a conventional lattice with periodic boundary conditions for the timedirection.The organisation of this first part of this thesis is the following: In chapter4 we present a detailed description of both discretisation schemes, empha-sising the advantages of each method. In chapter 5 we present the charac-terisation of the phases in the model; we dedicate part of this chapter tothe description of some technical aspects related to thermalisation problemsin Monte Carlo simulations. In chapter 6 we identify the phase diagram ofthe model analysing the scaling behaviour of the critical lines. In chapter7 we present the discussion of our results. The key point of this analysis isthe behaviour of the triple point under different limits. It reveals that inthe thermodynamic limit N −→ ∞ is it not possible to recover the Isinguniversality class due to the dominance of a phase that breaks the rotationalsymmetry spontaneously. We find that the UV-IR mixing predicted in theperturbative regime of the model appears in the strong coupling regime aswell. There are ways to remove those divergences. In the context of thescalar field theory λφ on the fuzzy sphere this is done by a suitable choiceof the action [22]. The UV-IR mixing exists in the non-perturbative regimee.g. Refs. [35]-[38] where it was detected as a matrix or striped phase whichhas no counterpart in the commutative theory.As a second point in this thesis we present an analytical part. We studya Scalar Field Theory on S F . S is a special 4-dimensional curved space,taking its radius to infinity, we arrive at R . S is not a phase space, henceits “construction” involves some complications that are explained in chapter9. To solve this problem we allow S to fluctuate into a larger space, this is C P . We present a review of the construction of C P as SU (4) ∼ = Spin (6)orbit. Nevertheless for moving on S , it is enough to preserve rotations in 5dimensions. We find that C P can also be constructed as a Spin (5) orbit,but demanding this less restrictive symmetry we construct a squashed C P .For our purposes we demonstrate that this construction has more advantagessince it allows us to identify C P as a fibre bundle over S with S as the fibre.We start chapter 10 with a short review of [39]. There, a Scalar Field Theoryon an squashed C P F is defined. Then a penalisation method for all the non S modes is introduced. This probabilistic method introduces an apparently“artificial” parameter h , such that h positive and large makes the non S configurations improbable. Now we give an interpretation to this parameterthrough a geometrical analysis of the proposed model. Using coherent statetechniques it is possible to extract the geometry. At the end we are able to“visualise” the penalisation method as a Kaluza Klein reduction of C P to S . h is interpreted in terms of the radius of the fibre S . art ISimulations of the λφ Modelon the Space S F × S hapter 2Generalities of the method The functional integral provides a powerful tool to study Quantum FieldTheories. It can be thought of as a generalisation of the path integral for-malism in Quantum Mechanics introduced by R. P. Feynman in the late 40’s(see e.g. [8] and [40]). The crucial idea behind the path integral is the su-perposition law. If we want to calculate the transition amplitude for goingfrom an initial state at τ (cid:48) to a final one at time τ (cid:48)(cid:48) , one has to consider asuperposition of all possible paths.To state this in a mathematical form, let us suppose the initial state at τ (cid:48) to be denoted by | ψ ( x (cid:48) ) (cid:105) and at time τ (cid:48)(cid:48) we have | ψ ( x (cid:48)(cid:48) ) (cid:105) . Then the transitionamplitude is given by (cid:104) ψ ( x (cid:48)(cid:48) ) |U ( τ (cid:48)(cid:48) , τ (cid:48) ) | ψ ( x (cid:48) ) (cid:105) (2.1)where U ( τ (cid:48)(cid:48) , τ (cid:48) ) = e − ı H ( τ (cid:48)(cid:48) − τ (cid:48) ) / (cid:126) , (2.2) H is the Hamiltonian of the system, which we assume to be time independent.We start slicing the time interval [ τ (cid:48) , τ (cid:48)(cid:48) ] into N subintervals of duration (cid:15) = τ i + i − τ i as in figure 2.1 . 90 2.1. Path integrals and functional integrals τ = τ (cid:1) τ τ τ N = τ (cid:1)(cid:1) τ N − x (cid:1) x x (cid:1)(cid:1) x N − ... x −→|| | x Figure 2.1: Schematic representation of the path.The time evolution operator can be broken into intervals: e − ı H ( τ (cid:48)(cid:48) − τ (cid:48) ) / (cid:126) = e − ı (cid:126) H ( τ N − τ N − + τ N − −···− τ + τ − τ + τ − τ ) , = (cid:0) e − ı H (cid:15)/ (cid:126) (cid:1) N . (2.3)where τ (cid:48) = τ and τ (cid:48)(cid:48) = τ N . H has the general form H = H + V , where H = p m . For (cid:15) → e − ı H ( τ (cid:48)(cid:48) − τ (cid:48) ) / (cid:126) ≈ (cid:0) e − ı H (cid:15)/ (cid:126) e − ı V (cid:15)/ (cid:126) (cid:1) N , (2.4)where we have used Trotter’s formula. Eq. (2.4) holds if H and V aresemibounded. The next trick is to insert between each term e − ı H (cid:15)/ (cid:126) e − ı V (cid:15)/ (cid:126) the set of complete states (cid:90) dx i | ψ ( x i ) (cid:105)(cid:104) ψ ( x i ) | = , (2.5)then eq. (2.1) can be written as the product of N terms (cid:104) ψ ( x (cid:48)(cid:48) ) |U ( τ (cid:48)(cid:48) , τ (cid:48) ) | ψ ( x (cid:48) ) (cid:105) ≈ (cid:90) dx N − (cid:90) dx N − · · · (cid:90) dx (cid:104) ψ ( x N ) | e − ı H (cid:15)/ (cid:126) e − ı V (cid:15)/ (cid:126) | ψ ( x N − ) (cid:105)×(cid:104) ψ ( x N − ) | e − ı H (cid:15)/ (cid:126) e − ı V (cid:15)/ (cid:126) | ψ ( x N − ) (cid:105) ... ×(cid:104) ψ ( x ) | e − ı H (cid:15)/ (cid:126) e − ı V (cid:15)/ (cid:126) | ψ ( x ) (cid:105) . Generalities of the method V depends on the position X and H depends on the momentum P . It is possible to demonstrate that (see e.g. Refs. [8],[41]-[42]): (cid:104) ψ ( x i +1 ) | e ı H (cid:15)/ (cid:126) e ı V (cid:15)/ (cid:126) | ψ ( x i ) (cid:105) ≈ (cid:90) dp i +1 π (cid:126) e ı (cid:126) ( p i +1 ( x i +1 − x i ) − (cid:15) H ( p i +1 , ( x i +1 + x i )) ) . (2.6)In eq. (2.6) we note that the argument in the exponential can be written as ı (cid:126) (cid:15) (cid:18) p i +1 x i +1 − x i (cid:15) − H ( p i +1 ,
12 ( x i +1 + x i )) (cid:19) , where we recognise a discrete version of Lagrangian in the interval [ τ i , τ i +1 ]times the duration of the interval (cid:15) , i.e., the action in such an interval. Takingthe product of the N -terms of the type in eq. (2.6) and taking N → ∞ wearrive at (cid:104) ψ ( x (cid:48) ) |U ( τ (cid:48)(cid:48) , τ (cid:48) ) | ψ ( x (cid:48) ) (cid:105) = (cid:90) x ( τ (cid:48)(cid:48) )= x (cid:48)(cid:48) x ( τ (cid:48) )= x (cid:48) [ Dx ( τ )] e ıS [ x ] / (cid:126) (2.7)where S [ x ] is the action S [ x ] = (cid:90) τ (cid:48)(cid:48) τ (cid:48) L ( x, ˙ x ) dτ, (2.8)and L is the Lagrangian of the system. x ( τ ) is a path that interpolatesbetween x (cid:48) and x (cid:48)(cid:48) , [ Dx ( τ )] is the functional measure, therefore “ (cid:82) x (cid:48)(cid:48) x (cid:48) [ Dx ( τ )]”denotes the integral over all paths between x ( τ (cid:48) ) = x (cid:48) and x ( τ (cid:48)(cid:48) ) = x (cid:48)(cid:48) .An important remark is that from eq.(2.7) we can beautifully recover theleast action principle noting that in the limit, (cid:126) →
0, the path of minimumaction dominates the integral since the phase e ıS/ (cid:126) of any path away fromthis fluctuates rapidly and different contributions cancel.The generalisation to quantum fields is a straightforward generalisation ofeq. (2.7). But before introducing its expression we would like to remark thatusing path-integral methods it is common to give the action an imaginarytime in order to simplify the calculations — the weight in the path integralis an exponential with real argument, which is easier to handle numerically2 2.1. Path integrals and functional integrals— and then return to a real action at the end. This can be done if theOsterwalder-Schrader axioms hold (see Refs. [43]-[44]).Besides simplification purposes, in imaginary time the paths away fromthe classical path are exponentially suppressed. This makes the path integralto converge much better than the phase rotation. This is crucial for numericalstudies since it allows us to have reliable results with a relatively modeststatistics.There is a deeper consequence of considering an imaginary time: it allowsus to establish a connection to Statistical Mechanics. In complex analysis, a branch of mathematics, analytic continuation is atechnique to be used in the domain of definition of a given analytic function.We can apply such techniques here to go from real time τ to the imaginarytime t called the Euclidean time . (For a formal treatment see Ref. [41]).Imaginary time and spatial coordinates play equivalent rˆoles. For exam-ple, in real time the D’Alembertian operator is given by: (cid:50) = ∂ ∂τ − ∂ ∂x − ∂ ∂x − ∂ ∂x . Under the Euclidean prescription we have∆ = − ∂ ∂x − ∂ ∂x − ∂ ∂x − ∂ ∂x , where we set x = t .In Quantum Mechanics we consider the possible particle positions at eachtime, given by functions x ( t ) (or (cid:126)x ( t ) in d = 3), and the path integral inte-grates over all these functions, i.e. over all possible particle paths (with thegiven end-point). This reproduces the canonical Quantum Mechanics, butspace and time are not treated in the same way. In field theory, one does treatthem in the same manner and introduces a functions of any space-time point x = ( (cid:126)x, t ), which are denoted as fields. The simplest case is a neutral scalarfield, where this field values are real, φ ( x ) ∈ R . The assignement of a field Generalities of the method x is called a configuration, and it takes overthe role of paths in Quantum Mechanics. Consequently, the functional inte-gral now runs over all field configurations, (cid:82) [ Dφ ]. The Lagrangian L is nowthe integral of a Lagrangian density at each point (cid:126)x , L ( φ ( x ) , ∂ µ φ ( x )), and theaction is obtained by an integral over the space-time volume, S = (cid:82) d x L .In imaginary time, the analog of eq. (2.7) for quantum fields is given by (cid:104) φ (cid:48)(cid:48) |T ( t (cid:48)(cid:48) , t (cid:48) ) | φ (cid:48) (cid:105) = (cid:90) [ Dφ ( x, t )] e − S [ φ ] . (2.9)In the Euclidean formulation the time evolution becomes a transfer ma-trix, T ( t (cid:48)(cid:48) , t (cid:48) ), and in the case that H is time independent eq. (2.2) becomes e −H t . Furthermore the quantum partition function Tr (cid:0) e − β H (cid:1) becomes thefunctional integral over paths that are periodic in Euclidean time of period β , where β = k B T with k B Boltzmann’s constant. There is also a secondinterpretation of the resulting functional integral as a functional integral instatistical field theory. Here one considers the Euclidean action as the energyfunctional of an analog statistical mechanical system with k B T = 1. As isconventional in lattice field theory it is the latter analogue that will be usedin this thesis. Then eq. (2.9) describes a statistical system in equilibrium. The expectation values of an observable F , denoted (cid:104) F (cid:105) , can be calculatedas follows: (cid:104) F (cid:105) = 1 Z (cid:90) [ Dφ ] F ( φ ) e − S [ φ ] . (2.10)where Z = (cid:90) [ Dφ ] e − S [ φ ] (2.11)is the partition function.The integration in eq. (2.10) involves all the possibles configurations inthe functional space.The problem is how to measure (or estimate) the value in (2.10). Here iswhere the importance sampling methods enters. The most popular approach We set (cid:126) = 1. representative samples is through random movesto explore the search space.In this thesis we use a variant of the Monte Carlo method called theMetropolis algorithm [45] to estimate the expectation values of the observ-ables defined in chapter 4. hapter 3A review of the 2 dimensional λφ model on a fuzzy sphere We devote this section to a review of some aspects of the 2-dimensional λφ model on a fuzzy sphere. We will discuss generic properties of fuzzy spacesby means of the most studied example: the fuzzy sphere. We will show thatthe fuzzy sphere can retain the exact rotational symmetry. λφ model A quite general scalar field theory on a 2 dimensional sphere is given by theaction s s ( φ ) := (cid:90) S (cid:20) φ ( x ) L R φ ( x ) + V [ φ ( x )] (cid:21) R d Ω (3.1)where d Ω = sin θdθdϕ , φ ( x ) is a neutral scalar field on the sphere. It dependson the coordinates x i ( θ, ϕ ) which satisfy: (cid:88) i =1 x i = R , (3.2)where R is the radius of the sphere. L = (cid:80) i L i , and L i are the angularmomentum operators. V [ φ ( x )] is the potential of the model.Eq. (3.2) describes S embedded in R . 156 3.2. The fuzzy sphere To obtain a fuzzy version of a continuous space we have to replace the algebraof the continuous space by a sequence of matrix algebras of dimension N , Mat N .In the case of the fuzzy sphere , the permitted values of N are L + 1 where L is the largest angular momentum (the cutoff), which can take the values L = 0 , , , · · · . The coordinates x i are elements in the algebra of functionsof S , C ∞ ( S ). They are replaced by the coordinate operators , X i , which aredefined as X i = 2 R L i √ N − , where L i , i = 1 , ,
3, are the SU (2) generators inthe N = ( L + 1)-dimensional irreducible representation.The coordinate operators satisfy the constraint (cid:88) i =1 X i = R · , (3.3)which can be interpreted as a matrix equation for a sphere, the analog toeq. (3.2). Note that the operators X i do not commute,[ X i , X j ] = ı(cid:15) ijk R √ N − X k . (3.4)Following the above prescription, the scalar field is represented by a her-mitian matrix Φ of dimension N . Just as in the standard case where φ canbe expressed as a polynomial in the coordinates x i , its fuzzy version Φ can bewritten as a polynomial in the fuzzy coordinates . The differential operators L i · are replaced by [ L i , · ] and the integral over S is replaced by the trace.Summarising the above: x i ∈ C ∞ ( S ) −→ X i ∈ M at N , (3.5) φ ( x ) ∈ C ∞ ( S ) −→ Φ ∈ M at N , (3.6) L i φ ( x ) −→ [ L i , Φ] , (3.7) R (cid:90) S φ ( x ) d Ω −→ πR N Tr (Φ) , (3.8) L · −→ ˆ L · := (cid:88) i =1 [ L i , [ L i , · ]] . (3.9) A review of the 2 dimensional λφ model on a fuzzy sphere φ ∈ R implies that Φ is hermitian and the choice of normali-sation in eq. (3.8) ensures that the integral of the unit function equals thetrace of the unit matrix, i.e.4 πR N Tr11 = 4 πR = R (cid:90) S d Ω . (3.10)Rotations on the fuzzy sphere are performed by the adjoint action of anelement U of SU (2) in the dimension N unitary irreducible representation. U has the general form U = e ıω i L i . The coordinate operators are then rotatedas U X i U † = R ij X j , R ∈ SO (3) (3.11)and the field transforms as Φ −→ Φ (cid:48) = U Φ U † . (3.12) Following [20], for the spatial part of our model (the fuzzy sphere) we have: • The commutative sphere limit S : N −→ ∞ , R fixed . (3.13) • The
Moyal plane limit R N −→ ∞ , R = N Θ2 , Θ constant . (3.14) • The commutative flat limit R N −→ ∞ , R ∝ N (1 − (cid:15) ) , > (cid:15) > . (3.15)The limit given by eq. (3.13) arises naturally from the fact that N −→ ∞ recovers C ∞ ( S ).A short way to deduce eqs. (3.14)-(3.15) is the following:8 3.2. The fuzzy sphereConsidering the north pole on the fuzzy sphere where X ∼ R
11 , wecan re-scale the coordinate X to X R ∼
11 , and re-write the commutationrelation (3.4) at the north pole as[ X , X ] = ı R √ N − X ≈ ı R N X R . (3.16)We propose R as a function in N . For the non-commutative plane we have:[ X , X ] = ı Θ , (3.17)then, comparing eq. (3.16) to (3.17) we obtain R N = Θ.We define the exponent (cid:15) in the relation R ∝ N − (cid:15) , > (cid:15) ≥ . (3.18)If (cid:15) = 0 we have Θ = const. For (cid:15) > N −→ ∞ . Note that in this limit we also require the commutator given byeq. (3.4) to vanish and this requirement is immediately satisfied for (cid:15) > The next step is to define our field theory on the fuzzy sphere. Implement-ing the replacements given by eqs. (3.5)-(3.9) in eq. (3.1) we arrive at thefollowing expression, s s [Φ] = 4 πR N Tr (cid:16)
12 Φ ˆ L R Φ + V [Φ] (cid:17) . (3.19)Eq. (3.19) is valid for any potential V [Φ]. For testing purposes it isconvenient to select a simple model. In Ref. [32] the λφ model on a fuzzysphere was studied, where the action is written as S [Φ] = Tr (cid:2) a Φ L Φ + b Φ + c Φ (cid:3) . (3.20)Φ is a Hermitian matrix of size N . After a suitable rescaling, the parameters b and c become the mass squared and the self-coupling, respectively.In Ref. [32] φ was rescaled to fix a = 1. The model in eq. (3.20) was alsostudied in Ref. [30] but in terms of a different convention of parameters a = 4 πN , b = arR , c = aλR . (3.21) A review of the 2 dimensional λφ model on a fuzzy sphere The model in eq. (3.20) has been studied numerically by several authors—see Refs. [30],[32]-[34]. We follow those results in Refs. [32], which aresummarised in figure 3.1 . c / N -b/N Disorder phase Non-Uniform Order phaseUniform Order phaseTriple point (0.80 ± ± -2 =(bN -3/2 ) /4 Figure 3.1: Phase diagram obtained from Monte Carlo simulations of themodel (3.20) in Ref. [32].
Figure 3.1 shows the existence of three phases: • Disordered • Non-uniform ordered • Uniform ordered We thank the authors of Refs. [32] for their permission to reproduce the graph here. N for the axes − bN / vs. cN . Two coexistence lines are: • Disordered - Ordered uniform: b = − . √ N c. (3.22) • Disordered - Non-uniform ordered: cN = ( bN − / ) . (3.23)The triple point is given by the intersection of the three critical lines( b T , c T ) = ( − . N / , . N ) . (3.24)The transition in eq. (3.23) is only valid for large values of c . Thereforeeq. (3.24) is not the intersection of eqs. (3.22)-(3.23).Although we discretised the spatial part of our three dimensional modelby means of a fuzzy sphere , our simulations will show that the propertiesdiffer from those of the two dimensional model in eq. (3.20). In certain limitsthe λφ theory on the fuzzy sphere can emerge as a limit of the 3-dimensionalmodel in eq. (4.7).The model in eq. (3.20) depends on the parameters N, R, m and λ —orequivalently it depends on N, a, b and c , see eq .(3.21)— but it effectivelydepends only on two out of three parameters a, b and c . For the threedimensional model in addition to the parameters in the 2-dimensional modelwe have as parameters the number of lattice sites, N t and the lattice spacing∆ t . We will fix in section 4.4.3 N t = N and the model will effectively dependon four parameters.The first question is if the three dimensional model has the phase of non-uniform ordering. We will see in chapter 5 that the answer to this questionis yes. hapter 4Description of the model In this Chapter we present the discretisation of the 3-dimensional modelcomposed by a 2-dimensional sphere plus a Euclidean time direction.We first recall the results of chapter 3 for the discretisation of the continu-ous model. We will apply them to the 3-dimensional model. After performingthe discretisation in the time direction we present the model to be studiedby Monte Carlo techniques. In section 4.4 we present the observables and abrief description of their meaning.
