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2+1d quantum field theories in large N limit Omid, Hamid 2017

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2+1d Quantum Field Theories in Large N limitbyHamid OmidM.Sc Theoretical Physics, The University of British Columbia, 2011B.Sc Theoretical Physics, Isfahan University of Technology, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinFACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University Of British Columbia(Vancouver)January 2017c© Hamid Omid, 2017AbstractIn Chapter 1, we present a brief introduction to the tight-binding model of grapheneand show that in the low-energy continuum limit, it can be modeled by reducedQED2+1. We then review renormalization group technique which is used in thenext chapters.In Chapter 2, we consider a quantum field theory in 3+1d with the defect ofa large number of fermion flavors, N. We study the next-to-leading order contri-butions to the fermions current-current correlation function 〈 jµ(x) jν(y)〉 by per-forming a large N expansion. We find that the next-to-leading order contributions1/N to the current-current correlation function is significantly suppressed. Thesuppression is a consequence of a surprising cancellation between the two con-tributing Feynman diagrams. We calculate the model’s conductivity via the Kuboformula and compare our results with the observed conductivity for graphene.In Chapter 3, we study graphene’s beta function in large N. We use the largeN expansion to explore the renormalization of the Fermi velocity in the screeningdominated regime of charge neutral graphene with a Coulomb interaction. Weshow that inclusion of the fluctuations of the magnetic field lead to a cancellationof the beta function to the leading order in 1/N. The first non-zero contribution tothe beta function turns out to be of order 1/N2.In Chapter 4, we study the phase structure of a φ6 theory in large N. Theiileading order of the large N limit of the O(N) symmetric phi-six theory in threedimensions has a phase which exhibits spontaneous breaking of scale symmetryaccompanied by a massless dilaton. In this chapter, we show that this “light dila-ton” is actually a tachyon. This indicates an instability of the phase of the theorywith spontaneously broken approximate scale invariance. We rule out the exis-tence of Bardeen-Moshe-Bander phase.In this thesis, we show that Large N expansion is a powerful tool which inregimes that the system is interacting strongly could be used as an alternative tocoupling expansion scheme.iiiPrefaceThis thesis is based on notes written by myself during my PhD program and alsopublications authored by my collaborators and me. Most of the calculations weredone by myself. The ideas were developed during several meetings between mysupervisor and myself. Chapter 2 is the study of AC conductivity of 2+1d Diracsemi-metal in the large N Limit. A version of this chapter is prepared to be pub-lished. Chapter 3 is the study of the beta function of charge neutral 2+1d DiracSemi-metal in the large N. A version of this chapter is also prepared to be pub-lished. In Chapter 4, we investigate φ6 theory in the large N limit. A version ofthis chapter is accepted to be published in Phys. Rev. D .ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . 101.3 Dimensional Reduction of Electromagnetism . . . . . . . . . . . 161.4 Outline and Results . . . . . . . . . . . . . . . . . . . . . . . . . 172 AC Conductivity of 2+1d Dirac Semi-metal in the large N Limit . . 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Next-to-Leading Order Contributions to 〈 jµ(x) jν(y)〉 . . . . . . . 282.2.1 Π1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2 Evaluation of Π1B . . . . . . . . . . . . . . . . . . . . . . 342.2.3 Evaluation of Π1A+Π1B . . . . . . . . . . . . . . . . . . 352.3 Evaluation of Π2 . . . . . . . . . . . . . . . . . . . . . . . . . . 36v2.4 Combining Π1 and Π2 . . . . . . . . . . . . . . . . . . . . . . . 372.5 The Current-Current Correlator in Presence of a Condensate . . . 402.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 472.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 The Beta Function of Charge Neutral 2+ 1d Dirac Semi-metal inthe Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Corrections to the Electron Propagator . . . . . . . . . . . . . . . 563.3 Electron Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Infrared Contributions to the Fermi Velocity Beta Function . . . . 663.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 723.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 φ6 Theory in the Large N Limit . . . . . . . . . . . . . . . . . . . . . 764.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . 834.3 Effective Action Technique . . . . . . . . . . . . . . . . . . . . . 864.4 Tachyonic Excitations in φ62+1 . . . . . . . . . . . . . . . . . . . 914.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110A Trace over Π1 Dirac Matrices . . . . . . . . . . . . . . . . . . . . . . 117B Recurrence Relations for Is . . . . . . . . . . . . . . . . . . . . . . . 121C Calculation of I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127D Instantaneous Limit of Graphene in Large N . . . . . . . . . . . . . 129E Dimensional Regularization of φ6 in Large N . . . . . . . . . . . . . 131E.1.1 ~j2 Term . . . . . . . . . . . . . . . . . . . . . . . . . . . 132viE.1.2 ∆(p) Term . . . . . . . . . . . . . . . . . . . . . . . . . . 132E.1.3 ∆(p)∆(−p) Term . . . . . . . . . . . . . . . . . . . . . . 133E.1.4 ∆(p)~j2 Term . . . . . . . . . . . . . . . . . . . . . . . . 134E.1.5 ∆(p)3 Term . . . . . . . . . . . . . . . . . . . . . . . . . 135E.1.6 All Terms Together . . . . . . . . . . . . . . . . . . . . . 136viiList of TablesTable B.1 Table of Recurrence Relations for Is . . . . . . . . . . . . . . . 126viiiList of FiguresFigure 1.1 A honeycomb lattice consists of two triangular sub-lattices(black and white atoms). . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Graphene spectrum has two bands with opposite signs. Bandsmeet at K points and form a Dirac cone. . . . . . . . . . . . . 5Figure 1.3 Scattering diagram for φ4 theory. . . . . . . . . . . . . . . . . 11Figure 2.1 Partial sum over fermionic loops to get the effective propaga-tor for the screening dominated regime. . . . . . . . . . . . . 20Figure 2.2 Fermionic bubble, the elementary ingredient for our partialsummation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.3 The Feynman diagram of the expansion of the fermion deter-minant is depicted. The series is even due to particle-hole andtime reversal symmetry. The Feynman integrals for diagramswith more than two legs are finite. . . . . . . . . . . . . . . . 27Figure 2.4 The next-to-leading order Feynman digrams that contribute tocurrent-current correlator to the next-to-leading order. . . . . 29Figure 2.5 The master diagram for the two-loop calculations of the current-current correlation function . . . . . . . . . . . . . . . . . . 29ixFigure 2.6 Adapted from [9]. The red line is the transmittance expected for non-interacting two-dimensional Dirac fermions, whereas the green curve takesinto account a nonlinearity and triangular warping of graphene’s electronicspectrum. The gray area indicates the standard error for the measure-ments. (Inset) Transmittance of white light as a function of the numberof graphene layers. . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.1 The Feynman diagram of the expansion of the fermion deter-minant is depicted. The series is even due to particle-hole andtime reversal symmetry. The Feynman integrals for diagramswith more than two legs are finite. . . . . . . . . . . . . . . . 54Figure 3.2 The leading contribution to the beta function in the large Nlimit comes from the Feynman diagram where the dotted lineis the relativistic large N propagator and the insertion into thephoton propagator is the tree-level classical Coulomb actionwhich is non-relativistic. This diagram is of order 1/N2. . . . 57Figure 3.3 We have plotted the beta function in [55] (purple) vs. Eq.(3.19) (orange).As one expects in the limit of v→ 1 (here we have chosen the units suchthat light velocity is our measure for velocity), we find that the Lorentzsymmetry prevents the Fermi velocity from running. The beta functionin [55] violates this condition as its Lorentz symmetry is violated by con-struction but not the presence of the two velocities in the Lagrangian. . . . 65Figure 3.4 We have plotted the beta function in [55](dashed lines) vs. Eq.