It is convenient to consider the Euclidean version of the model.As it was remark in chapter 2, the main advantage of working in thisformalism is that it allows to establish a connection to Statistical Physicsand the functional integral converges with a relatively modest statistics.The model to regularise is S ( φ ) := (cid:90) S dt (cid:90) S (cid:20) φ ( x, t ) (cid:18) L R − ∂ t (cid:19) φ ( x, t ) + m φ ( x, t ) + λ φ ( x, t ) (cid:21) R d Ω(4.1) φ ( x, t ) is a neutral scalar field on the sphere. It depends on time (euclidean)and on the coordinates x i ( θ, ϕ ) satisfying eq. (3.2), where R is the radius ofthe sphere. 212 4.1. Regularisation of the actionWe consider the integral over Euclidean time on a compact version, S ,which has circumference T .First we will explain how to discretise the spatial directions and then wewill perform the discretisation in time direction. Let us consider the spatial part of the action given in eq. (4.1), s ( φ, t ) := (cid:90) S (cid:20) φ ( x, t ) (cid:18) L R − ∂ t (cid:19) φ ( x, t ) + m φ ( x, t ) + λ φ ( x, t ) (cid:21) R d Ω . (4.2)Implementing the replacements given by eqs. (3.5)-(3.9) in eq. (4.2) wearrive at s [Φ , t ] = 4 πR N Tr (cid:16)
12 Φ( t ) (cid:32) ˆ L R − ∂ t (cid:33) Φ( t ) + m ( t ) + λ ( t ) (cid:17) . (4.3)Then, the action (4.1) “discretised” in the spatial directions is: S [Φ] = 4 πR N (cid:90) S dt Tr (cid:104)
12 Φ ( t ) (cid:32) ˆ L R − ∂ t (cid:33) Φ ( t )+ m ( t )+ λ ( t ) (cid:105) . (4.4)The model given by eq. (4.4) has the exact rotation symmetry of model (4.1)since any rotation on the sphere is allowed and the action (4.4) is invariantunder uniform rotations given by eq. (3.12). To discretise the time direction we take a set of N t equidistant points, then T = N t ∆ t .The changes to implement in eq. (4.1) are: (cid:90) S dt −→ N t (cid:88) t =1 ∆ t, (4.5) ∂ t φ ( x, t ) −→ φ ( x, t + ∆ t ) − φ ( x, t )∆ t . (4.6) Description of the model S [Φ] = 4 πR N ∆ t N t (cid:88) t =1 Tr (cid:104) R Φ ( t ) ˆ L Φ ( t ) + 12 (cid:18) Φ( t + ∆ t ) − Φ( t )∆ t (cid:19) + m ( t ) + λ ( t ) (cid:105) . (4.7)One configuration Φ corresponds to a set of matrices { Φ( t ) } , for t = 1 , . . . , N t .Alternatively we can write down eq. (4.7) in terms of the constants A , D , B and C defined in eqs. (4.8)-(4.11): A = 2 π ∆ tN , (4.8) D = 2 πR N ∆ t , (4.9) B = 2 πR m ∆ tN , (4.10) C = πR λ ∆ tN . (4.11)Then the action reads: S [Φ] = N t (cid:88) t =1 Tr (cid:104) A Φ ( t ) ˆ L Φ ( t ) + D (Φ( t + ∆ t ) − Φ( t )) + B Φ ( t ) + C Φ ( t ) (cid:105) . (4.12) As we mentioned in the previous section, we are representing a configuration ofthe field in our model by a set of matrices { Φ( t ) } , for t = 1 , . . . , N t . Every elementin this set can be expanded in the polarisation tensor basis Φ( t ) = N − (cid:88) l =0 l (cid:88) m = − l c lm ( t ) ˆ Y lm , (4.13)where c lm ( t ) are N coefficients. The polarisation tensors ˆ Y lm are N × N matri-ces that are the analog of the spherical harmonics, Y lm ( θ, ϕ ). Details about thepolarisation tensors are presented in appendix B.At the end, the quantities of interest can be expressed as expectation valuesor averages over the configurations. The expectation value of the observable F (Φ)was defined in eq. (2.10): (cid:104) F (cid:105) = (cid:90) [ D Φ] F (Φ) e − S [Φ] Z (4.14) where Z = (cid:82) [ D Φ] e − S [Φ] is the partition function. Quantities of interest will be (cid:104) Φ( t ) (cid:105) , (4.15) (cid:104) Φ( t )Φ( t (cid:48) ) (cid:105) (4.16)where eq. (4.15) is a condensate and eq. (4.16) is a correlation function. These canbe mapped to standard correlation functions by replacing the ˆ Y lm by Y lm ( θ, ϕ ).We can reduce the expressions (4.15)-(4.16) to combinations of the expectationvalues of the coefficients c lm ( t ) introduced in eq. (4.13) (cid:104) c lm ( t ) (cid:105) , (4.17) (cid:104) c ∗ lm ( t ) c l (cid:48) m (cid:48) ( t (cid:48) ) (cid:105) . (4.18)Now it is convenient to compute the quantities (4.17)-(4.18) after a Fourier trans-form in (Euclidean) time.Following Ref. [29], the complete Fourier decomposition of the field is given byΦ( t ) := (cid:88) l,m N t − (cid:88) k =0 c lm ( k ) e ı πktNt ˆ Y lm , (4.19)where c lm ( k ) := 1 N t (cid:88) t e − ı πktNt πN Tr (cid:16) ˆ Y † lm Φ( t ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) . (4.20) c lm ( t )It will sometimes prove convenient to defineΦ( k ) = 12 πN t (cid:88) t e − ı πktNt Φ( t ) . (4.21)In this space the correlator (4.18) is diagonal, (cid:104) c ∗ lm ( k ) c l (cid:48) m (cid:48) ( k (cid:48) ) (cid:105) = G lm ( k ) δ kk (cid:48) δ ll (cid:48) δ mm (cid:48) . (4.22) G lm ( k ) is the Green function in momentum space. Here the term “momentumspace” is used to include both angular momentum ( l, m ) and frequency k . Description of the model For all our simulations we are interested in taking the thermodynamic limit N −→∞ . For the time direction, we are interested on taking N t −→ ∞ .We now set to ∆ t = 1. Now, the tricky part is how to relate the parameter R ,the radius of the spheres, and N , the dimension of the matrices.For the limits of the spatial part of our model we follow section 3.2.1. We define the field averaged over the time lattice as: Φ := 1 N t (cid:88) t Φ( t ) . (4.23)The average over the time lattice of the coefficients c lm are: c lm := 1 N t (cid:88) t c lm ( t ) . (4.24)This picks out the zero frequency component of Φ, i.e. eqs. (4.23)-(4.24) areparticular cases of the equations (4.19)-(4.21) when k = 0.Some particular cases in eq. (4.24) are c := √ πN Tr Φ , (4.25) c m := 4 πN Tr (cid:16) ˆ Y † ,m Φ (cid:17) , (4.26)where ˆ Y ,m are given in eqs. (B.12)-(B.14) of appendix B. We want to measure the contributions of different modes to the configuration Φ .For this purpose we need a control parameter. This turns out to be the sum | c lm | , this quantity was called the full power of the field in Ref. [30] and itrepresents the norm of the field Φ ; it can be calculate as : ϕ all := (cid:88) l,m | c lm | = 4 πN Tr (cid:0) Φ (cid:1) . (4.27) Although (cid:104) ϕ all (cid:105) cannot play the rˆole of an order parameter, we will show thatit is useful to localise the region where the phases split into disordered and ordered.We expect (cid:104) ϕ all (cid:105) ∼ (cid:104) ϕ all (cid:105) (cid:29) ϕ l := (cid:118)(cid:117)(cid:117)(cid:116) l (cid:88) m = − l | c l,m | . (4.28)We can re-write eq. (4.27) in terms of the quantities in eq. (4.28) ϕ all := (cid:88) l ϕ l . (4.29)In the disordered phase we expect (cid:104) ϕ l (cid:105) ≈ l .Studying the contributions of the different modes to (cid:104) ϕ all (cid:105) can provide moreinformation about the phases. If (cid:104) ϕ l (cid:105) (cid:29) l > l = 0 and the first mode for l = 1 as representative ofthose modes where the rotational symmetry is broken.Choosing the particular case l = 0 in eq. (4.28) we have ϕ := | c | . (4.30)For m <
0, if the contribution of the fuzzy kinetic term to the action is notnegligible we can expect the kinetic term to select the zero mode as the leadingone, (cid:104) ϕ (cid:105) ∼ = (cid:104) ϕ all (cid:105) . As a consequence (cid:104) ϕ (cid:105) (cid:29) (cid:104) ϕ l (cid:105) is expected to be close to zero in the disordered phase.Its corresponding susceptibility is defined as: χ := (cid:104) ϕ (cid:105) − (cid:104) ϕ (cid:105) . (4.31)As the contribution of the kinetic term to the action reduces compared to thepotential contribution we can expect the system can undergo the condensation ofhigher modes. Let us consider the p-wave contribution to Φ, i.e. the contributionof the l = 1 mode. Using c m , m = 1 , , − −→ c := c , c , c , − . Description of the model With this vector we can define the order parameter ϕ , as a particular case l = 1 in eq. (4.28) we have ϕ := (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) m = − | c ,m | := |−→ c | (4.32)and its susceptibility , χ : χ := (cid:104) ϕ (cid:105) − (cid:104) ϕ (cid:105) . (4.33)Following Ref. [30] the ordered non-uniform phase is then characterised by (cid:104) ϕ (cid:105) (cid:29)
0. Note, however, that due to fluctuations we will always have (cid:104) ϕ (cid:105) > how large it has to be. We will give more details of how tocharacterise this phase in the next section.We can include contributions of the remaining modes generalising (4.32) and(4.33). In practice the study of the first two modes should be enough to understandthe behaviour of the system. The internal energy is defined as: E ( m , λ ) := (cid:104) S (cid:105) , (4.34)and the specific heat takes the form C ( m , λ ) := (cid:104) S (cid:105) − (cid:104) S (cid:105) . (4.35)These terms correspond to the usual definitions E ( m , λ ) = − Z ∂ Z ∂β and C ( m , λ ) = ∂E∂β where Z is the partition function. We separate the action (4.12) into its four contributions: S [Φ] = A (cid:88) t Tr (cid:16) Φ ( t ) ˆ L Φ ( t ) (cid:17) , (4.36) S [Φ] = D (cid:88) t Tr (cid:16) Φ( t + 1) − Φ( t ) (cid:17) , (4.37) S [Φ] = B (cid:88) t Tr (cid:16) Φ ( t ) (cid:17) , (4.38) S [Φ] = C (cid:88) t Tr (cid:16) Φ ( t ) (cid:17) , (4.39) β is proportional to the inverse of the temperature T , i.e. β = k B T , where k B is theBoltzmann constant. where A , B , C , D were defined in eqs. (4.8)-(4.11).The corresponding expectation values to eqs. (4.36)-(4.39) are E ( m , λ ) := (cid:104) S (cid:105) , (4.40) E ( m , λ ) := (cid:104) S (cid:105) , (4.41) E ( m , λ ) := (cid:104) S (cid:105) , (4.42) E ( m , λ ) := (cid:104) S (cid:105) . (4.43) • Eq. (4.7) is written in terms of the following parameters: a general temporallattice spacing ∆ t , the radius of the sphere R , the dimension of the matrices N , the number of points in the lattice N t ,the mass squared m and the self-coupling λ . • In order to simplify the simulations, we use the freedom to re-scale the fieldΦ to fix the value of one of the constants given in eqs. (4.8)-(4.11). For oursimulations we fix A = 2 π — see chapter 7 for more details. We defined thedimensionless parameters: ¯ R = R ∆ t , (4.44)¯ m = (∆ t ) m , (4.45)¯ λ = ∆ tλ. (4.46) • In all our simulations for the 3-dimensional model N t was taken equal to N . hapter 5Description of the differentphases in the model In this chapter we characterise the different phases present in this model. Accord-ing to the values of ¯ λ relative to a critical value ¯ λ T we will see that we can dividethe space of parameters into two regions. In both cases we can subdivide accordingto values of ¯ m :1. ¯ λ T > ¯ λ > m < ¯ m c we have a uniform ordering (Ising type).(b) For ¯ m > ¯ m c we have the disordered phase.2. ¯ λ > ¯ λ T .(a) For ¯ m < ¯ m c we have a non-uniform ordering.(b) For ¯ m > ¯ m c we have the disordered phase. ¯ λ T > ¯ λ > In the previous chapter we defined the observable ϕ all called the full power of thefield . It provides a control parameter since it represents a norm of the matrix Φ .We will see that (cid:104) ϕ all (cid:105) ≈ (cid:104) ϕ all (cid:105) (cid:29)
290 5.1. Behaviour of the system for ¯ λ T > ¯ λ > study the contribution from separate modes to (cid:104) ϕ all (cid:105) . In figure 5.1 we presenta typical case for ¯ λ T > ¯ λ where we show the partial contributions from the zeroand first mode to (cid:104) ϕ all (cid:105) . (cid:1) ϕ (cid:2)(cid:1) ϕ (cid:2)(cid:1) ϕ all (cid:2) ¯ m Figure 5.1: (cid:104) ϕ all (cid:105) , (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) vs. ¯ m at ¯ λ = 0 .
17, ¯ R = 4, N = 12. From figure 5.1 we can observe that for ¯ m > − . ϕ all ≈
0, ¯ m < − . ϕ all ∼ ϕ >
0, so the dominant mode turns out to be the zero mode. (cid:1) ϕ (cid:2)(cid:1) ϕ (cid:2) ¯ m Figure 5.2: (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) vs. ¯ m at ¯ λ = 0 .
17, ¯ R = 4, N = 12. Figure 5.2 shows the order parameters (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) . In order to determineprecisely where the phase transition occurs the standard way is to search themaximum in the susceptibility, in this case χ since the zero mode is relevant forthis phase transition. Description of the different phases in the model χ χ ¯ m Su s ce p t i b ili t i e s Figure 5.3: The susceptibilities χ and χ , in eqs. (4.31) and (4.33), at ¯ λ =0 .
17, ¯ R = 4, N = 12 χ peaks at ¯ m = − . figure 5.3 we can observe that the susceptibilityassociated to the first mode reveals a small response too. Figure 5.4 shows the internal energy for the same parameters as in figures5.1-5.3 . E E E E E ¯ m E n e r g y / V o l. Figure 5.4: Internal energy density E , see eq. (4.34), and its partial contri-butions, given in eqs. (4.40)-(4.43), at ¯ λ = 0 .
17, ¯ R = 4, N = 12.2 5.1. Behaviour of the system for ¯ λ T > ¯ λ > We observe in figure 5.4 that the leading contribution for ¯ m > − . E , is the one that comes from the kinetic fuzzy term in eq. (4.40).Note that the phase transition occurs at ¯ m where the potential contributions E and E deviate from zero. Figure 5.5 shows an archetypical behaviour of the specific heat for ¯ λ T > ¯ λ . ¯ m Sp ec i fi c H e a t Figure 5.5: Specific heat per volume at ¯ λ = 0 .
17, ¯ R = 4, N = 12. Itfollows from eq. (3.10) that for this values of the parameters the volume isthe constant 4 πR × N t = 192 π . The specific heat provides an alternative criterion to the susceptibilities to deter-mine where the phase transition occurs. It provides information about the orderof the phase transition. We prefer at this point to follow the specific heat criterionbecause it is a more universal quantity that does not distinguish the dominantmode. Since we expect that as we increase ¯ λ the dominant modes are higher thanthe zero and first mode, we cannot ensure that in that region the susceptibilitiesrelated to such modes χ and χ give a reliable prediction to the critical point.If we follow the criteria of the susceptibilities we have to take into account whichmode is the dominant one. Both criteria are theoretically supposed to detect thesame phase transition at the same parameters. For ¯ λ < ¯ λ T where the dominantmode in the ordered phase is the zero mode, this is confirmed comparing figure5.3 to figure 5.5 since the susceptibility of the dominant mode, χ , and the spe-cific heat peaks around the same value in ¯ m . We can also observe in figure 5.5 Description of the different phases in the model that there is a smaller response in the susceptibility of the first mode, χ . For¯ λ < ¯ λ T we will see that this situation is different since the susceptibilities of thenon-dominant modes do not peak at the phase transition.We call the value of ¯ m where the specific heat peaks ¯ m c , and for figure 5.5 ¯ m c = − . ± . In the case that for ¯ m ≥ ¯ m c the kinetic term is not leading, itappears as a small shift between both peaks. This happens when ¯ R is big enoughto have N ¯ R small , ¯ λ T > ¯ λ . Another observation is that in figure 5.3 the errorbars are smaller than in the case of the specific heat from figure 5.5 . The reasonis that in general more statistics is necessary for the specific heat than for thesusceptibilities. This phase transition is of second order as it is shown in figure5.5 In this section we want to present the typical behaviour of the observables for¯ m > ¯ m c and ¯ m < ¯ m c . ¯ m > ¯ m c First we discuss some aspects of the thermalisation procedure. We define thethermalisation time as the number of Monte Carlo steps necessary for an observableto stabilise around one value independently of the starting conditions. If ¯ m > ¯ m c we are in the disordered phase that is characterised by the property that thecoefficients c lm in eq. (4.24) are in average near to zero. We assume that thethermalisation of the coefficients c lm is similar, and we check if the coefficient c thermalises. Figure 5.6 shows the thermalisation of the action and the coefficient c for the point ¯ m = 0 in the figures 5.1- 5.5 . For practical purposes we had to estimate the error by referring to the spacing of the¯ m values that we simulated. During the run the values stored were c and from them we can trivially calculate ϕ . We prefer to present the histories and histogram of c rather than ϕ in order tocheck if the samples are symmetric under c −→ − c . λ T > ¯ λ > a c t i o n / V o l. c Figure 5.6: Example of thermalisation of the action and the coefficient c at ¯ m = 0, ¯ λ = 0 .
17, ¯ R = 4, N = 12. We chose a hot start for figure 5.6 . We simulate the same parameters witha cold start and we obtained results in agreement within the statistical errors.The thermalisation time for c in figure 5.6 is estimated to 1500 Monte Carlosteps.After the thermalisation we begin the measurement procedure. Figure 5.7 shows the histograms for the observables and parameters used in the simulationsof figure 5.6 : Action/Vol. 0.60.580.560.540.520.50.480.460.440.422520151050 c Figure 5.7: Histogram of the of the action and the coefficient c at ¯ m = 0,¯ λ = 0 .
17, ¯ R = 4, N = 12.¯ m < ¯ m c If ¯ m > ¯ m c we are in the ordered regime, for ¯ λ < ¯ λ T we have uniform orderingcharacterised by the property that in the expansion (4.23) all coefficients c lm for The starting configuration is a vector of hermitian matrices filled in with randomnumbers. For more details see appendix A. The starting configuration is a vector of matrices proportional to the unit. The area is normalised to 1. The number of bins is 500.
Description of the different phases in the model l > c thermalise. Figure 5.8 shows the thermalisation of the action and the parameter c for thepoint ¯ m = − . figures 5.1- 5.5 . a c t i o n / V o l. c Figure 5.8: Example of the thermalisation of the action and the coefficient c at ¯ m = − .
2, ¯ λ = 0 .
17, ¯ R = 4, N = 12. We chose a hot start for figure 5.8 . After 1500 Monte Carlo steps the valueof c oscillates around 0 . .
43. We also simulated at the same parameters with a cold start . We obtainedthe same results within the statistical error but the thermalisation time decreasesby more than 50% as we can observe in figure 5.9 . a c t i o n / V o l. (cid:1) c (cid:2) Figure 5.9: Example of thermalisation with cold starting conditions of theaction and the coefficient c at ¯ m = − .
2, ¯ λ = 0 .
17, ¯ R = 4, N = 12. Finally, after the thermalisation procedure we measure the expectation value ofthe observables. We present the histograms for the same parameters as in figure5.8 : λ > ¯ λ T Action/Vol. 0.520.50.480.460.440.420.40.380.360.342520151050 c Figure 5.10: Histograms of the action and the coefficient c at ¯ m = − . λ = 0 .
17, ¯ R = 4, N = 12. The peaks in the probability distribution of c are approximately located at (cid:113) π | ¯ m | N ¯ λ . In figure 5.10 the peaks are approximately located at c = ± .
95 and (cid:113) π × . × . = 1 .
12. As we move forward for a more negative ¯ m this prediction ismore accurate. This is shown in figure 5.11 . c / (cid:1) π × / (12 × .
17) 1.510.50-0.5-1-1.5706050403020100
Figure 5.11: Histogram of coefficient c at ¯ m = −
2, ¯ λ = 0 .
17, ¯ R = 4, N = 12. We re-scale the x -axis by the factor (cid:113) π | ¯ m | N ¯ λ which in this casetakes the value 3 . ¯ λ > ¯ λ T For this region of the phase diagram we observe two phases: for ¯ m > ¯ m c wehave the disordered phase characterise by (cid:104) ϕ all (cid:105) ≈
0, for ¯ m < ¯ m c we have theordered phase characterised by (cid:104) ϕ all (cid:105) >
0. For ¯ m (cid:28) ¯ m c there are thermalisationproblems, we will discuss these difficulties in section 5.2.1. Description of the different phases in the model In Figure 5.12 we present the partial contributions from the zero and firstmode in (cid:104) ϕ all (cid:105) . (cid:1) ϕ (cid:2)(cid:1) ϕ (cid:2)(cid:1) ϕ all (cid:2) ¯ m Figure 5.12: (cid:104) ϕ all (cid:105) , (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) vs. ¯ m at ¯ λ = 1 .
25, ¯ R = 8, N = 12. We observe in figure 5.12 (cid:104) ϕ all (cid:105) ≈ m > − . m < − . (cid:104) ϕ all (cid:105) > − . > ¯ m > − . (cid:104) ϕ all (cid:105) ∼ (cid:104) ϕ (cid:105) . (cid:1) ϕ (cid:2)(cid:1) ϕ (cid:2) ¯ m Figure 5.13: (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) vs. ¯ m at ¯ λ = 1 .
25, ¯ R = 8, N = 12. For those values of ¯ m where ¯ m < − . (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) in figure 5.12 and in ϕ and ϕ in figure 5.13 sincethey do not grow monotonously. We will come back to this point at section 5.2.1.For the moment we focus on the region where the observables behave smoothly,i.e. ¯ m > − . λ > ¯ λ T χ χ ¯ m Su s ce p t i b ili t i e s Figure 5.14: Susceptibilities χ and χ at ¯ λ = 1 .
25, ¯ R = 8, N = 12. In figure 5.14 we can observe that χ indicates a phase transition for ¯ m (cid:39)− .
35 while χ cannot detect it since χ keeps growing as m decreases. Weconjecture that χ should peak for some value of ¯ m < − . E E E E E ¯ m I n t e r n a l e n e r g y / V o l. Figure 5.15: Internal energy E of eq. (4.34) and its partial contributionseqs. (4.40)-(4.43) at ¯ λ = 1 .
25, ¯ R = 8, N = 12. The specific heat in figure 5.16 indicates a phase transition at ¯ m = − . ± . Description of the different phases in the model ¯ m Sp ec i fi c H e a t Figure 5.16: Specific heat at ¯ λ = 1 .
25, ¯ R = 8, N = 12. As we can observe comparing figure 5.14 to figure 5.16 , there is a smalldifference in the critical value of ¯ m predicted by the susceptibility of the zeromode χ and the one given by the specific heat. We conjecture that this is dueto a finite volume effect, nevertheless both criteria are qualitatively the same. Formore details see appendix C. In this section we want to sketch the thermalisation problems.The program was designed to perform an arbitrary number of independent simulations in every run, n sim . In the case of figure 5.17 we performed tenindependent simulations, the first three of them with a cold start while for the lastseven simulations we chose a hot start . If we choose a hot start , they have different starting configurations. λ > ¯ λ T a c t i o n / V o l. c Figure 5.17: History of the action and the coefficient c for different startingconditions at ¯ m = − .
66, ¯ λ = 1 .
25, ¯ R = 8, N = 12. We can observe in figure 5.17 that the expectation value of the energy and thecoefficient c depend on the starting conditions. The interpretation of this phe-nomenon is that the effective potential of the system has several local minima withbarriers large enough to suppress tunnelling between them. For different startingconditions the system gets trapped in one of those minima. As a consequence wehave thermalisation problems (or practical ergocidity problems in the algorithm).They should disappear at an infinitely large Monte Carlo time, T MC . In figure5.17 we have a new simulation every 1 , ,
000 Monte Carlo steps from which thefirst 500 ,
000 steps where taken as thermalisation time.The large error bars in the observables at ¯ m = − .
66 in figures 5.12 and can be explained because the different simulations give different results. For cold starts the action of the system oscillates around 0 .
05 – this is the value of theenergy in the absolute minima – and the trace c of the sampled configurationsfluctuate around 0 .
17; for hot starts the energy of the system oscillates around 0 . c of the sampled configurations fluctuate around zero. Description of the different phases in the model Figure 5.18 shows the histograms corresponding to figure 5.17 . Action/Vol. 0.250.20.150.10.050-0.05181614121086420 c Figure 5.18: Histogram of the action and the coefficient c at ¯ m = − . λ = 1 .
25, ¯ R = 8, N = 12. It is possible to observe, from the Monte Carlo time evolution of the observable c in eq. (4.25) that there are no thermalisation problems and that its probabilitydistribution has several peaks. Monte Carlo steps c Figure 5.19: History of the coefficient c for ¯ m = 0 . λ = 0 .
75 ¯ R = 16, N = 12. The history corresponding of the action for the same parameters in figure5.19 is the following:
Monte Carlo steps a c t i o n / V o l. Figure 5.20: History of the action for ¯ m = − .
30 at ¯ λ = 0 .
75 ¯ R = 16, N = 12 In this case the fluctuations were large enough to jump from one minimum toanother. Nevertheless the configurations sampled belong to different subspaces ofthe spaces of configurations characterised by the different values of c it is shownin figure 5.19 .The histogram corresponding to figure 5.19 is presented in figure 5.21 . c / . Figure 5.21: Histogram of the coefficient c for ¯ m = − .
30 at ¯ λ = 0 . R = 16, N = 12. We divided the x -axis by the factor √ π | ¯ m | / ( N ¯ λ ) NN t (cid:39) . The estimated value for the triple point for N = 12, ¯ R = 16 is (cid:0) ¯ λ T , ¯ m T (cid:1) =(0 . , − . figures 5.19 - are in the region Description of the different phases in the model λ > λ T (i.e. in the ordered non-uniform phase). For this region the leadingcontribution comes from configurations characterised by a angular momentum l > ϕ all = ϕ + ϕ + · · · For the parameters simulated for figure 5.19 the contribu-tions to (cid:104) ϕ all (cid:105) split as follows (cid:104) ϕ all (cid:105) (cid:39) . (cid:39) .
014 + 0 .
23 + · · · .The several maxima in figure 5.21 reflect the existence of several minimain the effective action. If λ (cid:29) λ T it is expected that the potential provides theleading contribution to the action, turning into a pure potential model. Then, if λ is large enough, for this region of parameters the minima in the effective actionare given by the minima in the potential. The maximal number of local minima for our model can be obtained from themaximal number of local minima for the two-dimensional model. In Ref. [30] itwas conjectured that the minima in the 2-dimensional pure potential model aregiven by the disjoint orbits: O n = { − m λ U † ( n ⊕ s +1 − n U | U ∈ U (2 s + 1) / [ U ( n ) × U (2 s + 1 − n )]) } , (5.1)where n ≤ s + and n are n × n matrices (2 s + 1 = N ), i.e. we have N disjointorbits.In our case that we have N t lattice points. Note that in the expression (5.1) theconstant − m λ is in terms of the parameters of the 2-dimensional —see eqs. (3.20)-(3.21). For large values of ¯ λ we can establish their “equivalence” in the threedimensional model via eq. (6.51) —see section 6.5.2. We conjecture that the min-ima in the 3-dimensional model are at { Φ( t ) } N t t =1 , Φ( t ) = (cid:114) | ¯ m | N ¯ λ U † Λ U (5.2)where Λ is a diagonal matrix with eigenvalues 1 or − c .The coefficient c –see eq. (4.25)– can be expressed as c = √ πN N t Tr (cid:34) N t (cid:88) t =1 Φ ( t ) (cid:35) (5.3) From expression (5.2) we have
N N t + 1 possible values (cid:114) N ¯ λ ¯ m N t (cid:88) t =1 Tr(Φ( t )) = − N t N, − N N t + 2 , . . . , N N t − , N N t , (5.4)then (cid:112) π | ¯ m | / ( N ¯ λ ) N N t × c = Tr (cid:34) N t (cid:88) t =1 Φ ( t ) (cid:35) (5.5)takes integer values as in figure 5.21 where N × N t − k = − , − , , , N × N t + 1. For N = 12 = N t as in figure5.21 the maximal number of peaks is 145 but we just observe 5 of them.We can discuss intuitively why we cannot observe the maximal number ofpeaks. There is a single way to obtain the value (cid:113) N ¯ λ ¯ m (cid:80) N t t =1 Tr(Φ( t )) N × N t ineq. (5.4) –all the matrices Φ( t ), t = 1 , · · · , N t should be proportional to the identity.To get in eq. (5.4) the value N N t − t ) ∝ U † Diag (1 , , · · · , , − U ,Φ( t ) ∝ U † U for t (cid:54) = t . Since (cid:80) N t t =1 Tr(Φ( t )) is invariant under interchange of thelattice points and interchange of the eigenvalues on each matrix Tr(Φ( t )) there are N N t ways to obtain the value N N t − t )) = N N t , they are the most probable tofall in a simulation with hot starting conditions.Since the number of local minima is larger the measurement problems are moresevere than in the 2-dimensional case, see e.g. Ref [30]-[32]. The tunnelling betweendifferent minima depends on the size of its potential barrier and the size of thefluctuations.In general we can diagonalise only one of the N matrices { Φ( t i ) } Ni =1 . Thisis represented in figure 5.22 where the matrix Φ( t j ) is diagonalised though arotation U ∈ SU (2) to Λ j . For { Φ( t i ) } N t i =1 , i (cid:54) = j the matrices are not diagonal.Φ( t j ) ∝ Λ j it can be map to the continuum to f ( θ, ϕ )Φ( t j ) ∝ Λ j = N − (cid:88) l =0 f l ˆ Y l −→ f ( θ, ϕ ) = N − (cid:88) l =0 f l Y l ( θ, ϕ ) . (5.6)This is schematically shown in figure 5.22 with the signs ” ± ” in red on Φ( t j ). Y l ( θ, ϕ ) > π > θ > Y l ( θ, ϕ ) < π > θ > π . Description of the different phases in the model
Note that the model in eq. (4.7) can be brought to one of diagonal matrices ifthe contribution of fuzzy kinetics term in eq. (4.36) is negligible (see Ref. [48]-[49]).We have the picture it figure 5.23 . Figure 5.23: Schematic view of the equilibrium configuration when the fuzzykinetic term is negligible. hapter 6The scaling behaviour
In this chapter we focus on finding out the dependence of the transition curvesand triple point on the parameters of the system.
First we explore the transition curve for N = 12 , ¯ R = 8 and we consider severalvalues for ¯ λ . − . λ . − . λI − II : N = 12 , ¯ R = 8 II: Ordered Uniform I: Disordered ¯ λ m Figure 6.1: Transition curve from the disordered to ordered-uniform phasefor N = 12. 478 6.1. Phase transition disordered to ordered-uniform We observe in figure 6.1 that the transition curve is a line that crosses the origin,given by the equation ¯ m = − (0 . ± . λ. (6.1)In addition we tried a fit of the form m = a ¯ λ b and we found the expression¯ m = − (0 . ± . λ (1 . ± . . (6.2)Eqs. (6.1) and (6.2) are in agreement. We choose a linear fit for our transitioncurves.For general values of N and ¯ R we propose a transition line between the disor-dered phase and the ordered uniform phase of the form¯ m = f ( N, ¯ R )¯ λ. (6.3)To identify the slope f ( N, ¯ R ) our strategy is the following:1. First we extract the dependence on N varying N and keeping ¯ R fixed. Whatwe would expect is of the form¯ m = f ( ¯ R ) N δ ¯ λ. (6.4)2. Then, if δ does not depend on ¯ R we extract the dependence on ¯ R proposing f ( ¯ R ) = const. ¯ R δ . (6.5)3. Note that eq.(6.3) would still be valid if both sides of the expression aremultiplied by a common factor. We will use this factor to stabilise the triplepoint to a fixed value. This procedure will be explained in section 6.3.The slope f ( N, ¯ R ) could have a more complicated dependence on N or ¯ R ,nevertheless we will prove the choice f ( N, ¯ R ) = const.N δ ¯ R δ is a good ansatz. Step 1
Now, to find the collapse on N , we keep ¯ R = 8 and we consider N = 8 , , , , The scaling behaviour − . N . ¯ λN = 8; ¯ R = 8 N = 12; ¯ R = 8 N = 16; ¯ R = 8 N = 23; ¯ R = 8 N = 33; ¯ R = 8 II: Ordered uniformI: Disordered N . ¯ λ m Figure 6.2: Transition curve from the ordered-uniform phase to the orderednon-uniform phase for ¯ R = 8. In figure 6.2 we re-scale the x -axis by a factor of N . . We will see in section6.3 this factor stabilises the value of the triple point in N .The equation of the fit of figure 6.2 is:¯ m = − . N . ¯ λ. (6.6)The exponent on N , δ = 0 .
64 is optimal for this fit; to compare with anotherexponent we show the same data on the next figure 6.3 for δ = 0 . − . N . ¯ λN = 8; ¯ R = 8 N = 12; ¯ R = 8 N = 16; ¯ R = 8 N = 23; ¯ R = 8 N = 33; ¯ R = 8 II: Ordered uniformI: Disordered N . ¯ λ m Figure 6.3: Transition curve from the ordered-uniform phase to the orderednon-uniform phase for ¯ R = 8 for δ = 0 . As we observed comparing figure 6.2 to figure 6.3 , the fit for the exponent δ = 0 .
64 is better and we estimate the error on δ to the value 0 . R = 8, thisis δ ( ¯ R = 8) = 0 . ± . R = 4 in figure 6.4 seems to confirm the value for the exponent, δ ( ¯ R = 4) = 0 . ± . − . N . ¯ λN = 8; ¯ R = 4 N = 9; ¯ R = 4 N = 12; ¯ R = 4 N = 16; ¯ R = 4 N = 23; ¯ R = 4 N = 33; ¯ R = 4 II: Ordered uniformI: Disordered N . ¯ λ m Figure 6.4: Transition curve between the disordered phase and the ordered-uniform phase for ¯ R = 4. We illustrate the case ¯ R = 16 in figures 6.5 - . − . N . ¯ λN = 8; ¯ R = 16 N = 12; ¯ R = 16 N = 16; ¯ R = 16 N = 16; ¯ R = 16 II: Ordered uniform I: Disordered N . ¯ λ m Figure 6.5: Transition curve between the disordered phase and the ordered-uniform phase for ¯ R = 16 for δ = − . The scaling behaviour − . N . ¯ λN = 8; ¯ R = 16 N = 12; ¯ R = 16 N = 16; ¯ R = 16 N = 23; ¯ R = 16 II: Ordered uniform I: Disordered N . ¯ λ m Figure 6.6: Transition curve between the disordered phase and the ordered-uniform phase for ¯ R = 16 and the optimal exponent δ = 0 . We have evidence to believe the exponent δ depends on ¯ R . But for the momentwe content ourselves with choosing δ as 0 .
64 and we consider the error as 0 . δ = 0 . ± . N we can study the coefficient f ( ¯ R ) ineq. (6.3). Step 2 − .
31 ¯ R − . f ( ¯ R )¯ R f ( ¯ R ) Figure 6.7: Coefficients f ( ¯ R ) for ¯ R = 2 , , , ,
32 and the fit in eq. (6.7).2 6.1. Phase transition disordered to ordered-uniform
In the previous figure 6.7 we estimate the errors on the fit of f ( ¯ R ): f ( ¯ R ) = ( − . ± .
1) ¯ R − (0 . ± . . (6.7)Then the equation for the transition curve from the disordered phase to theordered-uniform phase is: ¯ m = ( − . ± . N . ± . ¯ R . ± . ¯ λ. (6.8)In terms of the constants A , B , C and D in eqs. (4.8)-(4.11) we re-write eq. (6.8) as B (cid:39) − . C ( AD ) . . (6.9)Checking the collapse on ¯ R for N = 12 we obtain figure 6.8 : − . N . ¯ R . ¯ λN = 12; ¯ R = 64 N = 12; ¯ R = 32 N = 12; ¯ R = 8 N = 12; ¯ R = 4 N = 12; ¯ R = 2 N = 12; ¯ R = 1 II: Ordered uniformI: Disordered N . ¯ R . ¯ λ N . ¯ R . ( ¯ R . ¯ m ) Figure 6.8: Transition curve from the disordered to ordered-uniform phasefor N = 12. We observed for ¯ R = 64 – where N ¯ R = 0 . (cid:28) R for N = 23, for N ¯ R >
1, see figure 6.9 . ¯ λ = CD , ¯ m = BD , N ¯ R − = π √ AD , and N ¯ R = 2 π DA The scaling behaviour − . N . ¯ R . ¯ λN = 23; ¯ R = 32 N = 23; ¯ R = 8 N = 23; ¯ R = 4 II: Ordered uniformI: Disordered N . ¯ R . ¯ λ N . ¯ R . ( ¯ R . ¯ m ) Figure 6.9: Transition curve from the disordered to ordered-uniform phasefor N = 23. Finally the collapse of data in eq. (6.10) considering all our data is shown in figure 6.10 . − . N . ¯ R . ¯ λ − . N . ¯ R . ¯ λ − . N . ¯ R . ¯ λN = 12; ¯ R = 64 N = 16; ¯ R = 64 N = 8; ¯ R = 32 N = 12; ¯ R = 32 N = 23; ¯ R = 32 N = 12; ¯ R = 16 N = 16; ¯ R = 16 N = 8; ¯ R = 8 N = 12; ¯ R = 8 N = 16; ¯ R = 8 N = 23; ¯ R = 8 N = 33; ¯ R = 8 N = 8; ¯ R = 4 N = 12; ¯ R = 4 N = 16; ¯ R = 4 N = 23; ¯ R = 4 N = 33; ¯ R = 4 N = 8; ¯ R = 2 N = 12; ¯ R = 2 N = 16; ¯ R = 2 N = 23; ¯ R = 2 II: Ordered uniformI: Disordered ( N ¯ R ) . ¯ λ ( N ¯ R ) . ( ¯ R . N . ) ¯ m Figure 6.10: Collapse of data for the disorder to the order-uniform phasetransition.
We observe that for N ¯ R < .
375 and N ¯ R > . The case N = 23, ¯ R = 4 could fall in this class, but is not possible to conclude due tothe poor resolution for these points. We conclude that the range where eq. (6.10) gives a good approximation is N ¯ R ∈ (0 . , N ¯ R / ∈ (0 . ,
8) we haveto reconsider f ( N, ¯ R ) defined in eq. (6.3) as a more complicated function on N ¯ R .For the moment we exclude the data that are not in the interval (0 . ,
8) and wepresent the figure 6.11 . − . N ¯ R ) . ¯ λN = 16; ¯ R = 64 N = 12; ¯ R = 32 N = 23; ¯ R = 32 N = 12; ¯ R = 16 N = 16; ¯ R = 16 N = 8; ¯ R = 8 N = 12; ¯ R = 8 N = 16; ¯ R = 8 N = 23; ¯ R = 8 N = 33; ¯ R = 8 N = 8; ¯ R = 4 N = 12; ¯ R = 4 N = 16; ¯ R = 4 N = 23; ¯ R = 4 N = 33; ¯ R = 4 N = 8; ¯ R = 2 II: Ordered uniformI: Disordered N . ¯ R . ¯ λ ( N ¯ R ) . ( ¯ R . N . ) ¯ m Figure 6.11: Transition curve from the disordered phase to the ordered-uniform phase.
To conclude the present section, the expression of the collapse in figure 6.11 is ¯ m c = ( − . ± . N . ± . ¯ R . ± . ¯ λ. (6.10) The scaling behaviour First we explore this transition curve for N = 12 , ¯ R = 8. y ( N = 12 , ¯ R = 8)I-III III: Ordered non-uniformI: Disordered ¯ λ m Figure 6.12: Transition curve from the disordered phase to the ordered non-uniform phase for N = 12 , ¯ R = 8. We observe that the transition curve shows curvature. The most natural fit wecan propose is a polynomial where the coefficients are functions that depend on ¯ R and N . In figure 6.12 we used a polynomial fit of degree 4: y ( N = 12 , ¯ R = 8) = − . − . λ + 0 . λ − . × − ¯ λ + 3 . × − ¯ λ . (6.11)In the figure 6.12 we covered a large range of ¯ λ . Nevertheless, to predict thetriple point we can concentrate on “small” values of ¯ λ but above ¯ λ triple . For thisrange of values the transition curve can be approximated by a polynomial of asmaller degree. In figure 6.13 we present a linear fit for an interval of figure6.12 : − . − . λ I-III
III: Ordered non-uniformI: Disordered ¯ λ m Figure 6.13: An interval of the transition curve from the disordered phase tothe ordered non-uniform phase for N = 12 , ¯ R = 8. First we explore a linear fit for the transition curves. In the subsequent section 6.4we will compare it with the results obtained for a fit using a polynomial of seconddegree.The fit of figure 6.13 is:¯ m = − (0 . ± . − (0 . ± . λ. (6.12)For general N, ¯ R we propose a linear fit for the transition curve from thedisordered phase to the ordered non-uniform phase of the form¯ m = h ( N, ¯ R ) + h ( N, ¯ R )¯ λ (6.13)and we further make a factorisation ansatz and h i ( N, ¯ R ) = h i ( N )˜ h i ( ¯ R ) , i = 0 , N , keeping ¯ R = 8 and varying N =12 , , , The scaling behaviour − . Nλ − . N = 33 , ¯ R = 8 N = 23 , ¯ R = 8 N = 16 , ¯ R = 8 N = 12 , ¯ R = 8 N = 8 , ¯ R = 8 III: Ordered non-uniform
I: Disordered N ¯ λ m Figure 6.14: Transition curves from the disordered phase to the ordered non-uniform phase for ¯ R = 8 The fit for figure 6.14 is:¯ m c = − . ± . N ¯ λ − . . (6.14)Here we conclude h ( N ) ∝ N , h ( N ) = const. Now we want the stabilise the triple point, at least the value of ¯ λ T as we didin the previous section. Then if we re-scale the x -axis in figure 6.14 by a factor N . we get the figure 6.15 : N = 33 , ¯ R = 8 N = 23 , ¯ R = 8 N = 16 , ¯ R = 8 N = 12 , ¯ R = 8 N = 8 , ¯ R = 8 III: Ordered non-uniformI: Disordered N . ¯ λ N − . ¯ m Figure 6.15: Approximate collapse of the slope for the transition curves fromthe disordered phase to the ordered non-uniform phase for ¯ R = 8. To stabilise the value of ¯ m T it is not sufficient to re-scale the y -axis. We fit the transition curves for ¯ R = 4 , ,
32 for different N by eq. (6.13) inorder to find the coefficient ˜ h ( ¯ R ). N = 23 , ¯ R = 2 N = 16 , ¯ R = 2 N = 12 , ¯ R = 2 N = 8 , ¯ R = 2 III: Ordered non-uniformI: Disordered N . ¯ λ N − . ¯ m Figure 6.16: Approximate collapse of the slope for the transition curves fromthe disordered phase to the ordered non-uniform phase for ¯ R = 2. N = 33 , ¯ R = 4 N = 23 , ¯ R = 4 N = 16 , ¯ R = 4 N = 12 , ¯ R = 4 N = 8 , ¯ R = 4 III: Ordered non-uniformI: Disordered N . ¯ λ N − . ¯ m Figure 6.17: Approximate collapse of the slope for the transition curves fromthe disordered phase to the ordered non-uniform phase for ¯ R = 4. The scaling behaviour N = 33 , ¯ R = 16 N = 23 , ¯ R = 16 N = 16 , ¯ R = 16 N = 12 , ¯ R = 16 N = 8 , ¯ R = 16 III: Ordered non-uniformI: Disordered N . ¯ λ N − . ¯ m Figure 6.18: Collapse of the slope for the transition curves from the disor-dered phase to the ordered non-uniform phase for ¯ R = 16. N = 23 , ¯ R = 32 N = 16 , ¯ R = 32 N = 12 , ¯ R = 32 N = 8 , ¯ R = 32 III: Ordered non-uniformI: Disordered N . ¯ λ N − . ¯ m Figure 6.19: Collapse of the slope for the transition curves from the disor-dered phase to the ordered non-uniform phase for ¯ R = 32. Now that we get the collapse on N as h ( N ) ∼ N we can determine thecoefficients, we propose: h ( N, ¯ R ) = − N ˜ h ( ¯ R ) (6.15) Figure 6.20 shows ˜ h ( ¯ R ) and its fit. .
064 ¯ R − . ˜ h ( ¯ R )¯ R Figure 6.20: Coefficients ˜ h ( ¯ R ) for ¯ R = 2 , , , ,
32 and the fit 0 .
064 ¯ R − . . Then we have h ( N, ¯ R ) = − (0 . ± . N ¯ R − . ± . . (6.16)Eq. (6.16) is the slope of the coexistence line from the disordered phase to theordered non-uniform phase. To solve the equation of this coexistence line we usethe form: (cid:0) ¯ m − ¯ m T (cid:1) = h ( N, ¯ R ) (cid:0) ¯ λ − ¯ λ T (cid:1) . (6.17)If we want to collapse the transition line completely, it is not enough to re-scale the y -axis. It is necessary to shift ¯ m , for example, by substituting ¯ m −→ ¯ m + . R . in figure 6.15 . This leads to figure 6.21 . N = 33 ¯ R = 8 N = 23 ¯ R = 8 N = 16 ¯ R = 8 N = 12 ¯ R = 8 N = 8 ¯ R = 8 III: Ordered non-uniformI: Disordered N . ¯ λ N − . ( ¯ m + . R − . ) Figure 6.21: Collapse of transition curve from the disordered phase to theordered non-uniform phase for ¯ R = 8. The scaling behaviour As another example for ¯ R = 4, form figure 6.17 we obtain figure 6.22 . N = 33 , ¯ R = 4 N = 23 , ¯ R = 4 N = 16 , ¯ R = 4 N = 12 , ¯ R = 4 N = 8 , ¯ R = 4 III: Ordered non-uniformI: Disordered N . ¯ λ N − . ( ¯ m + . R − . ) Figure 6.22: Collapse of transition curve from the disordered phase to theordered non-uniform phase for ¯ R = 4. We conclude that for ¯ λ around ¯ λ T the transition curve from the disorderedphase to the ordered non-uniform phase obeys eq. (6.18).¯ m c = − (0 . ± . N ¯ R − . ± . ¯ λ + (cid:0) . N . − . (cid:1) ¯ R − . . (6.18) As we mentioned in section 6.1 we have the freedom to re-scale both axes of thephase diagram by a common factor. We want to use this factor to fix the triplepoint, i.e. we want to find a function of N and ¯ R , k ( N, ¯ R ), such that k ( N, ¯ R )¯ λ T = const. To identify the triple point (cid:0) ¯ λ T , ¯ m T (cid:1) ( ¯ R, N ) our strategy is the following:1. First we extract the dependence on ¯ R varying ¯ R and keeping N fixed. Whatwe would expect is of the form¯ λ T = Z ( N ) ¯ R e ( N ) , (6.19)¯ m T = M ( N ) ¯ R e ( N ) . (6.20)For each N we estimate the exponents e ( N ) , e ( N ) and the coefficients Z ( N ) , M ( N ) using the package “gnu-plot”.
2. In general the exponents can depend on N , but we will show they do notand fix e , e to a certain value. Since our final expression of the triple pointmay strongly depend on the choice of e , e , we will compare the expressionsfor two different sets of exponents. Both fits for e in the case of ¯ λ T – or e in the case of ¯ m T – will be presented in the corresponding figures. Weinclude the case when we vary e and Z ( N ) – or e and M ( N ) in the caseof ¯ m T .3. We extract the dependence on N proposing Z ( N ) = const. · N d , (6.21) M ( N ) = const. · N d . (6.22) Step 0
First we explore the dependence on ¯ R . To do this we consider N = 12 andseveral values for ¯ R and we estimate the intersection of both transition curves. Figure 6.23 is an example of this procedure.
Linear fit for I − III
Prediction of I − II III : O r d e r e dn o n - un i f o r m ph a s e U n i f o r m II : O r d e r e d I : D i s o r d e r e dph a s e ¯ λ m Figure 6.23: Transition curves for N = 16 , ¯ R = 8. The linear fit for figure 6.23 is given by the equation:¯ m = − (0 . ± . − (0 . ± . λ. (6.23) We will choose e = − . , e = − .
92 vs. e = − . , e = − .
89 and its correspond-ing coefficients Z ( N ) , M ( N ). The scaling behaviour The intersection of both curves is estimated in ¯ λ T = 0 . ± .
02, ¯ m T = − . ± . Step 1
Figure 6.24 shows the estimated values of ¯ λ T for different ¯ R fixing N = 12and a fit for these points: .
34 ¯ R − . .
20 ¯ R − . .
26 ¯ R − . ¯ R λ T Figure 6.24: Estimation of ¯ λ T for N = 12 and three different fits. The fitswere obtained via a function ¯ λ T = Z ( N = 12) ¯ R e . In the first case we fit bothparameters Z ( N = 12) , e and we got ¯ λ T = (6 . ± .
64) ¯ R ( − . ± . . In thesecond and third case we fixed the exponent e and vary Z ( N = 12). Forthe second fit we chose e = − .
28 and we fit Z ( N = 12) as in eq. (6.24) andfinally in the third case we fixed e = − .
25 to get Z ( N = 12) = 8 . ± . The fit for
Figure 6.24 is: ¯ λ T = (8 . ± .
31) ¯ R − . ± . . (6.24)Then, if we multiply both axes in the phase diagram by a factor on ¯ R such thatthe x -axis is ¯ R . ¯ λ , we stabilise the triple point to the value of ¯ λ T = 8 . m T with the eq. (6.10): We tried another exponent and we got an acceptable fit for ¯ λ T = 9 .
11 ¯ R − . . − .
06 ¯ R − . − .
61 ¯ R − . .
24 ¯ R − . ¯ R ¯ m T Figure 6.25: Estimation of ¯ m T for N = 12 and three different fits. The fitswere obtained via a function ¯ λ T = M ( N = 12) ¯ R e . In the first case we fitboth parameters M ( N = 12) , e and we got ¯ m T = ( − . ± .
66) ¯ R − . ± . .In the second case we fixed e = − .
92 and we fit Z ( N = 12) as in eq. (6.25)and finally we fixed e = − .
89 to get Z ( N = 12) = − . ± . The equation of the collapse is:¯ m T = ( − . ± .
20) ¯ R ( − . ± . . (6.25)Then, if we re-scale the y -axis by a factor ¯ R . we stabilize the value of ¯ R . ¯ m T = − . N = 12 the equation of the triple point reads¯ λ T ( N = 12) = (8 . ± .
31) ¯ R − . ± . , (6.26)¯ m T ( N = 12) = ( − . ± .
2) ¯ R ( − . ± . . (6.27)Now, to estimate the dependence on N we apply the same procedure as in thecase N = 12, for N = 8 ,
16 and N = 23. As in figures 6.24 - we will comparethree different fits. The scaling behaviour . R − . .
03 ¯ R − . .
20 ¯ R − . ¯ R λ T Figure 6.26: Estimation of ¯ λ T for N = 8 and three different fits. The first fitis ¯ λ T ( N = 8) = (9 . ± .
91) ¯ R − . ± . . The second fit is given by eq. (6.28). − .
91 ¯ R − . − .
85 ¯ R − . − .
91 ¯ R − . ¯ R ¯ m T Figure 6.27: Estimation of ¯ m T for N = 8 and three different fits. The firstfit is ¯ m T ( N = 8) = − (10 . ± .
88) ¯ R − . ± . . The second fit is given byeq. (6.29). We conclude that for N = 8 the equations of the triple point reads¯ λ T ( N = 8) = (12 . ±
3) ¯ R − . ± . , (6.28)¯ m T ( N = 8) = ( − . ±
2) ¯ R − . ± . . (6.29) .
82 ¯ R − . .
71 ¯ R − . .
87 ¯ R − . ¯ R λ T Figure 6.28: Estimation of ¯ λ T for N = 16. The first fit is ¯ λ T ( N = 16) =(2 . ± .
60) ¯ R − . ± . . The second fit is given by eq. (6.30). − .
42 ¯ R − . − .
76 ¯ R − . − .
92 ¯ R − . ¯ R ¯ m T Figure 6.29: Estimation of ¯ m T for N = 16. The first fit is ¯ m T ( N = 16) = − (4 . ± .
91) ¯ R − . ± . . The second fit is given by eq. (6.31). At N = 16 the equations of the triple point are¯ λ T ( N = 16) = (7 . ±
5) ¯ R − . ± . , (6.30)¯ m T ( N = 16) = − (9 . ±
5) ¯ R − . ± . . (6.31) The scaling behaviour .
45 ¯ R − . .
68 ¯ R − . .
77 ¯ R − . ¯ R λ T Figure 6.30: Estimation of ¯ λ T for N = 23. The first fit is ¯ λ T ( N = 23) =(10 . ± .
39) ¯ R − . ± . . The second fit is given by eq. (6.32). − .
92 ¯ R − . − .
69 ¯ R − . − .
73 ¯ R − ¯ R ¯ m T Figure 6.31: Estimation of ¯ m T for N = 23. The first fit is ¯ m T ( N = 23) = − (18 . ± . R − . The second fit is given by eq. (6.33).8 6.3. Stabilising the triple point At N = 23 the equations of the triple point are¯ λ T ( N = 23) = (7 . ±
3) ¯ R − . ± . , (6.32)¯ m T ( N = 23) = − (16 . ±
2) ¯ R − . ± . . (6.33)We summarise the coefficients Z ( N ) obtained in eq. (6.26)-(6.33) in Table 6.1 . Table 6.1: Coefficients Z ( N ) and M ( N ) considering (cid:0) ¯ λ T ( N ) , ¯ m T ( N ) (cid:1) = (cid:0) Z ( N ) ¯ R − . , M ( N ) ¯ R − . (cid:1) . N Z ( N ) M ( N )8 12 . ± − . ±
212 8 . ± − . ±
216 7 . ± − . ±
523 7 . ± − . ± Figure 6.32 shows the values of Z ( N ) in Table 6.1 and three different fits: . N − . . N − . . N − . N Z ( N ) Figure 6.32: Estimation of Z ( N ) for N = 8 , , , For ¯ m T we present a constant fit in figure 6.33 . The scaling behaviour N M ( N ) Figure 6.33: Estimation of M ( N ) for N = 8 , , , Then we get as estimation for the triple point the eqs.:¯ λ T = (45 . ± N − . ± . ¯ R − . ± . , (6.34)¯ m T = − (12 . ±
1) ¯ R − . ± . . (6.35)In eq. (6.35) we observe that ¯ m T seems not to depend on N .Let us consider the expression for the triple point fixing the exponents to e = − . , e = − .