(3.19) (solidlines). The orange, purple and green lines respectively correspond to N =4,10,100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 3.5 By zooming into p q regime, we only check that a given theory with aninfrared limit that corresponds to the ultraviolet regime of Eq.(3.21) willbe infrared divergent free. . . . . . . . . . . . . . . . . . . . . . 69xFigure 3.6 Adapted from [18]. (a) Cyclotron mass as a function of Fermi wave-vector. The dashed curves are the best linear fits with assumption thatmc ∼ n 12 . The dotted line is the behavior of cyclotron mass derived fromthe standard value of Fermi velocity. Graphene’s spectrum renormalizeddue to electron-electron interactions is expected to result in the dependenceshown by the solid curve. (b) Cyclotron mass plotted as a variable of vF . . 73Figure 3.7 Adapted from [20]. N = 1 to N = 6 LLs’ energy as a function of level num-ber for different values of carrier density and B= 2T . For fixed density thecurves are highly linear, resulting in a possible negligible renormalizationof the Fermi velocity. (Inset) Residuals from the linear fit showing verygood linearity in the LLs. . . . . . . . . . . . . . . . . . . . . . . 75Figure 4.1 N× the beta function of large N regime of g2(~φ2)3 theory inthree dimensions. The infrared fixed point is g2IR = 0 and theultra-violet fixed point occurs at g2UV = 192. The critical cou-pling where in the infinite N limit scale symmetry breakingoccurs is g2 = (4pi)2 ≈ 158. . . . . . . . . . . . . . . . . . . 82Figure 4.2 Spontaneous breaking of the internal rotation symmetry in φspace. The field φ chooses a ground state that violates theinternal U(2) symmetry in the potential V (φ) = φ∗φ . . . . . . 84Figure 4.3 Connected reducible three-point function in terms of irreduciblevertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.4 Connected reducible four-point function in terms of irreduciblevertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.5 The phase diagram of the Landau-Ginzburg potential in equa-tion (4.31). The tri-critical point O appears at the intersectionof a line of second order phase transitions and a line of firstorder phase transitions where the potential is equal to g26 (~φ2)3. 95xiAcknowledgmentsThe writing of this dissertation would not have been possible without the help andencouragement of many people.I am grateful to my supervisor Gordon. W. Semenoff, who let me followmy interests and helped me by his brightness and wide range of knowledge. Iwould like to thank my supervisory committee, Josh Folk, Marcel Franz, Philip.C. Stamp and Mark Van Raamsdonk for their criticism of my research, construc-tive feedback and guidance. Special thanks to Marcel and Philip in helping me todevelop my background in condensed matter physics. I would like to thank IanAffleck and Eric Zhitnitsky for their enthusiasm in training new physicist; Ian andEric always welcomed me and my questions. Special thanks to Ian Afleck, RyanMcKenzie, and Rocky So for proofreading this thesis.To the University of British Columbia for providing me the funding throughthe Graduate Entrance Fellowship, International Partial Tuition Scholarship, thePhD Tuition Fee Award and University Four Year Fellowship.I would like to thank my great friends around the world who made life moreinteresting. Special thanks to Shahzad Ghanbarian, Omid Nourbakhsh, Ali Na-rimani and Hamid Atighechi. And last but not least, I am deeply grateful to myparents for their endless love and devotion to their children.xiiChapter 1IntroductionQuantum field theories have been essential tools for studying various physical sys-tems in the past century. Quantum electrodynamics, the first achievement of thequantum field theory paradigm, is still one of the most successful theories usedwidely in many fields of studies. Quantum field theories equipped with renor-malization group techniques provide such a strong paradigm for studying vari-ous systems that they are used in many fields outside of physics including socialnetworks and financial markets. We owe our understanding of the fundamentalconstituents of matter, the elementary particles, to the Standard Model of particlesphysics, which is a quantum field theory in 3+1 d. On the other hand, quantumfield theories can be used to model phenomena such as superconductivity in a verydifferent energy regime and field of physics, condensed matter physics.In this thesis, we will focus on the quantum field theories in 2+ 1d. Quan-tum field theories in 2+ 1d are of particular interest as they are between 1+ 1dquantum field theories that enjoy infinite tower of symmetries and 3+ 1 dimen-sional quantum field theories that mean-field-theory approximation works well at.Quantum field theories in 2+1d have received a lot of attention recently. The ex-perimental discovery of the two-dimensional cousin of graphite, graphene in 2004by Novoselov et al. [1], was a seminal event in electronic materials science. The1novel features of graphene such as its emergent relativistic dispersion relation hasattracted a lot of attention. The fundamental ingredients of graphene interact viaexchanging photons that live in 3+1 d. The interactions turn out to be strong andas a consequence of the low density of electrons in graphene, the usual methodsof expansion in terms of particle density seems invalid. Most of our knowledgeof quantum field theories come from perturbative solutions. In contrast to weaklycoupled systems, where one can use the coupling as a perturbation parameter, instrongly coupled field theories there is no obvious candidate to be used as the per-turbative parameter, a parameter that should be adequately small.In this chapter, we provide an introduction to the tight-binding model of grapheneand its continuum limit as a quantum field theory in 2+ 1d. We then present abrief introduction to the renormalization group technique that turns out to be theoil of our machinery. We finish the introduction by presenting the outline of theremainder of this thesis and a brief introduction to the following chapters.1.1 GrapheneGraphene is made of a honeycomb lattice with two distinct type of atoms. Ahoneycomb lattice can be described in terms of two triangular sub-lattices, A andB, Fig.1.1. A unit cell contains two atoms, one of type A and one of type B.Assuming that the lattice constant is a ' 1.48A˚, the primitive vectors a1 and a2are given bya1 = a(12,√32),a2 = a(12,−√32). (1.1)We define the following δ vectors such that they connect different sites to each-2Figure 1.1: A honeycomb lattice consists of two triangular sub-lattices(black and white atoms).other.δ1 =13(a1−a2)δ2 =13(a1+2a2)δ3 =13(−2a1−a2) (1.2)The sp2 hybridization of the atomic s orbital with two p orbitals leads to thetrigonal planar structure [hybridization of the atomic 2s, 2px, and 2py orbitals],and to the formation of so called σ bond between the neighboring carbon atoms.This σ bond is responsible for holding the carbon atoms in two dimensions andthe robustness of graphene. The remaining p orbitals (pz) are perpendicular tothe plane of this planar structure and have a weak overlap. The resulting covalentbond between pz orbitals of the neighbouring atoms lead to the formation of piband. Therefore, we model our material, graphene, using a tight-binding model3for the pi orbitals. The Hamiltonian would be given by the following operatorH =−t ∑i,δ ,sa†i,sbi+δ ,s+h.c., (1.3)where t ' 2.8eV is the nearest neighbour hopping parameter, ai,s and bi+δ ,s arethe Fermi operators of the electrons with the spin s on the A and B sub-lattices.The next nearest-neighbor hopping energy is around 0.1eV and can be ignored. Inthe momentum representation the Hamiltonian readsH =∑s∫B.Z.d2k(2pi)2ψ†s (k)(0 ϕ(k)ϕ(k) 0)ψs(k)=∑s∫B.Z.d2k(2pi)2ψ†s (k)H ψs(k), (1.4)whereϕ(k) =−t∑δeik·δ=−t(exp(ikya√3)+2exp(− ikya2√3)coskxa2). (1.5)The function ϕ(k) can be decomposed into real and imaginary parts correspond-ing to its magnitude and phase. AsH 2 = ϕϕ∗1, the magnitude of ϕ(k) is indeedthe absolute value of the energy eigenvalues and is given byε(k) = t√1+4cos2kxa2+4coskxa2cos√3kya2. (1.6)Then the eigenvalues of the Hamiltonian are simplyE±(k) =±ε(k). (1.7)4Figure 1.2: Graphene spectrum has two bands with opposite signs. Bandsmeet at K points and form a Dirac cone.We plot this spectrum in Fig.1.2. As the lattice is a bipartite lattice, we observethat there are two energy bands. Discrete symmetries of the original Hamiltoniancould indeed be used to show that for any band, there is another band with theopposite sign in the energy.As it can be observed from Fig.1.2, the valence and conduction band meet at sixpoints, the K points. Two of these