89. The coefficients Z ( N )and M ( N ) are slightly different tothose obtained with e = − . , e = − .
92, they are summarise in the
Table6.2 . Table 6.2: Coefficients Z ( N )and M ( N ) considering (cid:0) ¯ λ T ( N ) , ¯ m T ( N ) (cid:1) = (cid:0) Z ( N ) ¯ R − . , M ( N ) ¯ R − . (cid:1) . N Z ( N ) M ( N )8 11 . ± − . ±
212 8 . ± − . ±
216 6 . ± − . ±
323 6 . ± − . ±
40 6.4. Testing the fit of the transition curve I − II The values of Z ( N ) in Table 6.2 can be fitted via the function Z ( N ) =(37 . ± N − . ± . , then the expression for ¯ λ T reads¯ λ T = (37 . ± N − . ± . ¯ R − . ± . . (6.36)For ¯ m T we observe the values of M ( N ) in Table 6.2 are practically the samethat in
Table 6.1 . We take as prediction of ¯ m T eq. (6.37):¯ m T = − (12 . ±
1) ¯ R − . ± . . (6.37)For simplicity from all the fits presented in this section we choose those wherethe exponents can be written as integer multiples of 0 . (cid:0) ¯ λ T , ¯ m T (cid:1) = (cid:0) (41 . ± N − . ± . ¯ R − . ± . , − (12 . ±
1) ¯ R − . ± . (cid:1) . (6.38)Note eq. (6.38) is in agreement with eq. (6.10) since if we substitute ¯ λ T ineq. (6.36) into eq. (6.10), the expected ¯ m T turns out to be:¯ m T = − .
99 ¯ R − . , (6.39)eq. (6.37) coincides with the expression for ¯ m T in eq. (6.38) within the errors. I − I I
In this section we want to check the viability of a linear fit for the disordered toordered non-uniform phase transition curve for values of ¯ λ slightly above ¯ λ T .We study the significance of a quadratic term in ¯ λ . We propose¯ m c = g ( N, ¯ R ) + g ( ¯ R, N )¯ λ + g ( ¯ R, N )¯ λ . (6.40)From the previous section we adopt the assumption g ( N, ¯ R ) ∝ ˜ g ( ¯ R ) and g ( N, ¯ R ) ∝ N ˜ g ( ¯ R ) and we propose g ( N, ¯ R ) = g ( N )˜ g ( ¯ R ).We observe g ( N ) = const . To find ˜ g i ( ¯ R ) , i = 0 , ,
2, we keep N = 12 fixedand fit the curve (6.40) for the set of data with different ¯ R . Figure 6.23 is anexample for the procedure.Now we present the plots and fits for ˜ h ( ¯ R ) , ˜ h ( ¯ R ) and ˜ g i ( ¯ R ) , i = 0 , , The scaling behaviour Fit for ˜ g ˜ g ( ¯ R )˜ h ( ¯ R )¯ R Figure 6.34: Coefficients ˜ h ( ¯ R ) defined in eq. (6.13) and ˜ g ( ¯ R ) in eq. (6.40)for ¯ R = 4 , , , , h ( ¯ R ) ≈ ˜ g ( ¯ R ) for ¯ R >
4. The fitfor ˜ g ( ¯ R ) is given in eq. (6.41). The fit for figure 6.34 is :˜ g ( ¯ R ) = 18 .
33 ¯ R − . − . R − . . (6.41) Fit for ˜ g ˜ g ( ¯ R )˜ h ( ¯ R )¯ R Figure 6.35: Coefficients ˜ h ( ¯ R ) defined in eq. (6.13) and ˜ g ( ¯ R ) in eq. (6.40)for ¯ R = 4 , , , , R >
4, ˜ h ( ¯ R ) ≈ ˜ g ( ¯ R ). Thefit for ˜ g ( ¯ R ) is given in eq. (6.42). The fit for figure 6.35 is :˜ g ( ¯ R ) = − .
195 ¯ R − . − .
004 ¯ R − . . (6.42) As we can observe the coefficients obtained by a fit with a polynomial of degree1 and 2 are very similar. Thus the linear fit should be a good approximation forthe region around the triple point. Fit¯ R ˜ g ( ¯ R ) Figure 6.36: Coefficients g ( ¯ R ) for ¯ R = 4 , , , , The fit for figure 6.36 is :˜ g ( ¯ R ) = 0 . R − . + 1 . · − ¯ R . . (6.43)The final expression for this transition curve is:¯ m = (18 .
33 ¯ R − . − . R − . ) − (0 .
195 ¯ R − . + 0 .
004 ¯ R − . ) N ¯ λ +144 · (0 . R − . + 1 . × − ¯ R . )¯ λ + O (¯ λ ) . (6.44) ¯ λ < ¯ λ T For this range of ¯ λ we find that rescaling the x -axis by the factor ¯ λ − the suscep-tibilities χ and χ collapse. We show an example in figure 6.37 . We chose the fit of the form ˜ h ( ¯ R ) = a ¯ R e + a ¯ R e since we could not find a fit ofthe form ˜ h ( ¯ R ) = a ¯ R e good enough to reproduce our set of data. The scaling behaviour ¯ λ = 6¯ λ = 5¯ λ = 4¯ λ = 3¯ λ = 2¯ λ − ¯ m ξ λ = 6¯ λ = 5¯ λ = 4¯ λ = 3¯ λ = 2¯ λ = 1¯ λ − ¯ m ξ Figure 6.37: Collapse of χ and χ for ¯ R = 8, N = 8. The related graph of figure 6.37 for (cid:104) φ all (cid:105) is: ¯ λ = 6¯ λ = 5¯ λ = 4¯ λ = 3¯ λ = 2¯ λ − ¯ m (cid:1) ϕ a ll (cid:2) Figure 6.38: Collapse of (cid:104) φ all (cid:105) for ¯ R = 8, N = 8. ¯ λ = 6¯ λ = 5¯ λ = 4¯ λ = 3¯ λ = 2¯ λ = 1¯ λ − ¯ m Sp ec i fi c H e a t Figure 6.39: Collapse of the Specific Heat in eq. (4.35) for ¯ R = 8, N = 8. We can see from figures 6.37 - the equivallence between the Specific Heatcriteria and the susceptibilities criteria at this regime. This subject is more widelydisccussed in appendix C. ¯ λ (cid:29) ¯ λ T If ¯ λ is sufficiently large we expect that the relevant contributions to the action ineq. (4.12) are those of the potential terms. Depending on the value of ¯ R one orboth of the kinetic terms (fuzzy and time derivatives) can be negligible.This leads us to reduced models , i.e. models that effectively depend on lessparameters than those in our model in eq. (4.12).If the time derivative terms are negligible we would have a chain of N -indepen-dent fuzzy spheres. If the fuzzy kinetic term is negligible the reduced model is a chain of matrix models , see Refs. [46]-[47]. If both kinetic terms are negligible twoparameters are redundant, this lead us to the 1-matrix model (see Ref. [49]).We devote this section to the study of our model for ¯ λ (cid:29) ¯ λ T . We concentratedour analysis in the collapse of the phase transition and some observables.The transition curve for this region of the space of parameters can be wellfitted by the expression (6.45): ¯ m c = w ( N, ¯ R )¯ λ w . (6.45)The exponent w typically oscillates in the range [0 . ,
1] as it can be appreciatedin Table
D.1 in appendix D.As an example we present the transition curve disordered — ordered non-uniform for N = 8 , ¯ R = 16 and its proposed fit in Figure 6.40 : − . λ / I-III
III: Ordered non-uniformI: Disordered ¯ λ ¯ m Figure 6.40: Transition curve from the disordered phase to the ordered non-uniform phase for N = 8 , ¯ R = 16. The value of the exponent w in eq. (6.45)was fixed to . The scaling behaviour A characteristical behaviour of the specific heat (4.35) for the this region oflarge ¯ λ is shown in figure 6.41 for λ = 625, the last point in figure 6.40 ( N ¯ λ =5000): ¯ m Sp ec i fi c H e a t / V o l. Figure 6.41: Specific heat at ¯ λ = 625, ¯ R = 16, N = 8. The critical point isestimated at ¯ m = − ± . The corresponding internal energy (4.34) and its partial contributions (4.40)-(4.43) to figure 6.41 are shown in figure 6.42 E E E E E ¯ m E n e r g y / V o l. Figure 6.42: Internal energy E in eq. (4.34) and its partial contributionseqs. (4.40)-(4.43) at ¯ λ = 625, ¯ R = 16, N = 8.6 6.5. Collapse of observables Figure 6.42 shows that the leading contributions to the internal energy isfrom the potential. Although small, the contributions of the kinetic terms are notnegligible.The next step would be to try to collapse in N and ¯ R the transition curves inthis region of large ¯ λ . As we have mentioned, the exponent w typically oscillatesin the range [0 . , w strongly depends on the consideredregion of ¯ λ –see Table D.1 in appendix D.In section 6.2 we focused in the region around the triple point, there we con-sidered the transition curve a straight line, i.e. w = 1. As we increase the rangeof ¯ λ the exponent w = 1 decreases. The N -matrix model If the fuzzy kinetic term is negligible we arrive at a chain of matrix model .Ref. [47] studied the large N limit of this model, with a potential gφ . Therea phase transition was predicted at the critical value (6.46) g c = ( − µ ) / π , (6.46)where g is the critical value of the coupling and µ is the squared mass parameter.Under the appropiate translation to our parameters in eqs. (4.44)-(4.46), eq. (6.46)reads ¯ m c = − (cid:18) N
16 ¯ R (cid:19) / ¯ λ / . (6.47)In terms of the coefficient w ( N, ¯ R ) and exponent w in eq. (6.45) the predictionfor the transition curve disordered ordered non-uniform at large ¯ λ is: w ( N, ¯ R ) = − (cid:18) N
16 ¯ R (cid:19) / , w = 23 . (6.48)The transition in figure 6.40 obeys eq. (6.47) (the fit in figure 6.40 is ¯ m c = − . λ / and the prediction from eq. (6.47) is ¯ m c = − . λ / ).Some other examples that have the transition in eq. (6.47) are shown in figures6.43 - . The scaling behaviour − . λ / I-III
III: Ordered non-uniformI: Disordered ¯ λ ¯ m Figure 6.43: Transition curve from the disordered phase to the ordered non-uniform phase for N = 16 , ¯ R = 16. − . λ / I-III
III: Ordered non-uniformI: Disordered ¯ λ ¯ m Figure 6.44: Transition curve from the disordered phase to the ordered non-uniform phase for N = 23 , ¯ R = 16. In figure 6.44 we observe that the fit ¯ m c = − . λ / works well for ¯ λ >
20 (theprediction for w (23 ,
16) in eq. (6.48) is − . λ >
20 for this case the estimated value for w would be w > . Thiscan explain the different estimated values for w in Table D.1 in appendix D. Asecond possibility is that for those cases in
Table D.1 where w < is that wehave a transition of a different nature. This is analysed in the following section6.5.2. The 1-matrix model
In the 1-matrix model a transition is expected at m c = − √ N λ, (6.49)according to the notation in Ref. [30] where it was confirmed numerically, or b c = − √ N c (6.50)according to the notation in Ref. [32] –see eq. (3.21).The question is if our model has such a transition. Theoretically for very largevalues of ¯ λ and small value of ¯ R , the dominant term in the action is the potential.Then the model should effectively depend on less parameters. Following Ref. [50]we obtain the relevant parameters in our model form those in the 2-dimensionalmodel: m d = N m d , λ d = N λ d . (6.51)As a next step we verify if our model has a transition at¯ m c ∝ (cid:112) N ¯ λ. (6.52)With this intention we present figures 6.45-6.46 . − . √ N ¯ λ + 5 . N = 33 N = 23 N = 16 N = 12 N = 8 √ N ¯ λ m Figure 6.45: Phase transition to the disordered phase for several values of N and ¯ R = 8. The scaling behaviour − . √ N ¯ λ + 2 . N = 33 N = 23 N = 16 N = 12 N = 8 √ N ¯ λ m Figure 6.46: Phase transition to the disordered phase for several values of N and ¯ R = 16. For the points with larger ¯ λ in figure 6.45 the N = 16 data can be fitted bya line eq. (6.53): ¯ m c ( ¯ R = 8) = − . (cid:112) N ¯ λ + 5 . , (6.53)but a substantial difference is that it does not cross the origin as in eq. (6.52).A similar situation occurs for ¯ R = 16 in figure 6.46 for N = 16 ,
23 where thefit is given by eq. (6.54): ¯ m c ( ¯ R = 16) = − . (cid:112) N ¯ λ + 2 . . (6.54)Note that the coefficients in eq. (6.53) for ¯ R = 8 are approximately doubled com-pared to eq. (6.54) for ¯ R = 16.We conclude that we cannot confirm the phase transition in eq. (6.51) for the2-dimensional model. A possible explanation is that in the data obtained we havenot reached a sufficiently large ¯ λ .Now we focus on the collapse of other observables. Collapse of φ all In this section we investigate the collapse of the norm of the field in eq. (4.27) forlarge ¯ λ . This quantity is of interest since for the ordered non-uniform phase for large ¯ λ the main contribution to (cid:104) φ all (cid:105) comes form higher modes, and the otherquantities we are measuring are related to the lowest modes. In addition it isknown from the 1-matrix model studied in Ref. [30] that for a region where thekinetic term is negligible, (cid:104) φ all (cid:105) has the simple form: (cid:104) φ all (cid:105) = − πrλ , (6.55)where r is the squared mass parameter. It would be interesting to check if ourmodel, under the appropriate translation of parameters, can reproduce the 1-matrix result as a limiting case.For ¯ m < ¯ m c we found that the norm of the field, (cid:104) φ all (cid:105) — eq. (4.27) —, canbe well fitted by a line. A important difference from the 1-matrix model studiedin Ref. [30] is that the line does not crosses the origin. We propose eq. (6.56) (cid:104) φ all (cid:105) = v ( N, ¯ R, ¯ λ ) + v ( N, ¯ R, ¯ λ ) ¯ m . (6.56)A concrete example is presented in figure 6.47 for the same parameters as in figures 6.41 - . (cid:1) ϕ (cid:2)(cid:1) ϕ (cid:2)− . m − . (cid:1) ϕ all (cid:2) ¯ m -7-7.5-8-8.5-9-9.5-100.0140.0120.010.0080.0060.0040.0020 Figure 6.47: (cid:104) ϕ all (cid:105) and a estimation vs. ¯ m at ¯ λ = 625, ¯ R = 16, N = 8.We observe that the contributions from the zero and the first mode, (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) respectively to (cid:104) ϕ all (cid:105) are small as we expected. The critical valueis ¯ m c = − . ± . The scaling behaviour Note:
There are some technical difficulties in the measurement of v and v :The coefficients v and v must be measured on an appropriate range of ¯ m , where v and v stabilise. This typically happens for ¯ m < ¯ m c , i.e. in the orderednon-uniform phase where we have thermalisation problems. For the parameters in figure 6.47 they appear for ¯ m < −
15. Then in figure 6.47 the value of v and v stabilise for − < ¯ m < − . chain of matrix models . In the chain of matrix models the fuzzy kinetic termshould be negligible while in the 1-matrix model both kinetic terms are negligible.We conjecture that the non-vanishing coefficient v ( N, ¯ R, ¯ λ ) is related to thechain of matrix models and therefore should reduce to zero for the limiting case ofthe 1-matrix model. Therefore the coefficients v ( N, ¯ R, ¯ λ ) and v ( N, ¯ R, ¯ λ ) shoulddepend differently on N and ¯ R .Finally we present the attempts to collapse (cid:104) φ all (cid:105) for ¯ λ > ¯ λ T in figures 6.48-6.49 . N ¯ λ = 2500 N ¯ λ = 2000 N ¯ λ = 1800 N ¯ λ = 1500 N ¯ λ = 100 N ¯ λ = 40 N ¯ λ = 30 N ¯ λ = 24 N ¯ λ = 20 N ¯ λ = 16¯ λ − . ¯ m (cid:1) ϕ a ll (cid:2) -0.01-0.015-0.02-0.025-0.03-0.035-0.04-0.045-0.050.160.140.120.10.080.060.040.020 Figure 6.48: Collapse of (cid:104) φ all (cid:105) for ¯ R = 8, N = 8. We re-scale the x -axis bythe factor ¯ λ − . . For values of ¯ λ slightly above ¯ λ T the collapse works but notfor ¯ λ ≥ In figure 6.48 the x -axis is re-scaled by the factor ¯ λ η . The optimal value of η depends on the range of ¯ λ . η = − . ≥ ¯ λ > ¯ λ T , it gives andacceptable collapse for ¯ λ <
100 in the range η ∈ [ − . , − . > ¯ λ > η = − figure 6.49 . N ¯ λ = 2500 N ¯ λ = 2000 N ¯ λ = 1800 N ¯ λ = 1500 N ¯ λ = 100 N ¯ λ = 40 N ¯ λ = 30 N ¯ λ = 24 N ¯ λ = 20 N ¯ λ = 16¯ λ − ¯ m (cid:1) ϕ a ll (cid:2) Figure 6.49: Collapse of (cid:104) φ all (cid:105) for ¯ R = 8, N = 8. We re-scale the x -axis bythe factor ¯ λ − . For values of ¯ λ ≥ We conclude that the collapse of (cid:104) φ all (cid:105) depends on the range of ¯ λ . For theSpecific Heat we have a similar situation a for (cid:104) φ all (cid:105) .For ¯ λ ≤
100 we have in figure 6.50 the collapse of the Specific Heat re-scalingthe x axis by the factor ¯ λ − η with η = − . figure 6.48 . N ¯ λ = 100 N ¯ λ = 40 N ¯ λ = 30 N ¯ λ = 24 N ¯ λ = 20 N ¯ λ = 16¯ λ − . ¯ m Sp ec i fi c H e a t -0.01-0.02-0.03-0.04-0.05-0.060.00160.00150.00140.00130.00120.00110.0010.00090.00080.00070.0006 Figure 6.50: Collapse of the Specific Heat in eq. (4.35) at ¯ R = 8, N = 8 for100 ≥ N ¯ λ ≥ In figure 6.51 we present the collapse of the Specific Heat for the same data thatin figure 6.50 , but re-scaling the x axis by the factor N − / ¯ R / ¯ λ − / . The scaling behaviour N ¯ λ = 2500 N ¯ λ = 2000 N ¯ λ = 1800 N ¯ λ = 1500 N ¯ λ = 100 N − / ¯ R / ¯ λ − / ¯ m Sp ec i fi c H e a t -0.15-0.2-0.25-0.3-0.35-0.40.00160.00140.00120.0010.00080.00060.0004 Figure 6.51: Collapse of the Specific Heat in eq. (4.35) at ¯ R = 8, N = 8 for N ¯ λ ≥ N ¯ λ = 6500 N ¯ λ = 6000 N ¯ λ = 5000 N ¯ λ = 3000 N ¯ λ = 2100 N ¯ λ = 1800 N ¯ λ = 1000 N − / ¯ R / ¯ λ − / ¯ m Sp ec i fi c H e a t -0.2-0.22-0.24-0.26-0.28-0.3-0.32-0.34-0.360.00150.00140.00130.00120.00110.0010.00090.00080.00070.00060.0005 Figure 6.52: Collapse of the Specific Heat in eq. (4.35) at ¯ R = 16, N = 8 for N ¯ λ ≥ From figures 6.51-6.52 we observe that re-scaling the x -axis by the factor N − / ¯ R / ¯ λ − / we fix the critical value of ¯ m c to N − / ¯ R / ¯ λ − / ¯ m ≈ − . figure 6.53 for N = 12: N ¯ λ = 2400 N ¯ λ = 2100 N ¯ λ = 1800 N ¯ λ = 1300 N ¯ λ = 1000 N − / ¯ R / ¯ λ − / ¯ m Sp ec i fi c H e a t -0.26-0.28-0.3-0.32-0.340.00060.000550.00050.000450.00040.000350.00030.00025 Figure 6.53: Collapse of the Specific Heat in eq. (4.35) at ¯ R = 16, N = 12for N ¯ λ ≥ We conclude the collapse of observables and the transition curve for ¯ λ > ¯ λ t does not lead to the 1-matrix model. Furthermore, the collapse of observablesindicates that the exponent of collapse — η for (cid:104) ϕ all (cid:105) as in figures 6.48 - —depends on the range of ¯ λ .For the largest values of ¯ λ considered our results agree to those predicted forthe chain of matrix models in Ref. [47] at large N , as it is shown in figures6.51-6.53 . hapter 7Discussion of the results As it was mentioned in section 4.3 we are interested in taking the limit N −→ ∞ .If we scale R in terms of N we can access different limiting models. The key pointin our analysis is the behaviour of the triple point and the phase coexistence curvesunder those limits; it will decide which phases survive at the end.If the triple point remains finite, it indicates that the three phases exist. If thetriple point goes to infinity it would indicate the existence of the Ising type phaseswhile the ordered non-uniform phases would disappear.Now we proceed to analyse the model around the critical curves. Coming backto the discretised model given by eq. (4.12), it can be re-written as S [Φ] = N t (cid:88) t =1 Tr (cid:104) A Φ ( t ) ˆ L Φ ( t ) + D (Φ( t + ∆ t ) − Φ( t )) + B Φ ( t ) + C Φ ( t ) (cid:105) (7.1)where we repeat for conveniece the definitions (4.8)-(4.11) A = 2 π ∆ tN , (7.2) D = 2 πR N ∆ t , (7.3) B = 2 πR m ∆ tN , (7.4) C = πR λ ∆ tN . (7.5)We have the freedom to re-scale the field Φ in the following way:Φ −→ Φ (cid:48) = 1 √ z Φ (7.6) and the constants (7.2) - (7.5) change to A = 2 πz ∆ tN , (7.7) D = 2 πR zN ∆ t , (7.8) B = 2 πR m z ∆ tN , (7.9) C = πR λz ∆ tN . (7.10)As we mentioned in section 4.4.3 we can choose z such that we fix one of theconstants (7.7)-(7.10). We fix A = 2 π , i.e. z = N ∆ t . (7.11)Under the re-scaling (7.11) the constants (7.7)-(7.10) change to A = 2 π, (7.12) D = 2 πR (∆ t ) = 2 π ¯ R , (7.13) B = 2 πR m = 2 π ¯ R ¯ m , (7.14) C = πR λN ∆ t = πN ¯ R ¯ λ. (7.15)We re-write the obtained expression of the triple point of eq. (6.38) makingthe substitutions of eqs. (4.44)-(4.46) to get (cid:0) ∆ tλ T , (∆ t ) m T (cid:1) = (cid:32) . (cid:18) (∆ t ) N R (cid:19) γ , − . (cid:18) ∆ tR (cid:19) γ (cid:33) (7.16)with γ = 0 . ± . R = N β for β >
0, then R −→ ∞ as N −→ ∞ .Eq. (7.16) changes to eq. (7.17) (cid:0) ∆ tλ T , (∆ t ) m T (cid:1) = (cid:16) . t ) γ N − γ (2 β +1) , − . t ) γ N − γβ (cid:17) , (7.17) Choice of ∆ t . We choose ∆ t = 1 N κ , (7.18)with κ > Discussion of the results Then the time extension T is N t ∆ t = N − κ (7.19)for N t = N .If κ < T −→ ∞ as N −→ ∞ .Implementing eq. (7.18) in eq. (7.17) we get (cid:0) ¯ λ T , ¯ m T (cid:1) = (cid:0) ∆ tλ T , ∆ t m T (cid:1) = (cid:16) . N − γ (2 β +2 κ +1) , − . N − γ ( β + κ ) (cid:17) . (7.20)We note that the dimensionless quantities in eq. (7.20) goes to zero as N −→ ∞ .Eq. (7.20) can be written as (cid:0) λ T , m T (cid:1) = (cid:16) . N κ (1 − γ ) − γ (1+2 β ) , − . N κ − γ ( β + κ ) (cid:17) . (7.21)To analyse the limit N −→ ∞ we focus on the exponents of N in eq. (7.20). For λ T the exponent is κ (1 − γ ) − γ (1 + 2 β ) and this is clearly negative if 1 − γ ≤ γ is quite large even taking the lower bound the exponent κ (1 − γ ) − γ (1 + 2 β ) is negative, which implies λ T −→ , N −→ ∞ . For m T the exponent of N is 2 κ − γ ( β + κ ), it is positive or negative depend-ing on the values of β and κ .In any case what we would have is the disappearance of the ordered uniformphase and therefore the phase diagram of the commutative theory cannot be re-covered from the one of our studied model (7.1).The tricritical action reads S T [Φ , N, R ] ≈ N (cid:88) t =1 Tr (cid:104) πN Φ ( t ) ˆ L Φ ( t ) + 2 πN (cid:18) R ∆ t (cid:19) [Φ( t + ∆ t ) − Φ( t )] − . πN Φ ( t ) + 41 . πN (cid:18) RN ∆ t (cid:19) Φ ( t ) (cid:105) . (7.22)We consider the particular case ∆ t = √ N and R = N β , The lower bound would be 0 . − . . We chose γ ≈ . S T [Φ , N, R ] ≈ N (cid:88) t =1 Tr (cid:104) πN Φ ( t ) ˆ L Φ ( t ) + 2 πN β [Φ( t + ∆ t ) − Φ( t )] − . πN Φ ( t ) + 41 . πN γ ( β − ) − Φ ( t ) (cid:105) . (7.23)The exponent of N for the Φ -term in eq. (7.23), γ ( β − ) −
1, is negative forthe values of β considered in the limits in section 4.3 –see eqs. (3.13)-(3.15). Weobserve that for N large the leading contribution to eq. (7.23) comes from thetemporal kinetic term. If we compare the powers on N multiplying the kinetic terms in eq. (7.23), wesee that the temporal kinetic term is N β +1 times larger than the fuzzy kineticterm. Under the limit N −→ ∞ the fuzzy kinetic term is negligible and then,because in some sense the geometry of the sphere is screened by the temporalkinetic term, the model behaves more like a “matrix chain” system interacting tofirst neighbours with the potential λφ (see Ref. [46]).To maintain the uniform ordered phase in the limit N −→ ∞ it is necessaryto reinforce the fuzzy kinetic term as in Refs. [22]-[29]. We will come back to thispoint in the conclusions. λφ on the fuzzy sphere We found the 3 phases present in the model in eq. (3.20) –see Refs. [30]-[32].A substantial difference from the 2-dimensional case studied in Refs. [30]-[32]is that the whole phase diagram in figure 3.1 collapses using the same scalingfunction of N . We do not have this situation since the scaling on N for thethe transition curves (6.10) (6.18) are different. The collapse in the 2 dimensionalstudies do not depend on the radius since it can be absorbed in the couplings. Thisis in contrast to the 3 dimensional case, where the radius plays an independentrˆole.In section 6.5.2 we studied the collapse of observables and of the transitioncurve disordered to ordered non-uniform. We conclude that we do not observe We will identify (cid:80) t Tr (cid:16) Φ ( t ) ˆ L Φ ( t ) (cid:17) / (cid:80) t Tr [Φ( t + ∆ t ) − Φ( t )] . Discussion of the results the same scaling behaviour that as in the 2-dimensional case. Furthermore, thecollapse of the transition curve I − III for large λ – see eq. (6.47)– indicates thatin this regime our model behaves as a chain of interacting matrices , where thefuzzy kinetic term is neglected Ref. [47]. In this section we want to compare our results with those obtained in Refs. [35],[36]and [37] where the following model was studied: S [Φ] = N Tr T (cid:88) t =1 (cid:104) (cid:88) i (cid:16) ˆ D i Φ( t ) ˆ D † i − Φ( t ) (cid:17) + 12 (Φ( t + 1) − Φ( t )) + m ( t ) + λ ( t ) (cid:105) . (7.24)Φ is an hermitian matrix. Eq. (7.24) describes a scalar field φ living on a non-commutative torus and interacting under the λφ potential.Comparing eq. (7.24) to eq. (7.1) we observe that, besides the different interpre-tation in each discretisation scheme, the substantial difference between the modelsis in the spatial kinetic term. The first term on the right-hand-side in eq. (7.24)corresponds to the energy due to spatial translations on a squared N × N latticewith lattice spacing a , while the first term on the right-hand-side in eq. (7.1) cor-responds to the energy due to translations (or rotations) over a discrete version ofthe sphere. It is to be expected that in the regime where the spatial kinetic termis negligible, both models describe the same physics. Now we want to comparethe phase diagram for each model, but first we denote the constants of the modelgiven by (7.24) as follows: A = N , (7.25) D = N , (7.26) B = N m , (7.27) C = N λ . (7.28) This is under the appropriate translation of parameters from the 2-dimensional modelto the 3-dimensional model, see eq. (6.51).
In Refs. [35]-[36] the phase diagram shows the existence of three phases: • Disordered phase • Uniform phase • Striped phaseThis phase diagram stabilises taking the axes as N λ vs. N m , this is, underthe re-scaling of the axes λ −→ N λ and m −→ N m the transition lines collapseas follows: • Disordered phase — Uniform phase coexistence line: N m ∼ = − . N λ. (7.29) • Disordered phase — Striped phase coexistence line: N m ∼ = − . N λ − , (7.30)and the triple point is given by( N λ, N m ) ∼ = (220 , − . (7.31)In the model (7.1) we found the existence of three phases: • I: Disordered phase • II: Ordered uniform phase • III: Ordered non-uniform phaseWe conjecture that the striped phase of the model in eq. (7.24) corresponds to theordered non-uniform phase in the model given by eq. (7.1).Coming back to the fuzzy model in eq. (7.1), their coexistence curves I − II and I − III stabilise under different scaling: • Disordered phase — Ordered uniform phase:¯ R γ ¯ m = − . N γ ¯ R γ ¯ λ. (7.32) Discussion of the results • Disordered phase — Ordered non-uniform phase:¯ R γ N γ − (cid:18) ¯ m + 12 . R γ (cid:19) = − . N ¯ R ) γ ¯ λ + 2 . N γ − . ¯ R γ (7.33)The triple point is given by the equations: (cid:0) ¯ λ T , ¯ m T (cid:1) = (cid:0) . N − . ± . ¯ R − . ± . , − (12 . ±
1) ¯ R − . ± . (cid:1) . (7.34)In the region around the transition curve I − II (to the ordered uniform phase)the kinetic term is relevant. Then, because of the difference in the nature of thekinetic term in the models in eq. (7.24) and eq. (7.1), we cannot expect that thereexists a re-scaling such that both collapses are compatible. In the region aroundthe transition curve I − III (to the ordered non-uniform phase) the leading termis the one due to the potential, then we could expect that there exists a re-scalingsuch that both collapses are compatible for the transition curve I − III . Now weconsider an special case of R and ∆ t . Case R = N , ∆ t = 1 . We consider R = N and ∆ t = 1. Implementing the substitution in eq. (7.32) weget R γ m = − . N γ λ. (7.35)Comparing eq. (7.29) to eq. (7.35) and considering 3 γ ≈
2, we conclude that thetransition line I − II for both models collapse with the same dependence in N .We have the same situation when we compare the triple points. Implementing thesubstitutions R = N and ∆ t = 1 in eq. (7.16) we get (cid:0) λ T , m T (cid:1) = (cid:18) .
91 1 N γ , − . N γ (cid:19) ≈ (cid:18) . N , − . N (cid:19) . (7.36)The dependence of the triple point on N is the same as in Refs. [35]-[36]. Thecritical action is: S T [Φ , N ] ≈ N (cid:88) t =1 Tr (cid:104) πN Φ ( t ) ˆ L Φ ( t ) + 2 πN [Φ( t + ∆ t ) − Φ( t )] − . πN Φ ( t ) + 41 . πN Φ ( t ) (cid:105) . (7.37)From (7.37) we observe that the dominance of the temporal kinetic term is evenstronger than in eq. (7.23). • Observation:
Choosing β = in eq. (7.23) we get the same critical action as in eq. (7.37),but the dependence on N in the triple point is not the same.The triple point (7.20) for β = is: (cid:0) λ T , m T (cid:1) = (cid:16) . N − γ , − . N − γ (cid:17) . (7.38)For the transition curve I − III the predictions from models (7.1)-(7.24) aredifferent. The main difference is that the transition curve in the model given by(7.1) is a curve as it is shown in figures 6.43 - , whereas in the model givenby eq. (7.24) this transition curve is a straight line. Nevertheless, we found thatin a range of parameters around the triple point the transition curve I − III inthe model (7.1) could be approximated by a line as in (7.24). We conjecture thatthis is the range of parameters studied in Refs. [35]-[36].Taking R = N and ∆ t = 1 in (7.33) we get N γ − (cid:18) m + 12 . N γ (cid:19) = − . N γ λ + 2 . N γ − . . (7.39)If we consider just the terms in m and λ in (7.39), we obtain N γ − m vs. N γ λ .We observed 3 γ ≈ γ − . ± .
8. We conclude that the dependenceon N in the eq. (7.39) could be the same as in eq. (7.30). hapter 8Conclusions from part I • We presented a numerical study of the λφ model on the 3-dimensionalEuclidean space which was regularised by means of: – the fuzzy sphere S F for the spatial coordinates – a conventional lattice with periodic boundary conditions for the timedirection.The obtained model was S [Φ] = 4 πR N ∆ t N t (cid:88) t =1 Tr (cid:104) R Φ ( t ) ˆ L Φ ( t ) + 12 (cid:18) Φ( t + ∆ t ) − Φ( t )∆ t (cid:19) + m ( t ) + λ ( t ) (cid:105) . (8.1)where Φ( t ) ∈ M at N , for t = 1 , . . . , N t . • We found the phase diagram of the model according to the specific heat.Following this criterion we determined the critical values of λ and m forfixed N and R denoted by an index “c”, λ c and m c . In addition we found thetricritical point ( m T , λ T ). λ T divides the phase diagrams into two regionsaccording to the behaviour of the observables. – λ T > λ >
0. In this domain we observe Ising type orderings. For m > m c we have a disordered phase. For m < m c we found auniform ordering. – λ > λ T . For m > m c we have a disordered phase. As in the two di-mensional model discussed in chapter 3, we found for m < m c there isa non-uniform ordering. For m (cid:28) m c we encountered thermalisationproblems, therefore we cannot conclude if there exists a boundary forthe non-uniform ordered phase as in the 2-dimensional case [32]. • We compared our results with those obtained by other criteria (two pointfunctions of the different modes) and we conclude both criteria are quali-tatively equivalent. We followed the specific heat criterion which is a moreuniversal quantity: it captures the phase transition without taking into ac-count which is the dominant mode. • We found the existence of three phases: – I : Disordered phase. – II : Ordered Uniform phase. – III : Ordered Non-Uniform phase.These three phases were also found in the 2-dimensional λφ model on afuzzy sphere studied in Refs. [30], [32]. • The phase of non-uniform ordering is characterised by the dominance ofseveral angular momenta for l >
0. In this phase the rotational invarianceis broken. • We get the transition curves of the model: – I − II : It turns out to be a line given by the equation:(∆ t ) m c = ( − . ± . N . ± . (cid:0) R ∆ t (cid:1) . ± . ∆ tλ. (8.2) – I − III : We can observe that the transition curve shows curvature. Themost natural fit we can propose is a polynomial where the coefficientsare functions that depend on R , N and ∆ t . Nevertheless, to predictthe triple point we can concentrate on values of λ slightly above λ T .For this range of values the transition curve can be approximated by aline. We conclude that for λ around λ T the transition curve from the Conclusions from part I disordered phase to the ordered non-uniform phase obeys the followingequation:(∆ t ) m c = − (0 . ± . N (cid:18) R ∆ t (cid:19) − . ± . ∆ tλ + (cid:0) . N . − . (cid:1) (cid:18) R ∆ t (cid:19) − . . (8.3) – We conjecture the existence of a transition curve II − III as in Ref. [32],but due to thermalisation problems it was not possible to measure it. – The effective action has several minima, and the thermalisation prob-lems appear when it is not possible for the algorithm to tunnel betweenthose minima. We sketched the main technical features of these ther-malisation problems. • We obtained the equation for the triple point: (cid:0) ∆ tλ T , (∆ t ) m T (cid:1) = (cid:32) . N − . ± . R − . ± . , − (12 . ± (cid:18) R ∆ t (cid:19) − . ± . (cid:33) . (8.4) • Different limits of the fuzzy sphere can be obtained scaling R as a functionin N and taking the limit N −→ ∞ , – Commutative sphere: R = const., N −→ ∞ – Quantum Plane: R ∝ N – Continuum flat limit. It requires: R ∼ N − (cid:15) , > (cid:15) > . (8.5)We analysed the behaviour of the scalar model in (8.1) under the differentlimiting spaces above. • Our numerical results reveal that the triple point goes to zero under thelimit N −→ ∞ , this is valid for all cases considered in the previous point. • In other words, in the limit N −→ ∞ the non-uniform ordered phase domi-nates the phase diagram. This result is as consequence of the UV-IR mixing:integrating out high energy in the loop produces non-trivial effects at lowexternal momenta [20]-[21]. A perturbative analysis of the action (our eq. 4.4 of chapter 4 ) : S [Φ] = 4 πR N (cid:90) S dt Tr (cid:104)
12 Φ ( t ) (cid:32) ˆ L R − ∂ t (cid:33) Φ ( t )+ m ( t ) + λ ( t ) (cid:105) , (8.6)is presented in Ref. [29]. It shows that the action given by eq. (8.6) doesnot reproduce the commutative continuum limit. This result was obtainedfrom a expansion to two loops. Our results show that the UV-IR mixing ispresented for all values of λ . • Following Refs. [22] and [29] the action (8.6) should be corrected by modi-fying the action.This modification in the action is equivalent to reinforcing the fuzzy kineticterm and it can be achieved adding a term Φ( L ) Φ / (Λ R ) where Λ is amomentum cutoff. • We compare our results to those obtained in a numerical study of the λφ potential on a non-commutative torus (see Ref. [35]-[36]). Our results are inagreement if the parameters R and ∆ t are scaled appropriately. • The axes cannot be chosen consistently for all regimes to lead to a sta-ble phase diagram for large N . This is a significant difference to the 2-dimensional formulation which has the “privileged” property of leading to aphase diagram. • We compare our results at large coupling λ to those of the chain of matricesat the large N -limit in Refs. [47]-[64] (where this model is known as the c = 1model in string theory). Our simulation results fully agree. The disorderedto ordered non-uniform phase transition in this regime obeys the predictedtransition in the model c = 1 in the large N -limit. To be precise, in ref. [29] discusses S F × R . art IIScalar Field Theory on S hapter 9From fuzzy C P to a fuzzy S F S is an especially important example since it is the most natural replacement of R in studies of Euclidean quantum field theory. Therefore our motivation to proposea fuzzy approximation to S , namely S F . Nevertheless S cannot be quantised inthe strict sense since it is not a phase space. It is important to clarify how we canwe obtain a matrix approximation to it. S F in analogy to S F Having in mind the seminal example of fuzzy space, we search for five matrixcoordinates , X a , a = 1 . . .
5, which fulfil a matrix equation of a 4-sphere in R : (cid:88) a =1 X a X a = R . (9.1)We propose X a = R √ a , (9.2)with Γ a ∈ M at the Dirac matrices including γ . They obey the algebra: { Γ a , Γ b } = 2 δ ab . (9.3)Eq. (9.1) follows directly from eqs. (9.2) and (9.3).The next step is to propose a sequence of matrices approximating S . Toachieve this we note that Γ a (which are 4-dimensional matrices) give the represen-tation ( , ) of Spin (5). We can therefore consider the irreducible representation S F in analogy to S F obtained from the L fold symmetric tensor product of this representation i.e. the Spin (5) representation ( L , L ). It will contain a set of five matrices: J a , a = 1 . . . L copies of the Γ matrices in the Spin (5) fundamental representation: J a = Γ a ⊗ ⊗ · · · ⊗ (cid:124) (cid:123)(cid:122) (cid:125) L − terms + ⊗ Γ a ⊗ · · · ⊗ + · · · + ⊗ ⊗ · · · ⊗ Γ a sym . (9.4)The subscript sym indicates that we are projecting onto the irreducible totallysymmetrised representation. The dimension of the matrices defined in eq. (9.4) is d L = ( L + 1)( L + 2)( L + 3)6 , (9.5)and J a ∈ M at d L . They satisfy the relation: J a J a = L ( L + 4) . (9.6)Now we can generalise the definition of the matrix coordinates (9.2) to eq. (9.7): X a = R (cid:112) L ( L + 4) J a . (9.7)The definition of the matrix coordinates (9.7) guarantees that the matrix equationof a 4-sphere (9.5) is fulfilled for matrices of dimension d L . In the limit L −→ ∞ the matrix coordinates in eq. (9.7) commute. In this limit we recover the algebraof function of continuous S , C ∞ ( S ) .However, a substantial difference to the fuzzy 2-sphere case is that for finite L the matrix coordinates X a do not provide a complete basis for the algebra offunctions, i.e. X a cannot provide a basis for M at d L .To clarify this point we analyse the lowest approximation, L = 1. We define σ ab proportional to the commutators of Γ a : σ ab = 12 i [Γ a , Γ b ] . (9.8)If F is a matrix representing a function on S , it will be of the form F = F + F a Γ a . (9.9) From fuzzy C P to a fuzzy S F However, a matrix product of two functions of the type in eq. (9.9) will involvenon-zero coefficients of σ ab , the matrices in (9.8), F (cid:48) = F (cid:48) + F (cid:48) a Γ a + F (cid:48) ab σ ab . (9.10)The 10 coefficients F (cid:48) ab in eq. (9.10) have no corresponding counterparts in theexpansion of functions on commutative S . In the language of Statistical Physicsthese coefficients constitute a set of extra degrees of freedom . On one hand, if weexclude them the involved algebra is not associative. On the other hand, if weincluded them, the approximated space is not exactly S .The first option, advocated by Ramgoolam [13], is to project out such terms,in which case one is left with a non-associative algebra. This involves additionalcomplications and does not seem particularly suited to numerical work. In additionthe necessary projector must be constructed. We will return to this point in chapter11 where we will, in fact, give the projector.An alternative is to include arbitrary coefficients of σ ab (demanding an asso-ciative algebra) and attempt to suppress such coefficients of unwanted terms, bymaking their excitation improbable in the dynamics. In this approach our algebrawill be a full matrix algebra and obviously associative.Including the extra degrees of freedom will lead us to work in a bigger space.This bigger space is C P , in section 9.2 will review its construction. C P The fuzzy version of C P N − denoted by the subindex “ F ”, C P N − F , is a matrixapproximation to the continuous C P N − . In this section we review the constructionof C P F following Ref. [18].A standard definition on C P N − is the space of all norm-1 vector in C N modulothe phase. For any unit vector | ψ (cid:105) we can define a rank-one projector, P ( ψ ) := | ψ (cid:105)(cid:104) ψ | . (9.11)Then C P N − can be defined as the space of all rank-one projectors C P N − := {P ∈ M at N ; P † = P , P = P , T r P = 1 } . (9.12)
02 9.2. Review of the construction of C P To construct the set of global coordinates for C P N − F we need a set of N hermitian matrices { , t µ } , µ = 1 . . . N −
1. The set { t µ } is a basis for the Liealgebra of SU ( N ), normalised as T r ( t µ t ν ) = δ µν . (9.13)For our case of C P we start with Spin (6) ∼ = SU (4). Let {J AB } , A, B = 1 . . .
Spin (6) generators. They are equivalent to the generators t µ of SU (4). As abasis for our algebra we take the set { Λ AB } , A, B = 1 . . . AB := 1 √ J AB , (9.14)their algebra is:Λ AB Λ CD = A AB ; CD √ (cid:15) ABCDEF Λ EF + (9.15) ı √ δ AC Λ BD + δ BD Λ AC − δ BC Λ AD − δ AD Λ BC ) . (9.16) A AB ; CD is defined as a two indexes Kronecker symbol, A AB ; CD = 12 ( δ AC δ BD − δ AD δ BC ) . (9.17)The algebra given by eq. (9.16) admits representations of dimension d L = ( L +1)( L +2)( L +3)6 for L integer. For the lowest non-trivial level L = 1 we havethe explicit form of the generators in appendix E.A projector P can be expanded in terms of the basis given by eq. (9.16): P = 14 + ξ AB Λ AB , (9.18) ξ AB are fifteen real coordinates. P = P implies that ξ AB obeys the restrictions: ξ AB ξ AB = 34 , (9.19)12 √ (cid:15) ABCDEF ξ AB ξ CD = ξ EF . (9.20)Taking contractions of eqs. (9.19) and (9.20) we get the three identities:12 √ (cid:15) ABCDEF ξ AB ξ CD ξ EF = 34 , (9.21)12 √ (cid:15) ABCDEF ξ AB = 2 ξ AB ξ CD − ξ AC ξ BD + 2 ξ AD ξ BC , (9.22) ξ AC ξ CB = − δ AB . (9.23) From fuzzy C P to a fuzzy S F Eqs. (9.19)-(9.20) describe how C P F is embedded on R . The global coordinates { ξ AB } allow us to describe the geometry of C P F . Following Ref. [18] we have thegeometrical structures K ± AB ; CD = 12 ( P AB ; CD ± ıJ AB ; CD ) , (9.24) P AB ; CD = 12 A AB ; CD + √ d EFABCD ξ EF − ξ AB ξ CD , (9.25) P ⊥ AB,CD = 12 A AB ; CD − √ d EFABCD ξ EF + 2 ξ AB ξ CD , (9.26) J AB ; CD = √ f EFABCD ξ EF . (9.27) K AB ; CD is the K¨ahler structure , J AB ; CD is the complex structure and P AB ; CD isthe metric .From eq. (9.16) we get the explicitly form of the normalisation constants,eq. (9.28), and the structure constants, eq. (9.29): d ABCDEF = 14 (cid:15)
ABCDEF , (9.28) f ABCDEF = 12 (cid:0) δ AC A BD ; EF − δ AD A BC ; EF + δ BD A AC ; EF − δ BC A AD ; EF (cid:17) . (9.29)From here to the end of the thesis, we used the Kronecker’s delta to arise and lowindexes unless the opposite is indicated. A simplification for eqs. (9.25)-(9.27) isthe following: P AB ; CD = 12 A AB ; CD − ξ AC ξ BD − ξ AD ξ BC ) , (9.30) P ⊥ AB ; CD = 12 A AB ; CD + ( ξ AC ξ BD − ξ AD ξ BC ) , (9.31) J AB ; CD = 1 √ δ AC ξ BD − δ AD ξ BC + δ BD ξ AC − δ BC ξ AD ) . (9.32)
04 9.2. Review of the construction of C P Some properties of the metric P AB ; CD and the complex structure J AB ; CD . P AB ; CD = P CD ; AB = − P BA ; CD , (9.33) P ⊥ AB ; CD = P ⊥ CD ; AB = − P ⊥ BA ; CD , (9.34) J AB ; CD = − J CD ; AB = − J BA ; CD , (9.35) P AB ; CD := P AB ; EF P EF ; CD = P AB ; CD , (9.36) (cid:16) P ⊥ AB ; CD (cid:17) := P ⊥ AB ; EF P ⊥ EF ; CD = P AB ; CD , (9.37) J AB ; CD := J AB ; EF J EF ; CD = − P AB ; CD , (9.38) J AB ; EF P EF ; CD = P AB ; EF J EF ; CD = J AB ; CD . (9.39) P AB ; CD , P ⊥ AB ; CD and K AB ; CD are projectors, their ranks are given in eqs. (9.40)-(9.42): P AB ; AB = 6 , (9.40) P ⊥ AB ; AB = 9 , (9.41) K ± AB ; AB = 3 . (9.42) P AB ; CD projects on to the tangent space of C P and P ⊥ AB ; AB onto the orthogonalcompliment in R . C P as orbit under Spin (6) . Now we want to perform an explicit construction for C P F , we will analyse the induced line element . C P F can be obtained taking one fiducial projector P androtating it with and element of Spin (6): P ( ψ ) = U ( ψ ) P U − ( ψ ) , U ( ψ ) ∈ Spin (6) . (9.43)We choose P as P = 14 + ξ AB Λ AB = 14 + 1 √ + Λ + Λ ) . (9.44) From fuzzy C P to a fuzzy S F Now we are interested on calculating the line element, ds , at the north pole ,defined by eq. (9.44). ds := (cid:88) AB dξ AB . (9.45) Spin (6) rotates Λ AB as a tensor, ξ AB = R AC R BD ξ CD . (9.46)The sum dξ AB in eq. (9.45) can be written in terms of an infinitesimal rotation R − dR : (cid:88) A,B dξ AB = − T r (cid:2) R − dR, ξ (cid:3) , (9.47)where ξ is the matrix of coefficients ξ AB and R − dR := − e AB L AB . (9.48)Eq. (9.48) is known as the Maurer-Cartan forms for the group of rotations on6 dimensions. L AB are the generators of this representation.The line element turns out to be: ds = 12 (cid:2) ( e − e ) + ( e + e ) + ( e − e ) + ( e + e ) +( e − e ) + ( e + e ) (cid:3) . (9.49)From eq. (9.49) we corroborate that C P is a 6-dimensional space. Tangent forms
We define the tangent forms as e || AB = P AB ; CD e CD . (9.50)At the north pole in eq. (9.44) we have the non-vanishing coordinates are ξ = √ , ξ = √ , ξ = √ and permutation of them. The tangent forms at thenorth pole turn out to be:
06 9.2. Review of the construction of C P e || = 0 , (9.51) e || = 12 ( e − e ) = − e || , (9.52) e || = 12 ( e + e ) = e || , (9.53) e || = 12 ( e − e ) = − e || , (9.54) e || = 12 ( e + e ) = e || , (9.55) e || = 0 , (9.56) e || = 12 ( e − e ) = − e || , (9.57) e || = 12 ( e + e ) = e || , (9.58) e || = 0 . (9.59)We note from eq. (9.49) ds = (cid:88) A,B (cid:16) e || AB (cid:17) . (9.60)Now we define the (anti)-holomorphic forms e ± AB = K ± AB,CD e CD e ± AB = 14 ( e AB + (cid:15) AC e CD (cid:15) DB ± ı (cid:15) AC e CB − e AC (cid:15) CB ))= 12 e || AB ± ı √ ξ AC e CB − ξ BC e CA ) (9.61) From fuzzy C P to a fuzzy S F At the north pole: e +12 = 0 (9.62) e +13 = 14 ( e − e + i ( e + e )) , (9.63) e +14 = 14 ( e + e + i ( e − e )) = − ıe +13 , (9.64) e +15 = 14 ( e − e + i ( e + e )) , (9.65) e +16 = 14 ( e + e + i ( e − e )) = − ıe +15 , (9.66) e +23 = 14 ( e + e + i ( e − e )) = − ıe +13 , (9.67) e +24 = 14 ( e − e − i ( e + e )) = − e +13 , (9.68) e +25 = 14 ( e + e + i ( e − e )) = − ıe +15 , (9.69) e +26 = 14 ( e − e − i ( e + e )) = − e +15 , (9.70) e +34 = 0 , (9.71) e +35 = 14 ( e − e + i ( e + e )) , (9.72) e +36 = 14 ( e + e + i ( e − e )) = − ıe +35 , (9.73) e +45 = 14 ( e + e + i ( e − e )) = − ıe +35 , (9.74) e +46 = 14 ( e − e − i ( e + e )) = − e +35 , (9.75) e +56 = 0 . (9.76)From eqs. (9.62)-(9.76) we note that only 3 holomorphic forms are independent: e +13 = 12 (cid:16) e || + ie || (cid:17) = ie +14 , (9.77) e +15 = 12 (cid:16) e || + ie || (cid:17) = ie +16 , (9.78) e +35 = 12 (cid:16) e || + ie || (cid:17) = ie +36 . (9.79)The line element can also be written as ds = e + AB e − AB , (9.80)where e − AB = ( e + AB ) ∗ . At the north pole ds = 2 (cid:88) A
08 9.2. Review of the construction of C P Eq. (9.81) reduces exactly to eq. (9.60). C P as orbit under Spin (5) . Note that Λ ab ∼ J ab , a, b = 1 . . . Spin (5) subalgebra of
Spin (6)while Λ a = √ J a transforms as a vector under Spin (5).We define Λ a = J a so we can write the projector (9.18) as: P = 14 + ξ a Λ a + ξ ab Λ ab . (9.82)The projector (9.43) takes the form: P = 14 + 12 Λ a + 1 √ + Λ ) . (9.83)Here we have an extra restriction: under Spin (5) rotations the norm of the vector ξ a is not affected. From eq. (9.83) we have (cid:80) a ξ a = .The induced line element is: ds = αdξ a + βdξ ab , (9.84)the constants α, β are arbitrary numbers until know. We will come back to thispoint at the end of this chapter. Spin (5) rotates Λ ab as a tensor: ξ ab = R ac R bd ξ cd (9.85)and it rotates Λ a as a vector: ξ a = R ab ξ b (9.86)In analogy to eq.(9.47), (cid:80) a,b dξ ab = − T r (cid:2) R − dR, ξ (cid:3) but now the trace runs overthe sub-indices a, b = 1 . . .