points are independent and we choose them tobe the following K± pointsK± =±4pi3a (1,0) (1.8)The linearized dispersion around these points get the form of the dispersion rela-tion for a Dirac cone. The Dirac cone turns out to be massless as a consequence ofthe discrete symmetries of the Hamiltonian including the spatial inversion sym-metry (P : (x,y)→ (x,−y)). These discrete symmetries protect the dispersionrelation from getting gapped.5Let us linearize the Hamiltonian around K points and study the low energy ex-citations of our model around those points. As graphene is half-filled, these pointsare naturally located on the zero-chemical potential line. We can approximate ϕand linearize it as ϕ(K±+k) =±vF(k1∓ ik2). As the name of the coefficient ofthe momenta vF suggests, vF turns out to be the emergent bare Fermi velocity.Then the linearized Hamiltonian around the two K points can be written asH =∑s∫ d2k(2pi)2[ψ†s (K++k)HK+ψs(K++k)+ψ†s (K−+k)HK−ψs(K−+k)],(1.9)where the Hamiltonian densities are given byHK+ = vF(τ1k1+ τ2k2),HK− = vF(−τ1k1+ τ2k2). (1.10)Here, the τ matrices are another set of Pauli matrices corresponding to the sub-lattice degrees of freedom. We can change our basis and write the Hamiltonian ina more familiar form of 4× 4 matrices, forming a reducible version of the DiracHamiltonian. The irreducible representation of the Dirac algebra in 2+1d is givenby 2× 2 matrices and there are two such irreducible representations. In aboveHamiltonian, we are using both of these representations. We can always combinetwo irreducible representations and make them into a reducible representation.We choose our new basis such that the fermionic field is given byΨs(k) =(ΨK+,s(k)ΨK−,s(k))=as(K++k)bs(K++k)bs(K−+k)as(K−+k) . (1.11)6In this basis the Hamiltonian gets the more conventional form of the Dirac Hamil-tonian given by the following HamiltonianH =∑s∫ d2k(2pi)2Ψ†s (k)0 k1− ik2 0 0k1− ik2 0 0 00 0 0 −k1+ ik20 0 −k1− ik2 0Ψs(k)=∑s∫ d2k(2pi)2Ψ†s (k)H (k)Ψs(k), (1.12)whereH can be written in terms of the α matricesH (k) = α1k1+α2k2, (1.13)where α matrices are given byα = (α1,α2) = τ3× (τ1,τ2)=(τ 00 −τ). (1.14)This representation is the original representation of the Dirac Hamiltonian thatwas found by Paul Dirac in a search of a relativistic version of the Schrdingerequation that was linear in momenta.The more familiar description of Dirac particles in quantum field theory isgiven by the Dirac representation of the Hamiltonian. We define β as τ1×1. Thewell-known gamma matrices that the Dirac Hamiltonian can be written in termsof and form a Clifford algebra are related to alpha matrices through the following7relationsγ0 = β ,γ = βα. (1.15)If we define Ψ¯≡Ψ†γ0, the Hamiltonian now can be written asH = vF∑s∫ d2k(2pi)2Ψ¯s(k)(γ1k1+ γ2k2)Ψs(k)= ivF∑s∫d2xΨ¯s(x, t)(γ1∂ 1+ γ2∂ 2)Ψs(x, t). (1.16)The Lagrangian formalism is a more natural gear for quantum field theories. Wecan infer the Dirac Lagrangian from our Hamiltonian. It can be found simply byapplying a Legendre transformation to the Hamiltonian. We introduce a new fieldthat is coupled to our fermionic field through minimal coupling to the fermionicfield. The added gauge field can be implemented through Peierls substitution inthe original tight-binding Hamiltonian [2]. Our final Lagrangian for the emergentexcitations in graphene coupled to a U(1) gauge field, A(x, t), is given by thefollowing well-known LagrangianL=∑s∫d2x i Ψ¯s(x, t)[γ0(∂ 0− i eA(x, t)0)+ vF~γ · (~∂ − i e~A(x, t))]Ψs(x, t).(1.17)The Lagrangian density of Eq.1.17 in a more formal notation using Feynman slashnotation, /A≡ γ0A0+ vF~γ ·~A, can be written asL =∑si Ψ¯s(x, t)(/∂ − i e/A(x, t))Ψs(x, t). (1.18)To consider the dynamics of the gauge fields, we need to add their kinetic term to8the Lagrangian. The interacting action for the excitations is given by the followingS=∫d3x[ψ¯aγ t(i∂t+At)ψa+ vF ψ¯a~γ · (i~∇+~A)ψa]+εc2e2∫d4x[1c~E2+ c~B2].(1.19)We can examine the relative strength of the above terms and investigate whetherwe can treat the Coulomb interaction as a perturbation. The ratio between thepotential term and the kinetic term is given byα =VK' e2h¯vF, (1.20)where we used the linear dispersion of the kinetic term and the usual Coulombpotential. In usual many-body systems, this ratio depends on n (the density ofthe particles) such that the interactions are suppressed in the high density limit.For example, in 2+ 1d materials with a quadratic dispersion relation, this ratiois proportional to n−1/2. However, this ratio for graphene is independent of nand is close to 2.2, a value which is not small. Consequently, the Coulomb inter-action term can not be investigated by using the usual coupling expansion scheme.In the next two chapters, we focus on the effects of electromagnetic inter-actions in Dirac semi-metals. In graphene, our favorite Dirac semi-metal, theCoulomb interaction term is the dominant interaction and based on the above es-timation, its strength is around few electron volts. Other interactions, includingphonon-electron (< 0.1eV), intrinsic spin-orbit (< 0.01meV), and Rashba spin-orbit (< 0.01meV) couplings are dominated by the Coulomb interaction [3, 4].In the following chapters, we will study the action (1.19) in more details. Weextend the number of fermionic species from four, two valleys times two spins,to an arbitrary large number of fermionic species and use that large number todevelop a well-defined perturbative method. In our calculations, this perturbative9method takes the place of the usual coupling expansion method.1.2 Renormalization GroupThe pioneers of quantum field theory were enormously puzzled by the divergentintegrals that they often encountered in their calculations of various physical prop-erties of quantum field theories. They spent the 1930’s and 1940’s struggling withthese infinities. As a result of these seemingly inconsistencies in quantum fieldtheories, many advocated abandoning quantum field theory altogether. Eventu-ally, a so called renormalization procedure was developed where the infinitieswere removed and finite physical results were obtained. However for many yearsmany physicists looked at renormalization theory suspiciously as a sleight of hand.Eventually, starting in the 1960’s a better understanding of quantum field theorywas developed through the efforts of Leo Kadanoff and Ken Wilson and manyothers. Renormalization group was used to infer observable consequences of thepresence of these singularities. Field theorists gradually came to realize that diver-gences in quantum field theory imply deep physical consequences that could notbe explained before the invention of renormalization group. Later, many differentrenormalization schemes where developed that each were more convenient thanthe others in different scenarios.The initial treatment of divergences in quantum field theories was based onthe fact that only observables need to be finite. Observable quantities in the lab-oratories are not the same as the bare coupling constants in Lagrangians but arerelated to them. The main idea was to change the bare coupling in a way that theobservable quantities turn out finite. This redefinition of the coupling constantsgenerally makes coupling constants a function of momenta. For example, a cou-pling constant expansion of the scattering amplitude, as shown in Fig.1.3, in aφ4 theory in terms of the cut-off and the external momenta can be written as the10Figure 1.3: Scattering diagram for φ4 theory.following [5]M =−λ −λ 2 [V (s)+V (t)+V (u)]+O(λ 3), (1.21)where s, t and u are Mandelstam variables that depend on external four-momentaass= (p1+ p2)2,t = (p1− p3)2,u= (p1− p4)2. (1.22)V (x) turn out to be divergent and is given by C ln Λ2x where C is a finite constantand Λ is the hard cut-off. If we assume that the scattering amplitude at a givenmomentum that corresponds to s0, t0 and u0 is λR, then at any other momenta itcan be found to beM =−λR−Cλ 2R[lnss0+ lntt0+ lnuu0]+O(λ 3R) (1.23)and now the scattering amplitude becomes finite for any momenta. Later, a more11systematic way of renormalization was suggested that would make calculationseasier for higher order calculations. To do so, one would need to add a finite num-ber of infinite terms called counter-terms that would make the observables finite.As these counter-terms are not observable, one can remove infinities of renormal-izable theories consistently without putting the physicists in danger.In the limit where the Mandelstam ratio becomes large or small, we observethat Eq.(1.23) becomes an invalid approximation. Our perturbative analysis breaksdown as the coupling constant, the perturbative parameter, becomes large. To dealwith this issue, one can use the renormalization group equation that is known asthe Callan-Symanzik equation. This equation is based on the observation that thebare couplings in our Lagrangian are independent of the renormalization criteriawhich we impose. Callan-Symanzik showed that the renormalized Green’s func-tions of our theory would satisfy the following differential equation[µ∂∂µ+β∂∂λ+nγ(λ )]G(n)(x1, . . . ,xn;µ,λ ) = 0, (1.24)where µ is the renormalization scale, β and γ are functions of the coupling con-stant only dictated by the Lagrangian and n corresponds to the number of fields inthe Green function.Using the Callan-Symanzik equation, one can find that the accurate version ofEq.(1.23) is given by the following expansion in the coupling constantM =− λR1−Cλ 2R[ln ss0 + lntt0+ ln uu0] +O(λ 3R), (1.25)and this equation is now valid in the infrared energy scale, or for small Mandel-stam ratios.The modern view is that quantum field theory should be regarded as an effec-12tive low energy theory, valid up to some energy scale. This view was suggestedby Wilson after the work that was done by Kadanoff to explain the scaling be-havior of systems at the critical points. In this scheme, one integrates out the fastmodes (UV) and leaves out the slow modes (IR). The question of interest is tofind a function for the renormalized couplings for the slow modes by starting theanalysis with a more fundamental model that consists of both fast and slow modes.Here we briefly discuss the Wilson renormalization scheme. We follow thepath integral treatment discussed in [6]. The scheme has three stages. We firsteliminate fast modes by integrating them out. In other words, we divide the modesinto two categories, fast and slow. Let’s assume our field representation in mo-mentum space is φ(k) thenφ< = φ(k)for 0 < k < Λ/s (slow modes),φ> = φ(k)for Λ/s≤ k ≤ Λ (fast modes). (1.26)The action would have three parts in themS(φ<,φ>) = S0(φ<)+S0(φ>)+Sint(φ<,φ>). (1.27)As we are interested in the low energy limit of our theory, we simply integrate φ>field out and find an effective action for φ< field. In Euclidean signature, we haveexp(Seff(φ<)) = eS0(φ<)∫[dφ>(k)]eS0(φ>)+Sint(φ<,φ>). (1.28)The beauty of the path integral formalism becomes more evident here as the taskof eliminating the fast modes in path integral language is the most natural.Although Seff provides a good description of the slow mode physics, the renor-malization group transformation has two more steps besides the above mode elim-ination. We need to rescale the momenta so that comparing our renormalized13couplings with the previous couplings makes sense. The previous theory was in-tegrated up to Λ while the effective theory is only integrated to Λ/s. We thendefine k′ = sk which run over the same range as k did before the elimination of thefast modes. However, theories that their actions are only different by a rescalingof the constituting fields are equivalent. To make sure that we incorporate thisobservation in our analysis, we rescale the field such that a certain coupling in thequadratic part of the action has a fixed coefficient. Explicitly, we rescale the fieldby a factor ξ and write the effective action in terms of renormalized fields φ ′(k′)given byφ ′(k′) = ξ−1φ<(k/s′). (1.29)The new couplings that describe the theory made of φ ′(k′)s are our effective cou-pling constants. In renormalizable field theories, we can use the same trick thatwe used in Eq.(1.25) and eliminate the dependencies on the cut-off, Λ. We are leftwith a consistent theory that is not sensitive to the cut-off and works well for slowdegrees of freedom. The flow in the coupling constants are typically describedusing their beta function. For a generic coupling constant, g, its beta function isdefined asβ (g) =dgd lns(1.30)In general, the beta function for a coupling constant at a given scale could behavein three different ways(1) β (g)> 0,(2) β (g)< 0,(3) β (g) = 0. (1.31)In the first class, the coupling constant increases by reducing the cut-off. In thisclass, if the sign of the beta function persists to stay positive, the coupling be-14comes larger after every step of reducing the cut-off and the perturbative analysisbreaks down at a given scale. The operators that are associated with these cou-pling constants are called relevant.In the second class, the coupling constant decreases by reducing the cut-off. Inthis case, if the coupling constant stays monotonic, the coupling becomes smallerafter every step of the cut-off reduction and loses its importance gradually. Theoperators that are associated with these coupling constants are called irrelevant aswe can ignore them in the strict infrared regime.In the third class, the coupling does not run (get renormalized) and is indepen-dent of the scale. The operators that are associated with these coupling constantsare called marginal.Of great importance are the points in which the beta function vanishes. Atthese points, the theory becomes scale independent. These points are called thefixed points of the theory and describe the critical behavior of the correspond-ing Lagrangian. At these points, the correlation length ξ either becomes zero orinfinite, corresponding to a trivial fixed point or critical fixed point respectively.Phase transitions happen when the correlation length tends to infinity. The factthat the theory becomes scale invariant at the critical point can be used to explainthe observed scaling behavior of critical systems in experiments.In the next two chapters, we will use the renormalization group technique tostudy low energy excitations of graphene. It turns out that renormalization groupmanages to somehow suppress the interactions in graphene. In the last chapter ofthis thesis, we use renormalization group to study the stability of a conjecturedphase in φ6 theory. Investigating this model and its relevance to graphene wouldbe argued in the last chapter.151.3 Dimensional Reduction of ElectromagnetismIn this section, we consider a gauge field that lives in 3+ 1d and is coupled to afield that lives in one lower dimension, 2+ 1d. We assume that the gauge fieldis coupled to the current of the lower dimensional field through minimal couplingand its kinetic term is described by the Maxwell Lagrangian. This model describesthe coupling of a 2+ 1d Dirac semi-metal, for example graphene, to the electro-magnetic field. The Euclidean action of the gauge field and its coupling to thecurrent is given by the following equationS=∫d3+1x[14e2FµνFµν +Aµ jµ], (1.32)where Aµ is the gauge field, Fµν is the electromagnetic field tensor and jµ is thecurrent of the lower dimensional field. The indices µ and ν run over {0,1,2,3}.As the lower dimensional field lives in 2+1d, j3 is zero and we can choose jµ tolive in z= 0 plane, in other-words jµ ∼ δ (z= 0).Due to the fact that Aµ is coupled to the other field through the jµ term and italways carries a delta function, we observe that every interaction vertex for themwould carry a delta function. In Fourier space, the delta function results in anintegration over the third component of the gauge field momenta. As a result,the effective propagator for the gauge field in 2+ 1d is given by the followingexpressionD(papa) =∫ ∞−∞dp2pi1papa+ p2z=121√papa, (1.33)where a runs over {0,1,2}.Now we can use the above propagator to find the reduced Lagrangian in 2+1d.16It can be written in the following form,Sreduced =∫d3+1x[12e2Fab1√−∂ 2Fab+Aa ja]. (1.34)1.4 Outline and ResultsIn Chapter 2, we consider a 2+1d defect in quantum field theory living in a 3+1dspace-time. We assume that our defect theory has a large number of fermionic fla-vors, N. The corresponding defect quantum field theory is of interest as it hasshown to be a successful model for 2+1d Dirac semi-metals, including graphene.We work in a coupling regime that the interactions are strong and can not be ig-nored. We investigate the linear response of our model to the electromagneticfields using a large N perturbative scheme. This investigation is motivated bythe observed experimental results which suggest that our model, despite beingstrongly-interacting can be approximated well by free quantum field theories. Westudy the next-to-leading order contributions to our model’s current-current corre-lation function 〈 jµ(x) jν(y)〉 by performing a large N expansion. We find that thenext-to-leading order contributions 1/N to the current-current correlation func-tion is significantly suppressed. The suppression is a consequence of a surprisingcancellation between the two contributing Feynman