5. For the vector part, dξ a we have: (cid:88) a dξ a = − T r (cid:18)(cid:104) (cid:126)ξ (cid:105) t R − dRR − dR(cid:126)ξ (cid:19) , (9.87)where ξ represents the matrix of coefficients ξ ab and (cid:126)ξ the vector ξ a . Nowwe can use the Maurer-Cartan forms for the group of rotations on 4-dimension: R − dR := − e ab L ab where L ab are the generators of this representation. For moredetails of these calculations see appendix F. From fuzzy C P to a fuzzy S F Finally, the line element at the north pole is: ds = α + β (cid:2) e + e + e + e (cid:3) + β (cid:2) ( e − e ) + ( e + e ) (cid:3) . (9.88)Eq. (9.49) reduces to eq. (9.88) if we ignore the rotations e a and we choose α = 1, β = 1. Then if we choose as a particular case α = β in eq. (9.88) we recoverthe Spin (6) symmetry.From eq. (9.88) we obtain geometrical information of C P . We focus into thefollowing two points:1. Isotropy group of the orbit.2. Local form of C P .
1) The isotropy group of the orbit
To find the isotropy group of the orbit, we have to identify the rotations that donot affect the projector given by eq. (9.83). Such rotations must be given by theorthogonal forms: e ⊥ AB = P ⊥ AB ; CD e CD . (9.89) e ⊥ = e , (9.90) e ⊥ = 12 ( e + e ) = e ⊥ , (9.91) e ⊥ = 12 ( e − e ) = − e ⊥ , (9.92) e ⊥ = 12 ( e + e ) = e ⊥ , (9.93) e ⊥ = 12 ( e − e ) = − e ⊥ , (9.94) e ⊥ = e , (9.95) e ⊥ = 12 ( e + e ) = e ⊥ , (9.96) e ⊥ = 12 ( e − e ) = − e ⊥ , (9.97) e ⊥ = e . (9.98)From eqs. (9.90)-(9.98) we have to restrict to rotations on 4-dimensions. The forms e ⊥ a , a = 1 · · ·
10 9.2. Review of the construction of C P From the remaining combination of coefficients we construct the correspondinggenerators of the isotropy of the orbit, T µ , µ = 1 , , , T = 1 √ − Λ ) = 14 ( σ − σ ) , (9.99) T = − √ + Λ ) = −
14 ( σ + σ ) , (9.100) T = 1 √ − Λ ) = 14 ( σ − σ ) , (9.101) T = 1 √ + Λ ) = 14 ( σ + σ ) . (9.102)The components T µ do not affect the fiducial projector in eq. (9.83) since:[ T µ , P ] = 0 , µ = 1 , , , . (9.103) T i , i = 1 , , SU (2)-subalgebra[ T i , T j ] = ı(cid:15) ijk T k . (9.104) T ∝ P generates U (1).The orbit turns out to be: C P (cid:39) Spin (5) / [ U (1) × SU (2)] . (9.105)Note that both prescriptions, developed in sections 9.2.1 and 9.2.2, give thesame orbit, C P , since we can rotate any point of C P as Spin (6) orbit and then unrotate it using an element of
Spin (5).If we follow the construction of C P demanding a Spin (6)-symmetry we obtaina rounded version of C P . Demanding the less restrictive Spin (5)-symmetry wehave a squashed version of C P . C P is locally of the form S × S . As a second observation, in eq. (9.88) we distinguish that the line element iscomposed by two parts: • A four dimensional part: (cid:80) a =1 e a . We identify it as a S -line element.Then α + β is related to the square of the radius of the S , namely R S . Itcan be re-written in terms of the (anti)-holomorphic forms as8 β (cid:0) e +15 e − + e +35 e − (cid:1) . From fuzzy C P to a fuzzy S F • A two dimensional part: ( e + e ) + ( e − e ) or in terms of the (anti)-holomorphic forms 4 αe +13 e − Using the same procedure we used in the calculation of the isotropy group,now we look at the combination of indices e ab that appeared in eq. (9.88).We associated τ i with the corresponding combination of generators:( e + e ) −→ τ = √ + Λ ) , ( e − e ) −→ τ = √ − Λ ) .τ is obtained by demanding that τ i , i = 1 , , τ i , τ j ] = ı(cid:15) ijk τ k .Summarising the above τ = √ + Λ ) , (9.106) τ = √ − Λ ) , (9.107) τ = √ + Λ ) , (9.108)generate a sphere S .Then the constant β in eq. (9.88) is related to the square of the radius ofthe S , R S . α + β is proportional to the square of the radius of the S , R S ds = R S R (cid:2) e + e + e + e (cid:3) + R S R (cid:2) ( e − e ) + ( e + e ) (cid:3) . (9.109)In eq. (9.88) the radius of C P , R is 2. Summarising, eq. (9.109) reflects thatlocally C P ∼ = S × S .It is known that C P is a fibre bundle with S as a fibre and S as a base space[51], in terms of diagrams we have S (cid:44) → C P ↓ S The construction of C P as an orbit under Spin (5) allows us to corroboratethis fact.
12 9.2. Review of the construction of C P As we mentioned in the introduction, in order to specify the geometry in fuzzyspaces we have to give a prescription for the Laplacian. In chapter 10 we will getthe exact relation between radius of the fibre S and a penalisation parameter h appearing in the scalar theory of S . hapter 10Decoding the geometry of thesquashed C P F Ref. [39] presents a prescription for a scalar field theory on a fuzzy 4-sphere.However, this theory depends on an additional parameter h and it is defined on a larger space, C P , instead S .The motivation for this chapter is to explore the feature of fuzzy spaces, toreflect the geometrical properties, and to explain why the prescription in [39] works.This chapter is divided into two parts: in section 10.1 we present a review ofthe results in [39]. In section 10.2 we extract the geometrical information from theLaplacian via the tensor metric. -sphere In this section we will summarise the prescription for working with a scalar fieldon S F following Ref. [39] . The action for the scalar field on a rounded C P isgiven by eq. (10.1): S [Φ] = R d L T r (cid:18) R [ J AB , Φ] † [ J AB , Φ] + V [Φ] (cid:19) , (10.1)where J AB are the Spin (6) generators defined in chapter 9.2. The constant infront of the trace, R d L represents the volume of C P F . C P is a 6-dimensional space. Since we are interested on retaining just the
Spin (5)-symmetry, we add toeq. (10.1) a SO (6) non-invariant but SO (5) invariant term given by eq. (10.2): S I [Φ] = R d L T r R (cid:18) [ J ab , Φ] † [ J ab Φ] −
12 [ J AB Φ] † [ J AB Φ] (cid:19) (10.2)so that we have the overall action for a squashed C P S [Φ] = T r (Φ∆ h Φ + V [Φ]) . (10.3)The expression for the overall Laplacian, ∆ h is∆ h · = 12 R (cid:18)
12 [ J AB , [ J AB , · ]] + h ([ J ab [ J ab , · ]] −
12 [ J AB , [ J AB , · ]]) (cid:19) . (10.4)or equivalently ∆ h = 12 R (cid:16) C SO (6)2 + h (2C SO (5)2 − C SO (6)2 ) (cid:17) (10.5)which gives a stable theory for all L if h ∈ ( − , ∞ ).This form (10.5) is an interpolation between SO (5) and SO (6) Casimirs. As aparticular case, the Laplacian is proportional to the SO (6) Casimir for h = 0 andthe SO (5) Casimir for h = 1.The probability of any given matrix configuration has the form: P [Φ] = e − S [Φ] − hS I [Φ] Z (10.6)where Z = (cid:90) d [Φ]e − S [Φ] − hS I [Φ] (10.7)is the partition function of the model.The values of h of interest to us are those large and positive since in thequantisation of the theory, following Euclidean functional integral methods, thestates unrelated to S then become highly improbable.Note that we have not specified the potential of the model since the aboveprescription is independent of the potential. The most obvious model to considerwould be a quartic potential, since this is relevant to the Higgs sector of thestandard model. Decoding the geometry of the squashed C P F The non-commutative product of matrices induces a non-commutative product onfunctions, this is the (cid:63) -product . This useful tool allows us to access the continuumlimit. Let (cid:99) M , (cid:99) M be two matrices of dimension d L , and M ( ξ ), M ( ξ ) are thecorresponding functions obtained by the mapping: M ( ξ ) := T r (cid:16) P L ( ξ ) (cid:99) M (cid:17) . (10.8) P L ( ξ ) is called the projector at L -level and it is contracted taking the L -fold tensorproduct of P defined in (9.18). P L ( ξ ) carries the coordinates ξ .The definition of the (cid:63) -product is given by( M (cid:63) M ) ( ξ ) := T r (cid:16) P L ( ξ ) (cid:99) M (cid:99) M (cid:17) . (10.9)For C P N − the (cid:63) -product can be written as a sequences of derivatives on thecoordinates ξ . For our proposes we will just follow the prescription given in Ref.[18]. Now we present the formal definitions of the set of coordinates ξ in eq. (9.18): ξ AB := T r ( P L Λ AB ) , (10.10)where Λ AB are proportional to J AB as in eq. (9.14). Note eq. (10.10) is consistentwith eq. (9.18) via eq. (9.16).A first interesting fact is that the commutator of J AB maps into the covariantderivative: L AB M ( ξ ) := T r (cid:16)(cid:104) J AB , (cid:99) M (cid:105)(cid:17) , (10.11)= ı √ J AB ; CD ∂ CD M ( ξ ) . (10.12)The quadratic Casimir operators are defined as:[ J AB , [ J AB , · ]] = C SO (6)2 · (10.13)[ J ab , [ J ab , · ]] = C SO (5)2 · (10.14)
16 10.2. Analysing the geometry encoded in the Laplacian
What we want to calculate is the image of ∆ h , eq. (10.5), under the (cid:63) -productmap. First we define the image of eqs. (10.13)-(10.13) in eqs. (10.15)-(10.16): C SO (6)2 (cid:99) M = (cid:104) J AB , [ J AB , (cid:99) M ] (cid:105) −→ C (6) M, (10.15) C SO (5)2 (cid:99) M = (cid:104) J ab , [ J ab , (cid:99) M ] (cid:105) −→ C (5) M, (10.16)we can rewrite C (6) as C (6) = − κ (10.17) C (5) = − κ (10.18)where κ = J AB,CD ∂ CD ( J AB,EF ∂ EF ) (10.19)= P CD ; EF ∂ CD ∂ EF + J AB ; CD ( ∂ CD J AB ; EF ) ∂ EF (10.20) κ = J ab,CD ∂ CD ( J ab,EF ∂ EF ) (10.21)A very useful expression to simplify the calculations is the contraction of thecomplex structure to the partial derivative J AB ; CD ∂ CD = √ ξ AC ∂ CB − ξ BC ∂ CA ) . (10.22)We are interested in extracting the metric tensor G comparing the relevant con-tinuous Laplacian with the general form: − L = 1 √ G ∂ µ (cid:16) √ GG µν ∂ ν (cid:17) (10.23)= G µν ∂ µ ∂ ν + ( ∂ µ G µν ) ∂ ν + 1 √ G G µν (cid:16) ∂ µ √ G (cid:17) ∂ ν . (10.24)For the case when we retain the full Spin (6)-symmetry we have the Laplacian C (6) . The associated metric tensor is just P AB ; CD as it can be verify from astraightforward calculation for κ : κ = 12 ∂ AB − ξ AC ξ BD ∂ AB ∂ CD − ξ AB ∂ AB . (10.25)For the Spin (5)-symmetry case the corresponding image to the Laplacianeq. (10.4) is defined proportional to L h Decoding the geometry of the squashed C P F T r (cid:16) ∆ h (cid:99) M (cid:17) := 12 R L h (10.26)then L h = κ + h (2 κ − κ ) . (10.27)Our guess for the tensor metric related to L h can be decomposed into a pure Spin (6)-symmetry part (given by P AB ; CD ) plus a Spin (5) invariant part denotedby X AB ; CD as in eq. (10.28): G AB ; CD = P AB ; CD + hX AB ; CD . (10.28) X is related to (2 κ − κ ), the term that breaks the Spin (6) symmetry.After a straightforward calculation we get:2 κ − κ = 12 ∂ ab − ξ ab ξ cd ∂ ac ∂ bd − ξ a ξ b ∂ ac ∂ bc − ξ ab ∂ ab (10.29)The tensor X AB,CD is obtained comparing the terms with second derivatives ineq. (10.29) to G AB ; CD ∂ AB ∂ CD X ab ; cd = 12 A ab ; cd − (cid:16) ξ ac ξ bd − ξ ad ξ bc (cid:17) − (cid:16) δ ac ξ b ξ d − δ ad ξ b ξ c + δ bd ξ a ξ c − δ bc ξ a ξ d (cid:17) (10.30) X a cd = 0 = X ab ; c = X a c (10.31)We can rewrite X as: X ab ; cd := P ab ; cd − M ab ; cd , (10.32)with M ab ; cd = 4 (cid:16) δ ac ξ b ξ d − δ ad ξ b ξ c + δ bd ξ a ξ c − δ bc ξ a ξ d (cid:17) . (10.33)Some properties of M ab ; cd and X ab ; cd are: M ab ; cd = M cd ; ab = − M ba ; cd , (10.34) X ab ; cd = X cd ; ab = − X ba ; cd , (10.35) M = M, (10.36) X = X. (10.37)
18 10.2. Analysing the geometry encoded in the Laplacian
Now the traces amount to M ab ; ab = 8 ( δ aa − ξ b ξ b = 4 . (10.38) X ab ; ab = 12 A ab ; ab + 2 ξ ab ξ ba − δ aa − ξ b ξ b = 5 + 2 ξ ab ξ ba − ξ b ξ b = 2 . (10.39)Notice X ab ; cd is a rank-2 projector while M ab ; cd is a rank-4 projector.In order to invert the tensor metric we calculate the products between projec-tors P AB ; CD , X ab ; cd and M ab ; cd : P ab ; ef P ef,cd = P ab ; ef − M ef,cd , (10.40) P ab ; EF X EF,cd = P ab ; ef X ef,cd = X ab ; cd . (10.41) P a EF X EF,cd = P a ef X ef,cd = 0 , (10.42) P ab ; EF M EF,cd = P ab ; ef M ef,cd = 12 M ab ; cd . (10.43) P a EF M EF,cd = P a ef M ef,cd = P a cd , (10.44) X ab ; ef M ef,cd = 0 . (10.45)The metric fulfils: G AB ; CD G CD ; EF = P AB ; EF . (10.46)Now we assume the covariant tensor metric to take the form G AB ; CD = P AB ; CD + αX AB,CD , (10.47)where α is a constant to determine.From eq. (10.46) we distinguish two cases: G ab ; CD G CD ; ef = = ( P + hX ) ab ; CD ( P + αX ) CD ; ef = P ab ; ef + ( αh + α + h ) X ab,ef (10.48)Comparing eq. (10.48) to eq. (10.46) we get α = h h . ⇒ G ab ; cd = P ab ; cd − h1 + h X ab , cd . (10.49) We have to demand h (cid:54) = − G AB ; CD . Decoding the geometry of the squashed C P F G a CD G CD ; ef = ( P + hX ) a CD ( P + αX ) CD ; ef = P a ef . (10.50)From eq. (10.50) we observe that for this case G CD ; ef does not depend on h . ⇒ G a6 ; cd = P a6 ; cd (10.51)Finally we summarise the results in eqs.(10.49)-(10.51) in eq. (10.52): G AB ; CD = P AB ; CD − hh + 1 X AB,CD . (10.52) Observations
Eq. (10.45) suggested that X ab ; cd and M ab ; cd project onto separated spaces. As wedemonstrate in section 9.2.2, C P is locally of the form S × S . X ab ; cd projects onto a 2-dimensional space that should be the fibre S while M ab ; cd projects onto a 4-dimensional that should be the base S . This informationcan be precisely obtained analysing the contributions to line element ds due tothe tensors X ab ; cd and M ab ; cd . This will be achieved in the subsequent section10.2.2.Let us define Ω, the symplectic structure as:Ω AB,EF = G AB,CD J CDEF (10.53)It is clear when h = 0 (full Spin (6)-symmetry case) Ω
AB,CD = J AB ; CD . In generalwe have Ω ab,c = P ab,c , (10.54)Ω ab ; cd = (1 + h ) P ab ; cd . (10.55) ds . The line element, ds at the north pole is defined as follows: ds := G AB ; CD e AB e CD , (10.56)
20 10.2. Analysing the geometry encoded in the Laplacian where e AB are defined as the coefficients of infinitesimal rotations under Spin (6),i.e. the Maurer-Cartan forms, R − dR = − e AB L AB , where L AB are generators forrotations in 5 dimensions.Note that in the case h = 0 (full Spin (6)-symmetry) eq. (10.56) reduces to ds = P AB ; CD e AB e CD (10.57)as it was expected.At the north pole we have M ab ; cd e ab e cd = 2( e c ) , (10.58) X ab ; cd e ab e cd = (cid:0) e − e (cid:1) + (cid:0) e + e (cid:1) . (10.59)From eqs. (10.58)-(10.59) we verify that M ab ; cd projects to the base space S and M ab ; cd projects to the fibre S . The line element at the north pole dependingon the parameter h is ds | northpole = (cid:0) e + e + e + e (cid:1) + 11 + h (cid:104) ( e − e ) + ( e + e ) (cid:105) . (10.60)We can compared eq. (10.60) to eq. (9.88). We identify the radius of S and S as R S R = 11 + h , (10.61) R S R = 1 . (10.62)As we saw in section 10.1, h takes values in the interval ( − , ∞ ). We analysesome particular cases of eq. (10.60):1. h = 0. We recover the Spin (6) symmetry. The radius of the rounded C P is R = 2.2. h −→ ∞ . In this limit R S R −→ h → ∞ corresponds to shrinking the S fibres to zero size.The radius of S remains finite: R S R = 1 . Decoding the geometry of the squashed C P F h −→ −
1. Note that h cannot take the exact value 1 since for h = 1 thetensor metric G AB ; CD in eq. (10.28) cannot be inverted. For this limit wehave R S R −→ ∞ , R S R = 1 . This limit corresponds to making the size of the fibre infinitely large. hapter 11Conclusions from part II • We reviewed the prescription given in Ref. [39] of the scalar field theory ona fuzzy • Since S is not a phase space the construction of its fuzzy version has somecomplications. The construction starts defining the matrix coordinates X a as X a = R (cid:112) L ( L + 4) J a , where J a belong to the ( L , L ) representation of Spin(5). They are matricesof dimension d L = ( L +1)( L +2)( L +3)6 with L an integer number. From J a J a = L ( L + 4) follows an equation which holds for the matrix coordinates of a4-sphere (cid:88) a =1 X a X a = R . In the limit L −→ ∞ the matrix coordinates commute, then we have amatrix approximation to S at algebraic level. • The complications emerge for a finite L where the five matrix coordinates X a , a = 1 , . . . , S , i.e. extra degrees of freedom . Then the constructed space is not S F , it turns out to be C P F . To recovera scalar theory on S F . We implemented a penalisation method for allthe non- S F modes in C P F .2. A second alternative consists of projecting out the non- S modes. Thisleads us to deal with a non-associative algebra. We found that followingthe previous alternative we got the projector of the non- S modesrequired in this second option. • The penalisation method introduced in the previous point consists of thefollowing: – We start defining an initial action on C P F , S [Φ].Then we modify S [Φ] by adding a term S I [Φ]. S I [Φ] gives a positivevalue for those field configurations associated to the non- S modes andit is zero for those of S . – Then we construct the overall action as S [Φ] = S [Φ] + hS I [Φ] (11.1)where h is a penalisation parameter in the interval ( − , ∞ ). – The probability of any given matrix configuration has the form: P [Φ] = e − S [Φ] − hS I [Φ] Z (11.2)where Z = (cid:82) d [Φ]e − S [Φ] − hS I [Φ] is the partition function. For h −→ ∞ the states unrelated to S become highly improbable. • The resulting action is S [Φ] = T r (Φ∆ h Φ + V [Φ]) , (11.3)where ∆ h = 12 R (cid:16) C SO (6)2 + h (2C SO (5)2 − C SO (6)2 ) (cid:17) (11.4)is written in terms of quadratic Casimir operator of the groups SO (5) and SO (6), C SO (5)2 and C SO (6)2 respectively. R is the square of the radius of C P . Conclusions from part II • Since the defined action S [Φ] for h large and positive contains only S modes,the resulting model describes a scalar field theory for S F . • The fuzzy spaces can well retain the geometrical properties of the discretisedspace. This nice feature allowed us to provide a exact geometrical interpreta-tion why the penalisation procedure described above works. The Laplacianis given in eq. (11.4). • C P can be constructed either as a Spin (6) orbit or as a
Spin (5) orbit. TheLaplacian in eq. (11.4) corresponds to C P F with Spin (5) symmetry. Werestore the
Spin (6) symmetry in a particular case of eq. (11.4) with h = 0. • It is known in the literature that C P is a non-trivial fibre bundle over S with S as the fibre. The construction of C P F as Spin (5) orbit clarified thisfact. • Using coherent states techniques we extracted the covariant tensor metricin eq. (11.4). Once we inverted the tensor metric we found the line element ds in terms of the parameter h as ds = 11 + h (cid:88) a,b,c,d =1 X ab ; cd e ab e cd + 12 (cid:88) a,b,c,d =1 M ab ; cd e ab e cd , (11.5)where e a,b are the Maurer-Cartan forms for Spin (5). X ab ; cd is a rank-2projector and M ab ; cd is a rank-4 projector. • X ab ; cd projects to the fibre S and M ab ; cd projects to the base space S . Theline element in eq. (11.5) shows that locally C P = S × S . • We identified the radius of the fibre and base space as R S R = 11 + h , (11.6) R S R = 1 . (11.7) • As h −→ ∞ we have R S R −→ . The meaning is that the S fibres shrink to zero size while the radius of S remains finite. • The suppression of the non- S states in the field theory on C P correspondsto a Kaluza-Klein type reduction of C P to S . • As we mentioned in section 9.1, following our construction of C P as a Spin (5) orbit we are able to give the prescription for the projector to S modes P S = L (cid:89) n =1 n (cid:89) m =1 C SO (5)2 − λ n,m C SO (6)2 − λ n,m = + L (cid:89) n =1 n (cid:89) m =1 C I C SO (6)2 − λ n,m . (11.8) hapter 12General conclusions andperspectives In this thesis we have presented a study of scalar field theory on fuzzy spaces.The main motivation of our work was to explore the fuzzy approach as a non-perturbative regularisation method of Quantum Field Theories. For this pur-pose we chose a hermitian scalar field theory in a three dimensional space, with λφ potential. This model is perturbatively super-renormalisable (i.e. there areonly a finite number of perturbative diagrams that require renormalisation). Ournon-perturbative regularisation consisted of a fuzzy two sphere for space anda lattice for Euclidean time. We performed Monte Carlo simulations and ob-tained the phase diagram of the model. The study of such model via a stan-dard lattice regularisation leads to a phase diagram consists of a disordered anda uniformly ordered phase separated by a continuous second order phase tran-sition that is governed by the Ising universality class. In contrast to the stan-dard lattice regularisation [9], in this new model we found three phases, twoare the disordered and uniformly ordered phases but they are separated by athird new phase of non-uniform ordering. This third phase is a property of thenon-commutativity of the regularised model and has arisen in other studies inthe literature and has variously been called a striped phase (Gubser and Shondi[26], Ambjørn and Catterall [28], Bietenholz et al. [35]-[37]) or a matrix phase byMartin [30]. We find that the three phases meet at a triple point characterisedby (cid:0) ¯ λ T , ¯ m T (cid:1) = (cid:0) (41 . ± N − . ± . ¯ R − . ± . , − (12 . ±
1) ¯ R − . ± . (cid:1) , see section 7, which inevitably runs to the origin as the matrix size is sent to infinity.The implication is that the model we have studied in the end does not capturethe non-perturbative physics of the flat-space λφ field theory. This was not un-expected since perturbative studies [29] found that this fuzzy sphere model suffersfrom Ultraviolet-Infrared (UV/IR) mixing, and the non-uniformly ordered phaseis a non-perturbative manifestation of this phenomenon in the neighbourhood ofthe ordering transition. It is conjectured (but still has not been demonstratedin numerical studies) that the introduction of another term into the action willsuppress UV/IR mixing and bring the model into the Ising Universality class. Wehave not pursued this issue in this thesis as the introduction of this additionalterm involves an additional parameter in the phase diagram and it was necessaryto proceed in steps. First it was essential to understand the phase structure of thisthree parameter model before studying how the phase diagram is deformed by theadditional parameters.The new model is naturally a non-commutative model and is of interest alsoas it is a non-perturbative regularization of a non-commutative field theory, thatof a scalar field theory on the Moyal-Groenewold plane. The new phase seems tobe a characteristic feature of such non-commutative theories. On the fuzzy sphereit is characterise by the dominance of non zero angular momenta (i.e. l >
0) inthe ground state of the model and implies that even though rotational invarianceis preserved in the regularisation process it is spontaneously broken by the groundstate.An advantage of the study is also that different limiting spaces (e.g. the com-mutative sphere, the Moyal plane and the commutative flat space) can be obtainedscaling R as a power on N and taking N −→ ∞ . Our analysis shows that in allthe limits we have considered so far the phase of non-uniform ordering survives.In particular (as mentioned above) the commutative flat model is not in the Isinguniversality class. However, we expect that a normal ordering in the vertex, i.e. theintroduction of a counter-term to cancel the tadpole diagrams, would return themodel to the Ising universality class. This modification in the action is equivalentto reinforcing the fuzzy kinetic term and can be achieved more simply adding aterm Φ( L ) Φ / (Λ R ) where Λ is a momentum cutoff to the action. An analogousprescription of adding higher derivative terms to the quadratic terms should beapplied to all fuzzy models, in such a manner that all diagrams are rendered finite, General conclusions and perspectives if the commutative theory is required in the infinite matrix limit.This three dimensional model is amenable to an alternative treatment as aHamiltonian on the fuzzy sphere where the Hamiltonian would be H = 4 πR N Tr (cid:18)
12 Π + 12 R Φ ˆ L Φ + 12 m Φ + λ (cid:19) with Π = ˙Φ.At strong coupling λ it is expected that the model effectively depends onless parameters. We found that our simulations in that regime reproduce thepredictions for the chain of matrices [47] where the contribution of the fuzzy kineticterm is neglected. This model is interesting from the String Theory point of view,it has been studied in Refs. [64]- [65] where it is known as c = 1 model .At the technical level, the simulations of scalar theories on fuzzy spaces areslower than in their lattice counterparts since the fuzzy models are intrinsically non-local . Nevertheless we have found that the simulations quickly converge asthe matrix size is increased and we can capture the characteristic behaviour of theobservables at very small matrix size. It is, however, expected that true advantagesof the fuzzy approach only emerges with the simulations involve a fermionic sectoror the models of interest are supersymmetric.A second, theoretical aspect of the thesis was the presentation of four dimen-sional models on a round fuzzy approximation to S , via a Kaluza-Klein reductionof C P . Though we did not perform numerical studies in this case, from the stud-ies that we have done so far, we are in a position to conjecture the structure ofthe phase diagram for the C P and S models. Since, the disordered-non-uniformordered transition line seems to be universal in the class of models where space-time is modelled by a fuzzy space, we expect it to arise in theses models also. Itcorresponds to the dominance of the potential term and the models become purepotential one matrix models (with, in our case, a φ potential). The large N limitof these models are solved in terms of the density of eigenvalues. This densityundergoes a transition from a connected to a disconnected density as the potentialwell is deepened by making the mass parameter, m , more negative. It is thisseparation of the eigenvalue spectrum that occurs at the disordered-non-uniformorder transition line. In the pure potential model (see eq. (6.50)) the transitioncurve is given by b c = − √ N c where b is the total coefficient of the quadratic po-tential term and c that of the quartic term. This disordered phase should give rise to a uniform ordered phase as the mass parameter is furthered decreased. Againall these models should exhibit UV/IR mixing. The study of S adds further com-plications due to the need to squash C P . A further complication is the need tointroduce the additional UV/IR suppressing term. However the principal conclu-sion of this thesis is that we see no insurmountable difficulty to the implementationof the scheme implemented in this thesis and the extensions outlined above as aregularisation scheme for quantum field theories. It has the distinct advantagethat it is also a natural regular method for non-commutative field theories. Thisconstitutes an interesting numerical experiment for the Kaluza-Klein reductionsin the fuzzy context.Although there is still a lot to do to reach the high acceptance status of LatticesField Theories, to the date the fuzzy approach have overcome the very first tests.We may take the present work as a starting point to continue the exploration ofthis possible alternative in future studies on fuzzy spaces. ppendix AA small description of theMonte Carlo method Let Φ be a configuration of the relevant field. The probability for to this configu-ration is given by P [Φ] = e − S [Φ] Z , (A.1)where S [Φ] is the Euclidean action of the system in the configuration Φ. Z iscalled the partition function , Z = (cid:90) D [Φ] e − S [Φ] , (A.2)where (cid:82) D [Φ] denotes the integration over all field configurations. The expectationvalue of the observable O is define by the expression: (cid:104) O (cid:105) = (cid:90) D [Φ] e − S [Φ] Z O [Φ] . (A.3)The idea of the Monte Carlo method is to produce a sequence of configurations { Φ i } , i = 1 , , ..., T MC and evaluate the average of the observables over that setof configurations. In this way the expectation value is approximated as (cid:104) O (cid:105) (cid:39) T MC T MC (cid:88) i =1 O i , (A.4)where O i is the value of the observable O evaluated in the i -sampled configuration,Φ i , i.e. O i = O [Φ i ]. T MC : Monte Carlo time The sequence of configurations obtained by Monte Carlo have to be represen-tative of the configuration space at the given parameters.
A.1 The Metropolis algorithm
The concrete way that the Metropolis algorithm works is the following: • Start with a configuration Φ init = Φ and generate another configurationΦ test . • Compute ∆ S := S [Φ test ] − S [Φ ]. – If ∆
S < test is accepted, i.e:Φ = Φ test . – Otherwise the number e − ∆ S is compared to a random number, ran ∈ [0 , e − ∆ S > ran the configuration Φ test is accepted, in the othercase it is rejected, i.e. Φ = Φ . • Set Φ init = Φ and compare it with another configuration Φ test .Before we measure it is necessary to perform a number of steps, in order to obtainstable values for the observables of interest. This is called thermalisation time . Inour study the thermalisation time was estimated from the history of the action . The standard way to propose the following configuration
The variation of the configuration Φ( t ) is performed element by element,Φ( t ) ij −→ Φ (cid:48) ( t ) ij = Φ( t ) ij + a ij (A.5)where a ij ∈ C is a random number. Its real and imaginary part are in the interval (cid:20) − N (cid:113) | m λ | , N (cid:113) | m λ | (cid:21) and the a ij are chosen so that we preserve hermiticity ofthe field, i.e. The picture obtained plotting the Monte Carlo step i vs. the action at the configu-ration Φ i , S [Φ i ]. In general the real and imaginary part are different random numbers.
A small description of the Monte Carlo method Φ (cid:48) ( t ) † = Φ (cid:48) ( t ).One Monte Carlo step correspond to updating all entries of each Φ( t ), t =1 , . . . , N t sequentially once. A.1.1 Modifying the Metropolis algorithm
The standard Metropolis algorithm works well when the observable fluctuatesaround one value. In our study we found that the value of the observable fluctuatesaround several values. This was interpreted in the sense that the effective poten-tial of the system has several local minima. It was also observed that for certainvalues of the parameters there is no tunnelling between the different minima for avery long history. Then the result depends on which minimum the system falls in,and this typically strongly depends on the starting conditions.
Independent simulations
As a first try to handle this dependence on the starting conditions we performmany independent simulations. The algorithm described in A.1 changes as follow: • Divide the T MC -Metropolis steps into sim parts. • Choose a starting configuration and thermalise . • Perform the loop describe in A.1 T MC /sim times. • Repeat the previous steps sim times until you collect T MC -configurations.This method is useful to check if the results do not depend on the startingconditions, but in some cases this method fails because the expectation values ofthe observables depend on the way that the sim different initial conditions arechosen.Let us characterise each minimum by the expectation value of the energy.In figure 5.17 of chapter 5 we showed that there is a clear difference betweenresults with cold start vs. hot start . Between simulations with a hot start notunneling is observed, as figure A.1 illustrates. We call it multilevel behaviour , some examples of it appears in appendix C.
34 A.1. The Metropolis algorithm
Monte Carlo steps a c t i o n / V o l. Figure A.1: History of the action for 6 different hot starting conditions at¯ m = −
24, ¯ λ = 1 .
25, ¯ R = 8, N = 12. We take a new start every 500 , In figure A.1 we observe the history of the energy for 6 independent simulations.We observe 4 different mean values in the sense of (A.4). This indicates that thespace of configurations is divided into separated subspaces since we do not observetunnelling in the same simulation.According to the mean value of the energy, in the second and third simulationthe system falls in the same minimum, characterised by the central value (cid:104) S (cid:105) ≈− (cid:104) S (cid:105) ≈ − . A.1.2 An adaptive method for independent simula-tions
A second attempt to sample the configurations takes into account the size of theminima . To do this we stick the independent simulations in the following way: • Divide the T MC -Metropolis steps into sim parts A small description of the Monte Carlo method • Choose a starting configuration and then do the loop described in sectionA.1 for T MC /sim times. Keep the last configuration, this will be the Φ init for the next loop. • Choose a new starting configuration and thermalize to get Φ test , and test itas in A.1 with Φ init obtained from the previous independent simulation.
Figure A.2 shows the Specific Heat for N = 12, ¯ λ = 22 /
12, ¯ R = 4 obtainedby the three different methods of measurement discussed in this appendix. Adaptive MetropolisIndependent simulationsStandard Metropolis m Sp ec i fi c H e a t -0.6-0.8-1-1.2-1.4-1.6-1.8-2-2.214000120001000080006000400020000 Figure A.2: The Specific Heat measured by the different methods at ¯ λ = 1 . R = 4, N = 12. As we can observe, different methods give different results and this is because thevalue of the observables strongly depends on the way of measuring. As an exampleof the statement above we present in figure A.3 the histogram for the action at¯ m = − . figure A.2 :
36 A.1. The Metropolis algorithm
Adaptive MetropolisIndependent simulations
Energy
Figure A.3: Histograms for the action at ¯ m = − .
7, ¯ λ = 1 .
83, ¯ R = 4, N = 12. Figure A.4 shows the histogram for Tr(Φ) with the same setting as in figureA.3 . Adaptive MetropolisIndependent simulations
T r (Φ) 10.80.60.40.20-0.2-0.4-0.6-0.8-1121086420
Figure A.4: Histograms for
T r (Φ) at ¯ m = − .
7, ¯ λ = 1 .
83, ¯ R = 4, N = 12. The thermalisation problems reflect the fact that there exist a large potentialbarrier between the different subspaces. As consequence we cannot sweep all thespace of configurations, therefore the expectation values of the observables are notreliable.
A small description of the Monte Carlo method
A.2 Methods to estimate the error
In this section we give a brief explanation of the methods we used to estimate thestatistical errors in our simulations.We start by defining the mean ¯ O over a subset of { O i } T MC i =1 of n elements( n ≤ T MC ) ¯ O = 1 n n (cid:88) i =1 O i . (A.6)The standard deviation is defined as σ = (cid:115) n (cid:80) ni =1 (cid:0) O i − ¯ O (cid:1) n − (cid:114) n − (cid:16) O − ( ¯ O ) (cid:17) . (A.7)If the samples are statistically independent eq. (A.7) gives a good estimation ofthe error. If this is not fulfilled, then the correct expression for σ is (see Ref. [52]): σ = (cid:115) τ ∆ n n − (cid:16) O − ( ¯ O ) (cid:17) (A.8)where τ is the autocorrelation time and ∆ n is the Monte Carlo time interval atwhich the samples were taken. It is related to the total number of samples by theexpression n = T MC ∆ n .For large n and 2 τ (cid:29) ∆ n , eq. (A.8) turns into eq. (A.9): σ = (cid:114) τT MC (cid:16) O − ( ¯ O ) (cid:17) . (A.9)Now the problem is to estimate the autocorrelation time. The formal expres-sion for the autocorrelation time (see Ref. [53]) is τ = 12 + n (cid:88) k =1 A ( k ) (cid:18) − kn (cid:19) , (A.10)with A ( k ) = n (cid:88) i (cid:54) = j (cid:104) O i O i + k (cid:105) − (cid:104) O i (cid:105)(cid:104) O j (cid:105)(cid:104) O i (cid:105) − (cid:104) O i (cid:105)(cid:104) O j (cid:105) . (A.11)
38 A.2. Methods to estimate the error
A.2.1 Binning method
This method is also called blocking method. The idea behind this method isto divide the vector of measurements { O i } T MC i =1 into n b blocks (also called binnumber), then we evaluate the observable for each block to obtain a new vectorof measurements { ˜ O j } n b j =1 . The error is estimated as if these new measurementswere statistically independent, then it is obtained via eq. (A.7) replacing n by n b .The principal disadvantage of this method is that the error may strongly dependson the choice of n b . Then one should test several values of n b and keep the onewhere the error is maximal. A.2.2 Jackknife method
This procedure can be consider a re-sampling method. It also starts by dividing thevector of measurements into n b blocks. Then one forms N B large blocks cointainingall data but one of the previous binning blocks O jackknife j = T MC · ¯ O − k · ˜ O j T MC − j , j = 1 , ..., N B . (A.12)where ˜ O j is the average in the j block. k is the number of samples in each block.The error is then calculated as follows: σ = (cid:118)(cid:117)(cid:117)(cid:116) N B − N B N B (cid:88) j =1 (cid:16) O jackknife j − ¯ O (cid:17) . (A.13) A.2.3 Sokal-Madras method
This method is based on the estimation of the autocorrelation time given byeq. (A.10), see [54]. The error is given by eq. (A.9). In the case that the au-tocorrelation time turns out to be below 0 . ϕ all , ϕ , χ and E – eqs. (4.27), (4.30), (4.32) and (4.34)respectively – the errors given by the three methods are compatible.For the Specific Heat (or the susceptibilities of the different modes) the mostcareful estimate was given by the Sokal-Madras method (largest errors). A small description of the Monte Carlo method
A.3 Technical notes
The runs were performed on two different clusters: • Berlin cluster : – – – • Dublin cluster – . – . – C + + and uses the Message Passing Interface (MPI). ppendix BPolarisation tensors for S U (2)
In this section we want to describe some generalities of the polarisation tensorsˆ Y l,m , which form a basis for the matrices Φ in eq. (4.13). They are the eigenvectorsof the operator defined in eq. (3.9) –the fuzzy version of the angular momentumoperator L ˆ L · = (cid:88) i =1 [ L i , [ L i , · ]] (B.1)where L i ∈ M at L +1 . So we have the set { ˆ Y lm } l ≤ L,m ≤ l such thatˆ L ˆ Y lm = l ( l + 1) ˆ Y lm , (B.2)ˆ Y † l,m = ( − m ˆ Y l, − m . (B.3)Following Ref. [55], their algebra isˆ Y l m ˆ Y l m = (cid:114) L + 14 π (cid:88) l (cid:48) ,m (cid:48) ( − L + l (cid:48) (cid:112) (2 l + 1)(2 l + 1) × (cid:40) l l l (cid:48) L/ L/ L/ (cid:41) C l (cid:48) m (cid:48) l m l m ˆ Y l (cid:48) m (cid:48) (B.4) (cid:40) l l l (cid:48) L/ L/ L/ (cid:41) are the Wigner 3mj-symbols –see Ref.[56]-[57] – and C l (cid:48) m (cid:48) l m l m are the Clebsch-Gordan coefficients. SU (2) IRR of dimension d L Their normalisation is chosen as:4 πL + 1 Tr (cid:16) ˆ Y † lm ˆ Y l (cid:48) m (cid:48) (cid:17) = δ ll (cid:48) δ mm (cid:48) . (B.5) B.1 Explicit form of the generators of SU (2) IRR of dimension d L As we mentioned in section 3.2, L i are the generators of the SU (2) irreduciblerepresentations (IRR) of dimension N = L +1. They satisfy the Casimir constraint: C SU (2) := (cid:88) i =1 L i L i = 14 ( N − · . (B.6)There is a well established procedure to construct these generators in anarbitrary representation. It operates in the Cartan basis where we work with L + := L + ıL , L − := L − ıL and L z := L .[ L z ] ij = (cid:40) ( d L + 1 − i ) if i = j L + ] ij = (cid:40) (cid:112) i ( d L − i ) if i + 1 = j L − ] ij = (cid:40) (cid:112) j ( d L − j ) if i − j L + ) † = L − .The commutation relations read:[ L z , L + ] = L + , [ L z , L − ] = − L − , [ L + , L − ] = 2 L z . (B.10) In fact the operators defined in [55], T lm , are essentially our polarisation tensors upto a factor: T lm = (cid:114) πL + 1 ˆ Y lm Polarisation tensors for SU (2) 143 B.2 Explicit construction of the polarisationtensors ˆ Y lm can be constructed as traceless polynomials of order l on L i , for example, for l = 0, ˆ Y ∝
11. Demanding πL +1 Tr (cid:16) ˆ Y ˆ Y (cid:17) = 1, we arrive at ˆ Y = √ π
11 (alsosee Ref. [58]).For l = 1 we have ˆ Y ∝ L z , ˆ Y , ∝ L + and ˆ Y , − ∝ L − . The normalisa-tion can be found calculating the trace of the squared of L z , L + , L − . Followingour definitions (B.7)- (B.9) we found Tr (cid:16) L † z L z (cid:17) = L ( L +1)( L +2)12 , Tr (cid:16) L † + L + (cid:17) =Tr ( L − L +) = L ( L +1)( L +2)6 . Then we choose ˆ Y , +1 = e ıφ (cid:113) π √ L ( L +2) L + and so onfor the remaining m ’s. The phase e ıφ has to be fixed demanding eq. (B.3) to hold.Summarising: ˆ Y = 1 √ π l (B.11)ˆ Y , +1 = ı (cid:114) π (cid:112) L ( L + 2) L + (B.12)ˆ Y = (cid:114) π (cid:112) L ( L + 2) L z (B.13)ˆ Y , − = ı (cid:114) π (cid:112) L ( L + 2) L − (B.14) ppendix CAside results C.1 Criteria to determine the phase transi-tion
To sketch the phase diagram we compare the following two criteria: • The criteria of the susceptibilities. The first phase diagram was revealed bysearching for the values of m where the two-point function of the zero modehas its peak for a given value of λ . • Specific Heat Criterion. A second phase diagram was found searching forthe values of m where the specific heat peaks.We found that for values of ¯ λ < ¯ λ T for ¯ R fixed both criteria roughly coincide, asit can be seen in the figure C.1 for N = 16, ¯ λ = 0 .
44, ¯ R = 4 (cid:1) χ (cid:2)(cid:1) χ (cid:2) ¯ m Su s ce p t i b ili t i e s -0.26-0.28-0.3-0.32-0.34-0.36-0.380.0090.0080.0070.0060.0050.0040.0030.0020.0010 ¯ m Sp ec i fi c H e a t / V o l. -0.26-0.28-0.3-0.32-0.34-0.36-0.380.0001450.000140.0001350.000130.0001250.000120.000115 Figure C.1: Susceptibilities χ and χ and the specific heat at ¯ λ = 0 . R = 4, N = 16. 14546 C.1. Criteria to determine the phase transition We observe that the specific heat an the zero mode susceptibility χ in eq. (4.31)roughly peak at the same value of m . This can be explainrd analysing the partialcontributions to the action: E E E E E ¯ m E n e r g y / V o l. -0.26-0.28-0.3-0.32-0.34-0.36-0.380.50.40.30.20.10-0.1 Figure C.2: Partial contributions given by eqs. (4.40)-(4.43) to the internalenergy eq. (4.34) at ¯ λ = 0 .
44, ¯ R = 4, N = 16 We observe that the main contribution comes from the kinetics terms. Thekinetic fuzzy term selects the configuration where the zero mode is leading.
SusceptibilitySpecific heat¯ λ ¯ m Figure C.3: Comparison between the critical points obtained by the critera ofthe Specific Heat –eq. (4.35)– and of the zero mode susceptibility in eq. (4.31),at ¯ R = 8, N = 12. We observe that the critical points overlap within theerror bars. Aside results
C.2 Free field results
In the case λ = 0, following Ref. [29] the expression for the space correlator (4.22)for l = 0 is: (cid:104) c ∗ (0) c (0) (cid:105) := (cid:104) ϕ (cid:105) = 14 π ¯ R ¯ m . (C.1) Figure C.4 shows that our simulation results agree with this formula. π × × ¯ m (cid:1) ϕ (cid:2) ¯ m Figure C.4: (cid:104) ϕ (cid:105) for ¯ λ = 0, ¯ R = 4, N = 12. ppendix DTables Table
D.1 is organised as follows:1. For each pair ( N, ¯ R ) it gives the maximal value of ¯ λ simulated —for severalvalues of ¯ m — free of thermalisation problems around the phase transition.2. Next it indicates the critical value, ¯ m c . This is done with the purpose ofhaving a reference of the magnitude of the simulated values of ¯ m .3. Then it contains the corresponding parameters A, B, C and D defined ineqs. (7.13)-(7.15) (see chapter 7 for more details).4. Finally it present the proposed fit for the transition curve I - III of the form(6.45) (in some cases it indicates the range of ¯ λ for the proposed fit). Maximal critical Proposed fit forparameters the transition curve N = 8 ¯ R = 4 N ¯ λ = 1300 ¯ m c = − ± . m c = − . N ¯ λ ) . A = 2 π D = 100 . B c = − . C = 65345 . N = 8 ¯ R = 8 N ¯ λ = 2500 ¯ m = − . ± m c = − . N ¯ λ ) . A = 2 π D = 402 . B c = − . C = 502655 N = 8 ¯ R = 16 N ¯ λ = 5000 ¯ m = − ± . m c = − . N ¯ λ ) . A = 2 π D = 1608 . B c = − . C = 4 . × N = 12 ¯ R = 4 N ¯ λ = 1200 ¯ m = − ± . m c = − . N ¯ λ ) . A = 2 π D = 100 . B c = − . C = 60318 . N ¯ λ ≥ N = 12 ¯ R = 8 N ¯ λ = 2000 ¯ m = − . ± . m c = − . N ¯ λ ) . A = 2 π D = 402 . B c = − . C = 402124 for N ¯ λ ∈ [600 , N =12 ¯ R = 16 N ¯ λ = 1800 ¯ m = − ± . m c = − . N ¯ λ ) . A = 2 π D = 1608 . B c = − . C = 1 . × for N ¯ λ ≥ N = 12 ¯ R = 32 N ¯ λ = 1600 ¯ m = − . ± . m c = − . N ¯ λ ) . A = 2 π D = 6433 . B c = − . C = 5 . × N ¯ λ ≥ N = 12 ¯ R = 64 N ¯ λ = 60 ¯ m = − . ± .