diagrams and is not explicitlydependent on N being large. Through the computed current-current correlator inthis chapter, we study the optical conductivity of our Dirac semi-metal system. Wecalculate the model’s conductivity via the Kubo formula and compare our resultswith the observed conductivity for graphene. The results show a good agreementwith the observed slow running of the Fermi velocity in graphene.In Chapter 3, we study graphene’s beta function in large N. We use the largeN expansion to explore the renormalization of the Fermi velocity in the screen-ing dominated regime of the charge neutral graphene with a Coulomb interaction.17We find that in contrast to the instantaneous model in which the speed of lightis assumed to be infinite and acquires a beta function of order of 1/N for theFermi velocity, our model acquires a beta function of order 1/N2. We show thatinclusion of magnetic fluctuations restores the Lorentz symmetry to the leadingcontribution and results in a cancellation between electric and magnetic parts. Asthe next-to-leading order is of order 1/N2, the first non-zero contribution to thebeta function turn out to be of the order 1/N2. Consequently, the beta functionwould much smaller in large N than the values quoted in the current literatureand the Fermi velocity would renormalize significantly slower. We prove that theFermi velocity is gauge invariant given that the infrared regime is divergence-free.We perform a careful analysis of the possible infrared divergences and show thatthe superficial infrared divergences do not contribute to the beta function.In Chapter 4, we study the phase structure of the phi-six theory for large Ns.The leading order of the large N limit of the O(N) symmetric phi-six theory inthree dimensions has a phase which exhibits spontaneous breaking of scale sym-metry accompanied by a massless dilaton which is a Goldstone boson. At thenext-to-leading order in large N, the phi-six coupling has a beta function of or-der 1/N and it is expected that the dilaton acquires a small mass, proportional tothe beta function and the condensate. However, the stability of the phase is notguaranteed and needs further investigations. In this chapter, we show that this“light dilaton” is actually a tachyon. This indicates an instability of the phase ofthe theory with spontaneously broken approximate scale invariance. We rule outthe existence of the Bardeen-Moshe-Bander phase by showing that the vacuum isunstable in that phase. Our analysis suggests that for a phi-six vector model theonly non-trivial critical behavior is present at the Wilson-Fisher point.18Chapter 2AC Conductivity of 2+1d DiracSemi-metal in the large N Limit2.1 IntroductionIn this chapter, we use the large N technique to study the current-current correla-tor of a 2+1d defect theory. We study the AC conductivity of a two-dimensionalDirac semi-metal and model the latter using a defect quantum field theory con-sisting of N species of Dirac fermions occupying an infinite planar 2+1d defectembedded in 3+ 1d Minkowski space-time. The conductivity of materials is animportant quantity in probing their underlying physics and is relatively easy tomeasure in laboratories. Experimental measurements of graphene’s conductivityshow that despite the strong electromagnetic interactions in graphene, the AC con-ductivity of this material is very close to the AC conductivity of non-interacting2+ 1d fermion gases [7–9]. In this chapter, we provide a theoretical explana-tion for this surprising behaviour of graphene’s conductivity. We focus on thescreening dominated regime of the 2+ 1d defect by doing a partial sum over thefermion loops shown in Fig.2.1 and then study the next-to-leading order contribu-tions to the current-current correlator. The contributions from the electromagnetinteractions are suppressed by a factor of N, the number of fermions, and would19Figure 2.1: Partial sum over fermionic loops to get the effective propagatorfor the screening dominated regime.not contribute to our calculations in this chapter. This is due to the fact that weare working in the screening dominated regime in which the induced kinetic termfor the gauge fields would be of order N while the Maxwell Lagrangian is inde-pendent of N. This hierarchy in N can be observed by considering the followingfermion partial summation and observing that each fermionic species contributesindependently to the polarization tensor. The fermion loops, shown in Fig.2.2,contributing to the sum can be found by calculating the following one-loop inte-gral for a mass-less fermion.Πµνfermion-loop =∫ d3k(2pi)3tr[γµ1/k+ /pγν1/k]=∫ d3k(2pi)3tr[γµ/k+ /p(p+ k)2γν/kk2]. (2.1)The functional form of the polarization tensor is dictated by the Ward identitykµΠµνfermion-loop = 0. The only tensors that can appear in the polarization tensor arethe metric ηµν and kµkν . This implies that the polarization tensor is proportionalto the projection operator p2ηµν− pµ pν . We can find the exact form of the polar-ization tensor by performing the integral. We use dimensional regularization as itmakes the calculations easier here20Πµνfermion-loop(p) =∫ dD+1k(2pi)D+1tr[γµ/k+ /p(p+ k)2γν/kk2]= 2∫ dD+1k(2pi)D+1kµ(k+ p)ν + kν(k+ p)µ −ηµνk · (k+ p)k2(k+ p)2. (2.2)We use our projector to simplify the calculations. Projecting the polarization ten-sor using a transverse projector results in(p2ηµν − pµ pν)Πµνfermion-loop =∫ dD+1k(2pi)D+1−2p · k p · (k+ p)k2(k+ p)2= (p2)2∫ dD+1k(2pi)D+11k2(k+ p)2D+1→3==p38. (2.3)Then the polarization tensor is given by the following finite tensorΠµνfermion-loop(p) =116(pηµν − pµ pν/p). (2.4)We observe here that the polarization tensor is proportional to 1p in contrast to theusual 1p2 behavior. As we observe in the next chapter, the propagator for the gaugefield after the dimensional reduction to 2+ 1d acquires the same form and has apole proportional to p instead of p2.Now we can do the geometric series in Fig.2.1 and focus on the regime thatthe screening is dominant which corresponds to the regime in which the numberof fermions is large. The polarization tensor gets a contribution from any fermionspecies and it can be written asΠµνleading-order(p) =N16(pηµν − pµ pν/p), (2.5)21Figure 2.2: Fermionic bubble, the elementary ingredient for our partial sum-mation.in which N is the number of the fermionic species. As we will see soon, this par-tial sum of fermion loop can be done systematically by investigating the large Nexpansion of our theory.The large N expansion is used to be an alternative to a coupling constant ex-pansion in theories that the underlying degrees of freedom interact strongly. Inthese theories, the coupling constant expansion becomes inaccurate and loses itsvalidity [10]-[17]. Graphene is like a perfect laboratory for examining such largeN expansion methods as the coupling constant in graphene, α , is around 2.2 [18].The experimental results suggest that graphene, despite having strongly interact-ing underlying degrees of freedom, can be approximated by a non-interactingmodel [18]-[21] .There has been a number of studies using both large N and coupling constantexpansion methods to investigate various properties of graphene [22]-[45] . Theresults seem to agree with graphene flowing to a Fermi liquid fixed point withan approximate Lorentz symmetry. in this chapter, we investigate the theoreticalvalidity of the large N expansion by computing the next-to-leading order contri-butions to the current-current correlation function 〈 jµ(x) jν(y)〉. We use severalloop calculation techniques developed in quantum field theory to evaluate the twocontributing Feynman diagrams to the effective photon kinetic term.The current-current correlator corresponds to the AC conductivity of the ma-22terial through the well known Kubo formalism. The AC conductivity of graphenehas been measured in several experiments [9], and was found to be very close tothe value of e24h¯ , which also happens to be the value found for the system of non-interacting Dirac fermions at the half filling [46]. Here we first review the Kuboformalism and then use it to derive the conductivity for a Dirac semi-metal usingthe large N limit. Let us review the Kubo formula derivation for the conductivityfirst. We would like to find the conductivity of a given system. Assume that theexternal field couples to the current in the following mannerδS[A, j] =∫j A, (2.6)the integration could be done in any space, for example the momentum space,for that reason we did not write an explicit form for the integral measure or thevariables of j and A. We need to find the vacuum expectation value (VEV) of thecurrent, 〈 j〉. If the external field can be treated as a perturbation (probe field), weexpect the VEV to be linear in the external field and the proportionality constantwould be called the conductivity. Using the fact that 〈 j〉 can be generated via theaction in the following way, we can proceed further and find a formula for theconductivity〈 j(1)〉= 1Z∫Dψe−S[A, j;ψ] j(1)=−δ lnZ[A, j]δA(1)'−∫ δ 2 lnZ[A, j]δA(1)δA(2)|A=0 A(2). (2.7)In the first line, ψ is the collection of the fields and Dψ is the path integral measureof this collection; j(1) means that we are computing the field configuration at thepoint 1 of the integration space and the same for A(1). In last line, we usedperturbative expansion of Z, and the integration is over the coordinates of A(2).23We find that the linear response coefficient is given by the following expressionK(1,2) =− δ2 lnZ[A, j]δA(1)δA(2)|A=0. (2.8)The functional derivatives would produce two terms. Assuming that there is nocurrent in the absence of the external field, the term that is proportional to VEVof the current would vanish in absence of an external field and the response coef-ficient of our material, K, can be written asK(1,2) =− 1Zδ 2Z[A, j]δA(1)δA(2)|A=0. (2.9)We can write down the Kubo formula in terms of the current-current correlator bydoing the functional derivatives. Using the definition for the current, we find thatK(1,2) =− 1ZδδA(1)∣∣∣A=0∫Dψe−S[ j,A;ψ] j(2;A)=−〈δ (1−2)∂A j(A)|A=0〉+ 〈 j(1;A= 0) j(2;A= 0)〉 . (2.10)The first term in the above formula is known as the diamagnetic term and theminimally coupled fermionic field turns out to be proportional to the current ex-pectation value. In this chapter, we focus on the second term that is known as theparamagnetic term as we are interested in linear response in presence of a probefield and the diamagnetic term is not present in this limit.Th actual conductivity is a retarded response and one needs to be more carefulas in above we used the time-ordered correlation functions. To take this obser-vation into account, we first need to employ the general formalism to derive theimaginary-time response and then analytically continue to real frequencies. Wealso notice that the actual optical conductivity is the linear response to the electricfield, which is the time derivative of the gauge field considered above. The usualoptical conductivity is given by the following formula24σ(ω) =− lim~q→01ωK(q)∣∣∣iω→ω+i0. (2.11)The conductivity for graphene has been calculated using coupling constant ex-pansion formalism [42],[43]. The conductivity in the regime that the couplingconstant expansion is accurate gets the following formσ =e24(1+Cα+O(α2)), (2.12)in which α is the fine structure for graphene and C is a constant. There hasbeen a controversy on the exact value of C . A few calculations find C to beC = 19−6pi12 ' 0.01 [47, 48] while other authors find C = 22−6pi12 ' 0.26 [49, 50].It is suggested [43] that this controversy stems from the inaccurate use of the reg-ularizations and the accurate result is given by C = 19−6pi12 . As the fine structureconstant for graphene is around 2.2, the coupling constant expansion seems to beless reliable and for the latter value for C the next-to-leading order contributionmakes the expansion converge poorly.In this chapter, we will model graphene as an infinite 2+1d defect embeddedin 3+1d space-time. Graphene electrons are the degrees of freedom that are oc-cupying the the 2+1d sheet [51, 52]. As we discussed, the tight-binding model ofgraphene results in emergent massless electrons with an emergent relativistic dis-persion relations. The Lorentz symmetry of this emergent dispersion relation willsimplify the future calculations significantly. The electrons interact by exchang-ing photons which are allowed to propagate in the surrounding bulk of space-time,3+1d. We describe this theory by the defect quantum field theory with Euclidean25space actionS=∫d3x[ψ¯aγ t(i∂t+At)ψa+ vF ψ¯a~γ · (i~∇+~A)ψa]+εc2e2∫d4x[1c~E2+ c~B2].(2.13)The first integral is over the 2+1d defect with the integrand being the Lagrangiandensity of N species of two-component spinor fermions. The Lagrangian has anextra global U(N) symmetry. For graphene, N = 4 which corresponds to the twovalleys and the spin degree of freedom. In the following, we test the validity of thelarge N expansion in the quantum field theory described by the action in Eq.(2.13)in the limit where N→ ∞.In Section 4.61, we add short range interactions to the above Lagrangian andconsider their effect on the conductivity. Short range interactions can be studiedusing the usual Hubbard-Stratonovich transformation. We present a detailed re-view of this transformation in Chapter 4. Hubbard-Stratonovich transformationintroduces a new scaler field in the Lagrangian that is coupled to the fermionsthrough a Yukawa interaction term, gφψ¯ψ . Depending on the phase of systems,the Yukawa interaction term might survive the renormalization or flow to zero.We find that the phase of system affects the conductivity and the phase can bedistinguished by comparing the conductivity of system with the calculated con-ductivities for each phase.To implement the large N expansion, we integrate out the fermions to get theeffective field theory for the gauge fields. The fermions contribute terms in theeffective action which are given by the Feynman diagrams in the series depicted26Figure 2.3: The Feynman diagram of the expansion of the fermion deter-minant is depicted. The series is even due to particle-hole and timereversal symmetry. The Feynman integrals for diagrams with morethan two legs are finite.in Fig. 2.3. We obtain the following effective action for our Dirac semi-metalSeff =N16∫d3x~E · 1/vF√−~∇2− 1v2F∂ 2t~E+~B vF√−~∇2− 1v2F∂ 2t~B .+N∞∑n=4∫dx1...dxnAµ1(x1) . . .Aµn(xn)Γµ1...µn(x1, ...,xn)+εce2∫d3x[~E · 1/c√−~∇2− 1c2∂ 2t~E+~B c√−~∇2− 1c2∂ 2t~B](2.14)The first line in Eq.(2.14) is the expansion of the fermion determinant to thequadratic order in the gauge fields. It has a factor of N in front of it. This termcontains the leading order screening of the Coulomb interaction by the relativisticelectrons. The second line in Eq.(2.14) contains the higher order, multi-photon in-teraction terms coming from the fermion determinant. These terms are of order N.To emphasize this, we have explicitly extracted a factor ofN. The Γµ1...µn(x1, ...,xn)are connected irreducible multi-photon correlation functions arising from a sin-gle species of electrons, computed to one-loop order. Due to charge conjugation(particle-hole) symmetry, this is an even series, and it begins at order four (withphoton-photon scattering). In the third line of Eq.(2.14) , we have presented the27same Maxwell theory of the photon as in the second line of Eq.(2.13), but di-mensionally reduced to 2+ 1d, with the component of the photon field that isperpendicular to the defect eliminated using its linear equation of motion.The first and second lines are of order N and they dominate the large N limit.The vacuum kinetic term for the photon in the fourth line is ignored to the leadingorder in the large N limit. The theory obtained by retaining the first two linesof Eq.(2.14), which is a Lorentz invariant conformal field theory whose speed oflight is vF . In the following, we are interested in the evaluation of the second lineup to three-loops. To have a sensible expansion, the new contributions needs to besuppressed in comparison to the leading order contributions.2.2 Next-to-Leading Order Contributions to〈 jµ(x) jν(y)〉In this section, we shall compute the next-to-leading order contributions to thecurrent-current correlation function 〈 jµ(x) jν(y)〉. We would need to compute thecontributions coming from two different Feynman diagrams, as shown in Fig.2.4.The first diagram is computable by using the one-loop techniques as the compu-tation can be decomposed into steps, where each step adds a one-loop diagram.The second diagram is an irreducible two-loop diagram and the computations cannot be reduced to a one-loop calculation. Indeed the corresponding topology ofFig.2.5 is the most general topology in the two-loops calculations of our theoryand all other diagrams are equivalent to this topology. Our main task would befinding an exact form for a master diagram with the given topology in Fig.2.5.As we encounter many two-loop integrals in the remaining of this chapter, letus define the following convention. For the rest of this chapter, we define〈. . .〉28Figure 2.4: The next-to-leading order Feynman digrams that contribute tocurrent-current correlator to the next-to-leading order.as the following integral〈. . .〉