006 ¯ m c = − . N ¯ λ ) . A = 2 π D = 25735 . B c = − . C = 772078 for N ¯ λ ≥ N = 12 ¯ R = 100 N ¯ λ = 30 ¯ m = − . ± . m c = − . N ¯ λ ) . A = 2 π D = 62831 . B c = − . C = 942478 for N ¯ λ ≥ N = 16 ¯ R = 2 N ¯ λ = 120 ¯ m = − . ± .
25 ¯ m c = − . N ¯ λ ) . A = 2 π D = 25 . B = − . C = 1507 . N = 16 ¯ R =4 N ¯ λ = 1200 ¯ m =-27 ± m c = − . N ¯ λ ) . A = 2 π D = 100 . B = − . C = 60318 . N ¯ λ ≥ N = 16 ¯ R = 8 N ¯ λ = 1600 ¯ m = − ± . m c = − . N ¯ λ ) . A = 2 π D = 402 . B = − . C = 321699 N ¯ λ ∈ [200 , N = 16 ¯ R = 16 N ¯ λ = 2000 ¯ m = − . ± . m c = − . N ¯ λ ) . A = 2 π D = 1608 . B = − . C = 1 . × N = 16 ¯ R = 32 N ¯ λ = 400 ¯ m = − . ± .
05 ¯ m c = − . N ¯ λ ) . A = 2 π D = 6433 . B = − . C = 1 . × for N ¯ λ ≥ . N = 23 ¯ R = 4 N ¯ λ = 115 ¯ m = − . ± . m c = − . N ¯ λ ) . A = 2 π D = 100 . B = − . C = 5780 .
53 for N ¯ λ ≥ N = 23 ¯ R = 8 N ¯ λ = 180 ¯ m = − . ± .
05 ¯ m c = − . N ¯ λ ) . A = 2 π D = 402 . B = − . C = 36191 . N ¯ λ ≥ N = 23 ¯ R = 16 N ¯ λ = 1500 ¯ m = − . ± .
05 ¯ m c = − . N ¯ λ ) . A = 2 π D = 1608 . B = − . C = 1 . × for N ¯ λ ≥ ppendix ERepresentations and Casimiroperators E.1 Explicit form of the generators of SO (6) in the dimensional IRR Γ = −
10 0 − − − ; Γ = ı − ı ı − ı ; Γ = − − ;Γ = ı
00 0 0 ı − ı − ı ; Γ = − − ; σ = − − ; σ = ı − ı ı − ı ; σ = −
10 0 − ; σ = − ı − ı ı ı ; SU (4) σ = ; σ = − ı ı ı − ı ; σ = −
10 0 1 00 1 0 0 − ; σ = − − ; σ = − ı
00 0 0 ıı − ı ; σ = − − − − . The generators of SO (6) in the fundamental representation are proportionalto the former Γ a , σ a,b , a, b = 1 , · · · , J a = 12 Γ a , J ab = 12 σ ab . (E.1) E.2 Gell-Mann matrices of SU (4) As SO (6) ∼ = SU (4) we can find the relations between both bases. Before we givethe matrix of transformation between the bases we introduce the explicit form ofthe Gell-Mann matrices for SU (4).The fundamental representation of SU (4) is given by the fifteen matrices { λ i } i =1 λ = ; λ = − ı ı ; λ = − ; λ = ; λ = − ı
00 0 0 0 ı ; λ = ; Representations and Casimir operators λ = − ı ı ; λ = 1 √ − ; λ = ; λ = − ı ı ; λ = ; λ = − ı ı ; λ = ; λ = − ı ı ; λ = 1 √ − . Let M be the matrix of transformation between Gell-Man matrices and SO (6)generators such that: M Γ Γ Γ Γ Γ σ σ σ σ σ σ σ σ σ σ = λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ
54 E.3. SO ( N ) Casimir operators M has the form: M = − − − − − − √ √ − √ √ − − − − − √ − √ − − − E.3 SO ( N ) Casimir operators
Following Refs. [59]-[61] the p -order Casimir operator is defined as C p ( m , m , ..., m n ) = T r ( a p E ) (E.2)( m , m , ..., m n ) denote the highest weight vector of the involved representation. a ij is a matrix associated to m i .For classical Lie groups we haveC = 2 S (E.3) C = 2 S − (2 αβ + β −
1) where (E.4) S k = n (cid:88) i =1 ( l ki − r ki ) (E.5) l i = m i + r i , α, β are given in Table E.1 .The fold symmetric representations of O (2 n + 1) are labelled by highest weightvector m = ( f, , , . . . , . Representations and Casimir operators their p -order Casimir operator is C p ( f, , ...,
0) = ( f + 2 α ) p + ( − f ) p . (E.6)The quadratic Casimir operator readsC = 2 f ( f + 2 α ) . (E.7) Table E.1: The constants α, β for the classic groups.Algebra Group α β r i A n − SU ( n ) n − n +12 − iB n O (2 n + 1) n − n + ) (cid:15) i − iC n Sp (2 n ) n − n + 1) (cid:15) i − iDn O (2 n ) n − n(cid:15) i − i where (cid:15) i = 1 for i > (cid:15) = − i < (cid:15) = 0 if i = 0. SO (5) Casimirs operators SO (5) is a rank-2 algebra. For O (5) we have r = ( r , r ) = ( , )C ( m , m ) = 2 (cid:0) m + m + 3 m + m (cid:1) = 2 S , (E.8) C ( m , m ) = 2 S −
32 C where (E.9) S = m + m + 6 m + 2 m + 272 m + 32 m + 272 m + 12 m . (E.10) Fold symmetric case representations of SO (5) For O (2 n + 1) we have α = n − , (E.11) β = 1 . (E.12)
56 E.4. Dimension of representations of SO ( N ) and SU ( N ) For O (5): C ( f,
0) = 2 f ( f + 3) , (E.13) C ( f,
0) = 2 f ( f + 3)[ f + 3 f + 3] . (E.14) SO (6) Casimir operators SO (6) is a rank-3. For SO (6) we have r = ( r , r , r ) = (2 , ,
0) .The Casimiroperator reads C ( m , m , m ) = 2 (cid:0) m + m + m + 4 m + 2 m (cid:1) . (E.15)The involved representations in chapter 9 are ( n, n, SO (6) IRR; their Casimirtakes the value C ( n, n,
0) = 4 n ( n + 3) . (E.16) E.4 Dimension of representations of SO ( N ) and SU ( N ) Following Ref. [62] we have SO (2 n ) dim ( m , m , ..., m n ) = (cid:89) i The representation is denoted by the number of boxes at each line. The represen-tation in the example in eq. (E.19) is shown in figure E.4 . n (cid:122) (cid:125)(cid:124) (cid:123)(cid:124) (cid:123)(cid:122) (cid:125) n (cid:124) (cid:123)(cid:122) (cid:125) n Figure E.1: IRR (2 n, n, n ) of SU (4). Since dim ( n, n, SO (6) = (2 n + 3)( n + 1) ( n + 2) we conclude(2 n, n, n ) IRR SU (4) ≡ ( n, n, 0) IRR SO (6) . (E.20) ppendix FCalculation of the inducedmetric of C P as S pin (5) orbit In chapter 9 we introduce the expression for the induced line element of C P as a Spin (5) orbit (see eq. (9.84): ds = αdξ a + βdξ ab , (F.1)where the constants α, β are arbitrary numbers.We start the construction of the orbit choosing a fiducial projector P P = 14 + 12 Λ a + 1 √ + Λ ) . (F.2)We can place the coordinates of the projector in eq. (F.2) in a matrix of coefficients, ξ , for ξ ab and a vector (cid:126)ξ for ξ a ξ = 12 √ − − , (cid:126)ξ = .Spin (5) rotates the coordinates of the fiducial projector as follows (cid:126)ξ a = R ab (cid:126)ξ b ,ξ ab = R ac R bd ξ cd , then dξ a = dR ab [ (cid:126)ξ ] b , dξ ab = dR ac [ ξ ] cd R − db + R ac [ ξ ] cd dR − db . ( dξ a ) For ( dξ a ) we have( dξ a ) = [ dξ a ] t [ dξ a ]= (cid:88) a =1 (cid:104) dR ab (cid:126)ξ b (cid:105) t dR ac (cid:126)ξ c = (cid:88) a =1 (cid:104) (cid:126)ξ b (cid:105) t [ dR t ] ba dR ac (cid:126)ξ c . (F.3)= T r (cid:0) [ ξ ] t [ dR ] t [ dR ][ ξ ] (cid:1) = T r (cid:0) [ ξ ] t [ dR ] t RR − [ dR ][ ξ ] (cid:1) (F.4)Using R − = R t and R − R = we have dR t R = dR − R = − R − dR . Ineq. (F.4) we obtain ( dξ a ) = − T r (cid:16) [ (cid:126)ξ ] t [ R − dR ][ R − dR ] (cid:126)ξ (cid:17) . (F.5)A rotation in 4-dimensions has the form R = e ıe ab θ ab . θ ab = ıL ab are the generatorsand e ab the coefficients of the rotations as they were defined in eq. (9.48). L ab arethe generators in the fundamental representation with the explicit form[ L ab ] ij = 12 ( δ ai δ bj − δ bi δ aj ) . (F.6)The expression for R − dR involves the Maurer-Cartan forms [ R − dR ] ij = − [ e ab L a b ] ij = − e ij . (F.7)Substituting eq. (F.7) in eq. (F.5) we obtain( dξ a ) = 14 (cid:88) a =1 ( e a ) . (F.8) dξ ab For dξ ab we have dξ ab = dξ ab dξ ab = − dξ ab dξ ba = − T r [ dξ ab ] (F.9) Calculation of the induced metric of C P as Spin (5) orbit ( dξ ab ) = − (cid:88) a =1 (cid:0) dR ac ξ cd R − dn + R ac ξ cd dR − dn (cid:1) × (cid:16) dR ne ξ ef R − fa + R ne ξ ef dR − fa (cid:17) = − T r (cid:16) dRξ R − dRξ R − + dRξ R − Rξ dR − + Rξ dR − dRξ R − + Rξ dR − Rξ dR − (cid:17) = − T r (cid:16) [ R − dR ] ξ [ R − dR ] ξ − [ R − dR ] ξ ξ [ R − dR ] (cid:17) = − T r (cid:2) R − dR, ξ (cid:3) . (F.10)Substituting eq. (F.7) in eq. (F.10) we get( dξ ab ) = − (cid:0) e ik ξ km e mn ξ ni − e ik ξ km ξ mn e ni (cid:1) . (F.11)Some intermediate steps for eq. (F.11): e ik ξ kj = 12 √ e − e e − e − e e e − e − e e − e − e e − e e − e ,e ik ξ kj e jm ξ mi = 14 (cid:0) e + e − e e + 2 e e (cid:1) . (F.12) (cid:0) ξ (cid:1) = 18 − − − − .e ik (cid:0) ξ (cid:1) kl e lj = − e e e e − e e e e − e − e e e − e − e − e e − e − e − e − e e e e − e e e − e − e e − e − e − e − e − e − e − e . e ik (cid:0) ξ (cid:1) kl e li = 14 (cid:0) e + e + e + e + e + e (cid:1) + 18 (cid:0) e + e + e + e (cid:1) . (F.13)Substituting eqs. (F.12) (F.13) in eq.(F.11) we obtain( dξ ab ) = 12 (cid:2) ( e − e ) + ( e + e ) (cid:3) + 14 (cid:0) e + e + e + e (cid:1) . (F.14)Finally we arrive at ds = α + β (cid:2) e + e + e + e (cid:3) + β (cid:2) ( e − e ) + ( e + e ) (cid:3) . ibliography [1] J. Madore, Classical Gravity on Fuzzy Space-Time , Nucl. Phys. Proc. Suppl. 56B (1997) 183-190 [ gr-qc/9611026 ].[2] J. Castelino, S. Lee and W. Taylor, Longitudinal -branes as -spheres inMatrix theory , Nucl. Phys. B 526 (1998) 334-350 [ hep-th/ On the Polarization of unstable D0-branes into non- commutative odd spheres , JHEP hep-th/ Higher Dimensional Geometries from MatrixBrane Construction , Nucl. Phys. B 627 (2002) 266-288 [ hep-th/ The exotic Galilei group and the Peierls substi-tution , Phys. Lett. B 479, 284 (2000).M. Fogle, Stripe and bubble phases in quantum Hall systems , High MagneticFields: Applications in Condensed Matter Physics and Spectroscopy, pp. 98-138, ed. by C. Berthier, L.-P. Levy, G. Martinez, Springer-Verlag, Berlin,(2002). [ cond-mat0111001 ].[6] N. Read, Lowest-Landau-level theory of the quantum Hall effect: the Fermi-liquid-like state , Phys. Lett. B 58 (1998) 16262 [ cond-mat 9804294 ].[7] B. A. Campbell, K. Kaminsky, Noncommutative Field Theory andSpontaneous Symmetry Breaking , Nucl. Phys. B 581 (2000) 240-256[ hep-th/ Magnetic backgrounds and noncommu-tative field theory , Int. J. Mod. Phys. A19:1837-1862 (2004) e-Print:physics/0401142 [8] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena , Oxford Uni-versity Press, Second edition (1993).[9] I. Montvay and G. M¨unster, Quantum Field Theory on a Lattice , CambridgeUniversity Press, (1997).[10] A. Connes, Noncommutative geometry , Academic Press, San Diego (1994).J. Madore, An introduction to Noncommutative differential Geometryand its Applications , Cambridge University Press, Cambridge, (1995)[ gr-qc/9906059 ][11] J. Fr¨ohlich and K. Gawedzki, Conformal Field Theory and the Geometryof strings Proceedings of the conference on Mathematical Quantum Theory,Vancover, (1993) [ hep-th/ The fuzzy sphere , Class. and Quant. Grav. 9, (1992), 69.H. Grosse, C. Klimˇc´ık and P. Preˇsnajder, Towards finite quantum fieldtheory in noncommutative geometry , Int. J. Mod. Phys. 35 (1996) 231[ hep-th/ On spherical harmonics for fuzzy spheres in diverse dimen-sions , Nucl. Phys. B 610 (2001) 461-488 [ hep-th/ Fuzzy C P , J.Geom.Phys. 42 (2002) 28-53 [ hep-th/ Higher dimensional geometries related to fuzzy odd-dimensional spheres , JHEP hep-th/ A Fuzzy Three Sphere and Fuzzy Tori , JHEP hep-th/ Fuzzy Complex Grassmannian Spaces and their StarProducts , Int. J. Mod. Phys. A18 (2003) 1935-1958 [ hep-th/ Fuzzycomplex projective spaces and their star-products , J.Geom.Phys. 43 (2002) 184[ hep-th/ IBLIOGRAPHY 165 [19] S. Baez, A.P. Balachandran, B. Ydri and S. Vaidya, Monopoles and solitonsin fuzzy physics , Commun. Math Phys. 208 (2000) 787 [ hep-th/ Fuzzy actions and their con-tinuum limits , Int. J. Mod. Phys. A 16, (2001) 2577 [ hep-th/ Fuzzy physics , Ph. D. thesis, Physics Department, Syracuse Univer-sity (2001) [ hep-th/ Scaling Limits of the Fuzzy Sphereat one Loop , JHEP hep-th/ Pertubative Dynamics on the Fuzzy S and R P , Phys. Lett. B512 (2001) 403-411 [ hep-th/ Matrix φ models on the fuzzysphere and their continuum limits , JHEP 03 (2002) 013 [ hep-th/ On the Origin of the UV-IR Mixing in Noncommuta-tive Matrix Geometry , Nucl. Phys. B 671 (2003) 401-431 [ hep-th/ New scaling limit for fuzzy spheres , [ hep-th/ Noncommutative Gauge theory on fuzzy Four-Sphere and MatrixModel , Nucl. Phys. B 637 (2002) 177-198 [ hep-th/ A Gauge-InvariantUV-IR Mixing and The Corresponding Phase Transition For U(1) Fields onthe Fuzzy Sphere , Nucl.Phys. B704 (2005) 111-153 [ hep-th/ Quantum effective potential for U(1) fields on S L × S L , JHEP hep-th/ Noncommutative perturba-tive dynamics , JHEP hep-th/ Phase structure of non-commutative scalar field theories , Nucl.Phys. B 605 (2001) 395 [ hep-th/ A Note on UV/IR forNoncommutative Complex Scalar Field , (2000) [ hep-th/ Stripes from (Noncommutative) Stars , Phys.Lett. B 549 (2002) 253 [ hep-lat/ 66 BIBLIOGRAPHY [29] R. Delgadillo-Blando, Teor´ıa Φ en el espacio tiempo S F × R y su l´ımitecontinuo , Master thesis, CINVESTAV (M´exico D.F., 2002).[30] X. Martin, A matrix phase for the phi**4 scalar field on the fuzzy sphere , JHEP hep-th/ Field Theory Simu-lations on a Fuzzy Sphere - an Alternative to the Lattice , PoS(LAT2005) 263(2005) [ hep-lat/ Simulating the scalar fieldon the fuzzy sphere , PoS(LAT2005) 262 (2005) [ hep-lat/ Numerical simulations of a non-commutative theory: the scalarmodel on the fuzzy sphere , JHEP hep-th/ Finite temperature phase tran-sition of a single scalar field on a fuzzy sphere (2007) [ hep-th/ Numerical results on the Non-commutative λφ Model , (2003) [ hep-th/ Field theory on a non-commutative plane: a non-perturbativestudy , Ph.D thesis, Humboldt University (2003) [ hep-th/ Phase diagram and dispersionrelation of the non-commutative λφ model in d = 3, JHEP hep-th/ Simulating non-commutativefield theory , Nucl. Phys. Proc. Suppl. (2003) 941 [ hep-lat/ The non-commutative λφ model , Acta Phys. Polon. B 34 (2003) 4711[ hep-th/ Scalar Field Theory on Fuzzy S , JHEP hep-th/ Quantum mechanics and path integrals ,McGraw-Hill (1965). IBLIOGRAPHY 167 [41] J. Glimm and A. Jaffe, Quantum physics: A functional integral point of view ,Springer-Verlag (1981).[42] M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory ,Addison-Wesley Publishing Co. (1995).[43] K. Osterwalder and R. Schrader, Commun. Math. Phys. (1973) 83; Com-mun. Math. Phys. (1975) 281.[44] G. Roepstorff, Path Integral Approach to Quantum Physics , Springer (1994).[45] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth. A. H. Teller and E.Teller, Equation of State Calculations by Fast Computing Machines , J. Chem.Phys. 21 (1953) 1087-1092.[46] B. Eynard, Correlation functions of eigenvalues of multi-matrix models,and the limit of a time dependent matrix , J. Phys. A 31 (1998) 8081[ cond-mat9801075 ]; Large N expansion of the 2-matrix model JHEP hep-th/ On the phase structure of large-N matrix models and gaugemodels , Phys. Lett. B 108 (1982) 407.[48] P. Zinn-Justin and J. B. Zuber, On some integrals over the U(N) unitary groupand their large N limit , J.Phys. A36 (2003) 3173-3194 [ math-ph/0209019 ].[49] A. Matytsin, On the large N limit of the Itzykson-Zuber Integral , Nucl. Phys. B 411 (1994) 805 [ hep-th/ Topology and geometry for physicists , London AcademicPress (1983).[52] M. E. Newman and G. T. Barkema, Monte Carlo Methods in StatisticalPhysics , Oxford University Press (2002).[53] W. Janke, Statistical Analysis of Simulations: Data Correlations and Er-ror Estimation , published in Quantum Simulations of Complex Many-Body 68 BIBLIOGRAPHY Systems: From Theory to Algorithms, Lecture Notes , J. Grotendorst,D. Marx, A. Muramatsu (Eds.), John von Neumann Institute for Com-puting, J¨ulich,NIC Series, Vol. 10, ISBN 3-00-009057-6 (2002) 423-445,[ ].[54] N. Madras and A. D. Sokal, The pivot algorithm: a highly efficient MonteCarlo method for the self-avoiding walk , J. Stat. Phys. 50 (1988) 109.[55] D. A. Varshalovich, A. N. Moskalev and V. K. Khersonky, Quantum Theoryof Angular Momentum: Irreducible Tensors, Spherical Harmonics, VectorCoupling Coefficients, 3nj Symbols , Singapore. World Scientific (1998).[56] A. Messiah, Quantum Mechanics , Volume II, North-Holland Publishing Com-pany, (1963).[57] K. Schulten and R. G. Gordon, Recursive evaluation of j and j coefficients , Comput. Phys. Comm. 11 (1976) 269–278; Semiclassical approximations to j- and j-coefficients for quantum-mechanical coupling of angular momenta , J. Math. Phys. 16 (1975) 1971–1975.D. F. Fang and J. F. Shriner Jr., A computer program for the calculation ofangular-momentum coupling coefficients , Comput. Phys. Comm. 70 (1992),147–153.S. T. Lai and Y. N. Chiu, Exact computation of the j and j symbols , Comput.Phys. Comm. 61 (1990), pp. 350–360.[58] J. Hoppe and S. T. Yau, Some properties of matrix harmonics on S , Comm.Math. Phys. 195 (1998), no. 1, 67–77.[59] A. M. Perelemov, V. S. Popov Casimir Operators for the Orthogonal andSymplectic Groups J. of Nucl. Phys. (U.S.S.R.) (1966) 1127-1134.[60] A. M. Perelemov and V. S. Popov, Casimir Operators for U ( n ) and SU ( n ), J. of Nucl. Phys.(U.S.S.R.) (1966) 924-931.[61] A. O. Barut and R. Raczka, Theory of Group Representations and Applica-tions Second edition, Ed. Polish Scientific Publishers-Warszawa (1980). IBLIOGRAPHY 169 [62] W. Fulton and J. Harris, Representation Theory. A First course , Ed. SpringerVerlag, (1991).[63] H. Georgi. Lie Algebras in Particle Physics , Ed. The Benjamin/CummingsPublishing Co.,Inc. (1982)[64] D. Gross and I. Klebanov, On-dimensional string theory on a circle , Nucl.Phys. B 344 (1990), 475; Vortices and the non-singlet sector of the c=1matrix model , Nucl. Phys. B 354 (1991), 459; Fermionic string field theory ofc=1 two dimensional Quantum Gravity , Nucl. Phys. B 352 (1991) 671.[65] D. Gross and A. Matytsin, Some properties of large-N two-dimensional Yang-Mills theory , Nucl. Phys. B 437 (1995) 541 [ hep-th/9410054 ].[66] B. P. Dolan, J. Medina and D. O’Connor, In preparation.[67] H. Grosse, C. Klimˇc´ık and P. Preˇsnajder, Finite quantum field the-ory in noncommutative geometry , Commun. Math Phys. 180 (1996) 429[ hep-th/ Nonuniform symmetry breaking in noncommu-tative λ Φ theory , Phys. Rev. D 68 (2003) 065008 [ hep-th/ Renormalization group equations and the Lifshitzpoint in non-commutative Landau-Ginsburg theory , Nucl. Phys. B 622 (2002)189 [ hep-th/ Duality in Scalar Field Theory on Noncommu-tative Phase Spaces , Phys. Lett. B 533 (2002) 168-177 [ hep-th/ A guide to Monte Carlo Simulations in statisticalPhysics , Cambridge University Press, (2000).[71] A. A. Kirillov, Encyclopedia of Mathematical Sciences ,vol 4, p.230;B. Kostant, Lecture Notes in Mathematics , vol.170, Springer-Verlag, (1970)87.[72] F. A. Berezin, General Concept of Quantization , Commun. Math Phys. 70 BIBLIOGRAPHY [73] H. Grosse, C. Klimˇcik and P. Preˇsnajder, Field Theory on a SupersymmetricLattice , Commun. Math Phys. 185 (1997) 155-175 [ hep-th/ The Fuzzy Supersphere , J. Geom. and Phys. math-ph/9804013 ].A. P. Balachandran, S. K¨urk¸c¨uoˇglu and E. Rojas, The star product on thefuzzy supersphere , JHEP 07 (2002) 056 [ hep-th/ Finite Quantum Physics and Noncommutative Ge-ometry , Nucl. Phys. Proc. Suppl. 37C (1995) 20 [ hep-th/ Noncommutative mechanics, Landau levels, twistors and Yang-Mills amplitudes , (2005) [ hep-th/0506120 ].[77] D. Karabali, V. P. Nair and S. Randjbar-Daemi, Fuzzy spaces, the M(atrix)model and the quantum Hall effect , Nucl. Phys. B 697 (2004) 513-540[ hep-th/ Statistical Mechanis of Phase Transitions , Oxford UniversityPress (2002).[79] J. Medina, Modelo Matricial en S y Teor´ıa de Campo Escalar en este espacio ,Master thesis, CINVESTAV (M´exico D.F. 2002).[80] E. Br´ezin, C. Itzykson, G. Parisi and J.-B. Zuber, Planar Diagrams , Comm.Math. Phys. 59 (1978) 35. ist of publications • J. Medina and D. O’Connor, Scalar Field Theory on Fuzzy S , JHEP hep-th/ • J. Medina, W. Bietenholz, F. Hofheinz and D. O’Connor, Field Theory Simu-lations on a Fuzzy Sphere - an Alternative to the Lattice , PoS(LAT2005)263(2005) [ hep-lat/0509162].