≡∫R6d3p(2pi)3d3q(2pi)3. . . . (2.15)The next-to-leading order contribution to the current-current correlation functionhas a complicated form and can be written as a sum of two distinct terms, Πµν1 (k)and Πµν2 (k). Each corresponds to a distinct Feynman diagram in Fig.2.4.We can write down the algebraic form of each Feynman diagram following theFigure 2.5: The master diagram for the two-loop calculations of the current-current correlation functionusual Feynman recipe. Due to the presence of several vertices and propagators,29the integrals get a relatively complicated formΠµν(k) =Πµν1 (k)+Πµν2 (k), (2.16)Πµν1 (k) =−16〈Tr(/q+ /p)γµ(/q+ /p+/k)γρ(/q+/k)γν/qγρ(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉, (2.17)Πµν2 (k) =−32〈Tr/qγµ(/q+/k)γν/qγρ(/q+ /p)γρ(q+ p)2(q+ k)2[q2]2ps〉. (2.18)In the above expression, we used Πleading-order that we found previously and isdefined asΠρσleading-order(k) =16N(δρσ +κkρkσk2). (2.19)The minus sign comes from the presence of a fermionic loop in our Feynman di-agrams. The combinatoric factor is one for each diagram. The additional factorof two in Π2 is due to the fact that there are two diagrams. We are working in theFeynman gauge and have regulated the integrals by changing the photon propa-gator from 16Nδρσp to a new propagator given by16Nδρσps . This will be sufficient todefine the integrals properly and regulate the ultraviolet divergences that would bepresent in our integrals.As a consequence of the Ward identities, Lorentz invariance and dimensionalanalysis, the vacuum polarization gets a particular form. The vacuum polarizationgenerally has the form given by the following expressionΠµν(k) =Π(k)(δµν − kµkνk2), (2.20)which makes the computations simpler; as a result, we will compute Π(k) fromthe following trace,Π(k) =12Πµµ(k) = 16Π1+16Π2. (2.21)We perform our calculations in Euclidean space and can use the following defini-30tions for the Clifford Algebra in a space with arbitrary dimensions and Euclideansignature. We use the Feynman slash notation defined by /p= γµ pµ .{γµ ,γν}= 2δ µν ,{/p,/q}= 2/p/q. (2.22)With a small amount of manipulation, the above Clifford Algebra could beused to show that in 2+1d, the following identities hold. These identities need toget corrected in other dimensionsγµ/pγµ =−/p,tr(γµγνγργσ ) = 2(δ µνδρσ −δ µρδ νσ +δ µσδ νρ). (2.23)In the next two subsections, we use the above identities and compute Π1 and Π2.The computations are tedious and most of the technical computations are reportedin the appendices instead.2.2.1 Π1We begin with Π1, which is the more complicated contribution to Π. This contri-bution is a faithful two-loop calculation and could not be decomposed into multi-ple one-loop calculations.Π1 =−12〈Tr(/q+ /p)γµ(/q+ /p+/k)γρ(/q+/k)γµ/qγρ(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉(2.24)In Appendix A, we computed the numerator in equation Eq.(A.7) and simplifiedit using the symmetries of the integral that it belongs to. We would need to usethe identities that follow the Clifford Algebra and were listed previously. We copy31the results hereNum1 =Tr(/q+ /p)γµ(/q+ /p+/k)γρ(/q+/k)γµ/qγρ=+(q+ p+ k)2(8k2−4(q+ k)2+2q2+8p2−4(q+ p)2)−2k4−2p4−5k2p2. (2.25)Now, we insert the numerator into the Feynman integral and obtain the followingintegralΠ1(k) =−12〈− 2k4+2p4+5k2p2(q+ p)2(q+ p+ k)2(q+ k)2q2ps+(q+ p+ k)2(8k2−4(q+ k)2+2q2+8p2−4(q+ p)2)(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉≡Π1A+Π1B. (2.26)We have dividedΠ1 into two separate termsΠ1A andΠ1B. We shall compute eachterm in the following subsections.Π1AConsider the first contribution to Π1A,Π1A =−12〈8k2−4(q+ k)2+2q2+8p2−4(q+ p)2(q+ p)2(q+ k)2q2ps〉=+〈 −4k2(q+ p)2(q+ k)2q2ps+2(q+ p)2q2ps− 1(q+ p)2(q+ k)2ps− 4p2(q+ p)2(q+ k)2q2ps+2(q+ k)2q2ps〉. (2.27)The second and the last term have no dependence on external momenta, k, andconsequently must contribute zero to Π1A. Then we are left with the following32three termsΠ1A =〈 −4k2(q+ p)2(q+ k)2q2ps− 1(q+ p)2(q+ k)2ps− 4p2(q+ p)2(q+ k)2q2ps〉.(2.28)The p integral can be done using standard one-loop integrals [53]. Let us copy theformula here∫ d3p(2pi)31[p+q)2]A[p2]B=1(4pi)321[q2]A+B−3/2Γ[A+B− 32 ]Γ[32 −A]Γ[32 −B]Γ[A]Γ[B]Γ[3−A−B] .(2.29)We deploy Eq. (2.29) in Eq. (2.28) which results in another one-loop integral. Thenew integral can be found using the same formula as it is a new one-loop integral.After a few lines of algebra and applying the fundamental identity that definesGamma functions, Γ[A] = (A−1)Γ[A−1], we arrive at the following simple result.Π1A =+1(4pi)32Γ[ s2 − 12 ]Γ[12 ]Γ[32 − s2 ]Γ[ s2 ]Γ[2− s2 ]〈[−4k2(q+ k)2[q2]s2+12− 1(q+ k)2[q2]s2− 12]+1(4pi)32Γ[ s2 − 32 ]Γ[12 ]Γ[52 − s2 ]Γ[ s2 −1]Γ[3− s2 ]−4(q+ k)2[q2]s2− 12〉=−116pi3ks−2[Γ[ s2 − 12 ]Γ[12 ]Γ[32 − s2 ]Γ[ s2 ]Γ[2− s2 ]Γ[ s2 ]Γ[12 ]Γ[1− s2 ]Γ[ s2 +12 ]Γ[32 − s2 ]+Γ[ s2 − 12 ]Γ[12 ]Γ[32 − s2 ]4Γ[ s2 ]Γ[2− s2 ]Γ[ s2 −1]Γ[12 ]Γ[2− s2 ]Γ[ s2 − 12 ]Γ[52 − s2 ]+Γ[ s2 − 32 ]Γ[12 ]Γ[52 − s2 ]Γ[ s2 −1]Γ[3− s2 ]Γ[ s2 −1]Γ[12 ]Γ[2− s2 ]Γ[ s2 − 12 ]Γ[52 − s2 ]](2.30)33Π1A =116pi2ks−2[4(2− s)(1− s) +1(2− s)(3− s) +4(4− s)(3− s)]=116pi2ks−23(20−15s+3s2)(4− s)(3− s)(2− s)(1− s) . (2.31)2.2.2 Evaluation of Π1BNow, consider the other contribution to Π1 given by the following integralΠ1B =12〈2k4+2p4+5k2p2(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉= k4Is+52k2Is−2+ Is−4, (2.32)where we have defined Is asIs =〈1(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉. (2.33)In Appendix C, we evaluate I1 using alpha parameter representation. Here, I1 isan irreducible two-loop integral that turn out to be the master integral for the two-loop calculations. Other two-loop integrals, Is, can be deduced by the well-knownTriangle Relations [53]. Despite the complicated form of the integrals and thetedious calculation that is needed to compute I1, the result turns out to be given bya simple expression. Here we quote the result for I1I1 =164ks+2. (2.34)In the Appendix B, we derive identities that are similar to the Triangle identitiesand use the recurrence relations derived to find an expression for Is−2 and Is−4.34Moreover, Is−2 and Is−4 are given by the following equationIs−2 =− k2 1− s2− sIs−22− s132pi2ks[s2− s +3− s1− s](2.35)Is−4 =k4(3− s)(1− s)(4− s)(2− s)Is+116pi2ks−214− s[s(3− s)(2− s)2 +(3− s)2(2− s)(1− s) −5− s3− s +2− s4− s]. (2.36)Then, using Eq. (2.35) and Eq.(2.36), we obtain a formula for Π1B in terms of Is.In the following sections of this chapter, we use our knowledge of I1 to find anexplicit result for Π1B. Let us report the results for Π1B here, in terms of Is, Is−2and Is−4.Π1B =k4Is+52k2Is−2+ Is−4=[1− 521− s2− s +(3− s)(1− s)(4− s)(2− s)]k4Is+132pi2ks−2{24− s[s(3− s)(2− s)2 +(3− s)2(2− s)(1− s) −5− s3− s +2− s4− s]− 52− s[s2− s +3− s1− s]}. (2.37)2.2.3 Evaluation of Π1A+Π1BIn the previous subsections we foundΠ1A andΠ1B. We combine them here to findan implicit formal for one of the Feynman diagrams self energies, Π1. Althoughthe results seem complex, in the next sections we see that combination of bothdiagrams cancels most of that complexity and leaves a simple result. Let us findΠ1 by combining our results for Π1A and Π1B.35Π1 =Π1A+Π1BΠ1 =+[1− 521− s2− s +(3− s)(1− s)(4− s)(2− s)]k4Is+132pi2ks−2{24− s[s(3− s)(2− s)2 +(3− s)2(2− s)(1− s) −5− s3− s +2− s4− s]− 52− s[s2− s +3− s1− s]}+116pi2ks−2[4(2− s)(1− s)+1(2− s)(3− s) +4(4− s)(3− s)]. (2.38)2.3 Evaluation of Π2In this section, we find the last part of the puzzle and compute Π2. Fortunately,Π2 is reducible and can be found by one-loop methods. Let us recopy Π2 here,for the readers convenienceΠµν2 (k) =−2〈Tr/qγµ(/q+/k)γν/qγρ(/q+ /p)γρ(q+ p)2(q+ k)2[q2]2ps〉. (2.39)We find the trace of Πµν2 (k) and use it for calculating Πµν2 (k) in the last step. Thetrace of Πµν2 (k) is given by the following expressionΠ2(k) =−〈Tr/q(/q+/k)/q(/q+ /p)(q+ p)2(q+ k)2[q2]2ps〉=−2〈2q · (q+ k)q · (q+ p)−q2(q+ k) · (q+ p)(q+ p)2(q+ k)2[q2]2ps〉. (2.40)36First by introducing the Feynman parameter, we do the integral over p and thenperform the integration on the Feynman parameterΠ2(k) =−2Γ[ s2 +1]Γ[ s2 ]∫ 10dα(1−α) s2−1〈2q · (q+ k)q · (q(1−α)+ p)−q2(q+ k) · (q(1−α)+ p)(q+ k)2[q2]2[p2+α(1−α)q2] s2+1〉=−2Γ[s2 +1]Γ[ s2 ]∫ 10dα(1−α) s2〈 q·(q+k)(q+k)2q2[p2+α(1−α)q2] s2+1〉=− 2Γ[s2 +1]Γ[s2 − 12 ]Γ[ s2 ](4pi)32Γ[ s2 +1]∫ 10dα(1−α) s2〈q · (q+ k)(q+ k)2[q2]s2+12 [α(1−α)] s2− 12〉.(2.41)Now, the only variable that is left out to be integrated over is q. After a few stepsof algebra, we arrive at a relatively simple answer for Π2 which we use in the nextsection to evaluate the final form of Π, the full vacuum polarization tensorΠ2(k) =−2Γ[ s2 +1]Γ[ s2 ]Γ[ s2 − 12 ](4pi)32Γ[ s2 +1]Γ[32 ]Γ[32 − s2 ]Γ[3− s2 ]∫ d3q(2pi)3q · (q+ k)(q+ k)2[q2]s2+12=1(4pi)32Γ[ s2 − 12 ]Γ[ s2 ]Γ[32 ]Γ[32 − s2 ]Γ[3− s2 ]∫ d3q(2pi)3k2−q2− (q+ k)2(q+ k)2[q2]s2+12=1128pi2ks−2Γ[ s2 − 12 ]Γ[32 − s2 ]Γ[ s2 ]Γ[3− s2 ][Γ[ s2 ]Γ[1− s2 ]Γ[ s2 +12 ]Γ[32 − s2 ]− Γ[s2 −1]Γ[2− s2 ]Γ[ s2 − 12 ]Γ[52 − s2 ]]=− 18pi2ks−21(4− s)(3− s)(2− s)(1− s) . (2.42)2.4 Combining Π1 and Π2Finally, we have all of the pieces to put together and find the next-to-leading or-der correction to the current-current correlator. We simply add Π1(k) and Π2(k)37derived in the previous sections and deduce the next-to-leading order contributionwe were seekingΠ=Π1+Π2=[1− 5−5s4−2s +(3− s)(1− s)(4− s)(2− s)]k4Is+132pi2ks−2{24− s[ 3s− s2(2− s)2+(3− s)2(2− s)(1− s) −5− s3− s +2− s4− s]− 52− s[s2− s +3− s1− s]+[ 8(2− s)(1− s)+2(2− s)(3− s) +8(4− s)(3− s)]− 4(4− s)(3− s)(2− s)(1− s)}. (2.43)The residue of the pole at s→ 1 cancels out and leaves a finite result for thepolarization tensor. We can therefore combine the terms with the pole in order tocancel the singularity and obtain the following finite result at the region that s isclose to oneΠ=[1− 521− s2− s +(3− s)(1− s)(4− s)(2− s)]k4Is+132pi2ks−2{24− s[s(3− s)(2− s)2 −5− s3− s +2− s4− s]− 52− s[ s2− s]+[2(2− s)(3− s) +8(4− s)(3− s)]+−34+21s−3s2(4− s)(3− s)(2− s)}. (2.44)As it is clear, there is no pole at s= 1 anymore and the above expression is finite.After this recombination, we can safely take the limit s→ 1. Using the result ofI1 = 164ks+2 , we finally arrive at an explicit result for Π, which has a surprisinglysimple form and is given by the following expressionΠ=k64(1− 929pi2). (2.45)38Restoring the overall factors of 16N , the polarization tensor and N, we can find Πµνby using our explicit form for Π. The equation for Πµν is given by the followingΠµν =k4(1− 929pi2)(δµν − kµkνk2). (2.46)Now we can add our next-to-leading order result to the leading-order result andfind the corrected polarization tensorpiµν =[N16+14(1− 929pi2)+ . . .](kδµν − kµkνk). (2.47)For graphene, there are four fermion species in its Lagrangian, N = 4, and thenumeric factor of the leading order contribution is equal to a quarter. The next-to-leading order is suppressed by a factor of 0.03 relative to the leading order. Thissmall numeric factor for the next-to-leading order is a result of a miraculous can-cellation that we observed between the two Feynman diagrams which contributeto the next-to-leading order.Using the derived Kubo formalism, we can find the conductivity of the phasethat does not have a condensate. It is simply given by the following equation fora Dirac semi-metal with N fermion speciesσ =N16+14(1− 929pi2)+ . . . . (2.48)To compare our results with experimental results and the previously found resultsthrough theoretical works, we rescale the gauge field so that our units match thestandard units used by them, A→ eA. For graphene, we find that the conductivityis given byσ =e24(1−0.03)+ . . . . (2.49)392.5 The Current-Current Correlator in Presence ofa CondensateIn this section, we investigate the effect of a scaler field that is coupled to thefermions through a Yukawa term, gφψ¯ψ . We will find the new current-currentcorrelator of the fermions and deduce the conductivity of this phase. As discussedin the introduction, this model describes the phase of our system in which theshort range interactions introduce a condensate in the Lagrangian. After rescalingthe φ and replacing it with φ/g, the action that describes this phase is given byS=+∫d3x[ψ¯aγ t(i∂t+At)ψa+ vF ψ¯a~γ · (i~∇+~A)ψa]+εc2e2∫d4x[1c~E2+ c~B2]+∫d3xφψ¯ψ+ . . . , (2.50)where the kinetic term and the higher order interaction terms of φ field are droppedas they contribute to the higher orders of N. The fermions can be integrated outin the same way that we integrated them out and found Eq.(2.14). The propagatorfor the scaler field (after the same partial re-summation that we did for the vectorfield) would be given with a similar functional form but with a different numericalcoefficient,pileading-order =−N∫ d3k(2pi)3tr[1/k+ /p1/k]=−N∫ d3k(2pi)32k · (k+ p)(k+ p)2k2=−N∫ d3k(2pi)3(k+ p)2+ k2− p2(k+ p)2k2. (2.51)40The calculation of pileading-order is straightforward, we find thatpileading-order = Np2∫ d3k(2pi)31(k+ p)21k2=N(4pi)32pΓ[12 ]Γ[12 ]Γ[12 ]Γ[1]=N8p. (2.52)The next-to-leading contribution to the current-current correlator can be found byusing our results for the leading-order result above. The re-summed propagatorfor the scaler field is given by 8N1p . Using the usual Feynman rules, we find thatthe next-to-leading contributions are given bypiµν(k) = piµν1 (k)+piµν2 (k), (2.53)piµν1 (k) =−8〈Tr(/q+ /p)γµ(/q+ /p+/k)(/q+/k)γν/q(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉, (2.54)piµν2 (k) =−16〈Tr/qγµ(/q+/k)γν/q(/q+ /p)(q+ p)2(q+ k)2[q2]2ps〉. (2.55)As a consequence of the Ward identities, Lorentz invariance, and dimensionalanalysis, the vacuum polarization acquires a particular form. The vacuum polar-ization generally has the form given by the following expressionpiµν(k) = pi(k)(δµν − kµkνk2),which makes the computations simpler. As a result, we will compute pi(k) fromthe following tracepi(k) =12piµµ(k) = 8pi1+8pi2 ,41and then recover piµν(k). The traces of pi1 and pi2 arepi1(k) =−12〈Tr(/q+ /p)γµ(/q+ /p+/k)(/q+/k)γµ/q(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉, (2.56)pi2(k) =−〈Tr/qγµ(/q+/k)γµ/q(/q+ /p)(q+ p)2(q+ k)2[q2]2ps〉. (2.57)We use the trace identities to simplify the traces and write them in terms of innerproducts of the momenta,Num1 =Tr(/q+ /p)γµ(/q+ /p+/k)(/q+/k)γµ/q=+2Tr(/q+ /p)/q(/q+ /p+/k)(/q+/k)+Tr(/q+ /p)(/q+ /p+/k)(/q+/k)/q=+4(q+ p) ·q(q+ p+ k) · (q+ k)−4(q+ p) · (q+ p+ k)q · (q+ k)+4(q+ p) · (q+ k)(q+ p+ k) ·q+2(q+ p) · (q+ p+ k)q · (q+ k)−2(q+ p) · (q+ k)(q+ p+ k) ·q+2(q+ p) ·q(q+ p+ k) · (q+ k)=+6(q+ p) ·q(q+ p+ k) · (q+ k)−2(q+ p) · (q+ p+ k)q · (q+ k)+2(q+ p) · (q+ k)(q+ p+ k) ·q .(2.58)The numerator of pi1 can be written asNum1 =− (q+ p+ k)2(−q2−2(q+ p)2+2p2)− (q+ k)2(−2q2− (q+ p)2+2p2)− p2(2q2+2(p+q)2−2p2− k2) . (2.59)42Using the symmetries of the integral, we can simplify the numerator of pi1 andfindNum1 =− (q+ p+ k)2(−2q2−4(q+ p)2+8p2)+2p4+ p2k2 . (2.60)As before, we divide the terms into two groups, pi1A and pi1B, and evaluate eachindividually,pi1(k) = pi1A+pi1B,in whichpi1A =+12〈−2q2−4(q+ p)2+8p2(q+ p)2(q+ k)2q2ps〉=−〈 1(q+ p)2(q+ k)2ps+2(q+ k)2q2ps− 4p2(q+ p)2(q+ k)2q2ps〉. (2.61)The second term in the second line vanishes and we find thatpi1A =〈 −1(q+ p)2(q+ k)2ps+4p2(q+ p)2(q+ k)2q2ps〉. (2.62)43The evaluation of the remaining parts of pi1A is straightforward. The integrals canbe decomposed into 1-loop integrals, we find thatpi1A =−〈1(4pi)32Γ[ s2 − 12 ]Γ[12 ]Γ[32 − s2 ]Γ[ s2 ]Γ[2− s2 ]1(q+ k)2[q2]s2− 12+1(4pi)32Γ[ s2 − 32 ]Γ[12 ]Γ[52 − s2 ]Γ[ s2 −1]Γ[3− s2 ]−4(q+ k)2[q2]s2− 12〉=− 116pi3ks−2[+Γ[ s2 − 12 ]Γ[12 ]Γ[32 − s2 ]4Γ[ s2 ]Γ[2− s2 ]Γ[ s2 −1]Γ[12 ]Γ[2− s2 ]Γ[ s2 − 12 ]Γ[52 − s2 ]− Γ[s2 − 32 ]Γ[12 ]Γ[52 − s2 ]Γ[ s2 −1]Γ[3− s2 ]Γ[ s2 −1]Γ[12 ]Γ[2− s2 ]Γ[ s2 − 12 ]Γ[52 − s2 ]]=116pi2ks−2[1(2− s)(3− s) −4(4− s)(3− s)]=116pi2ks−23s−4(4− s)(3− s)(2− s) . (2.63)We need to take the limit in which s→ 1, around s= 1 it is given bypi1A =− k96pi2 . (2.64)Now, we consider the other part of pi1 which is given by the following integralpi1B =−12〈2p4+ p2k2(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉=−12k2Is−2− Is−4 . (2.65)44To evaluate pi1B, we use our results in Appendices B and C. Using the recurrencerelations that we found, pi1B can be written aspi1B ==+ k41− s4−2sIs+12− s132pi2ks−2[s2− s +3− s1− s]− k4 (3− s)(1− s)(4− s)(2− s)Is− 116pi2ks−214− s[s(3− s)(2− s)2 +(3− s)2(2− s)(1− s) −5− s3− s +2− s4− s]. (2.66)Again, we take the limit that s→ 1. Around s= 1, pi1B is given bypi1B =+k48pi2[1−1+ s]+k48pi2[76]. (2.67)We found an exact form for pi1 and the only missing piece is pi2, it is given bypi2(k) =−〈Tr/qγµ(/q+/k)γµ/q(/q+ /p)(q+ p)2(q+ k)2[q2]2ps〉=+〈Tr/q(/q+/k)/q(/q+ /p)(q+ p)2(q+ k)2[q2]2ps〉. (2.68)The above expression has the exact form of (2.57). We recopy the results here forreaders’ convenience.pi2(k) =18pi2ks−21(4− s)(3− s)(2− s)(1− s) (2.69)Around s= 1, it is given bypi2(k) =− k48pi2(s−1) (2.70)45Now, we have all of the pieces and we can add them to get pi(k),pi(k)8=pi1A(k)+pi1B(k)+pi2(k)=− k96pi2+k48pi2[1−1+ s]+k48pi2[76]− k48pi2[1−1+ s]=+k72pi2. (2.71)Recovering the tensor form of the self-energy, we find that piµν is given bypiµν =+19pi2(δµν − kµkνk2). (2.72)Now we can add our next-to-leading order result to the leading-order result andfind the corrected polarization tensor. We add the new contribution coming fromthe condensate to the previous results in Eq.(2.48) and find thatΠ¯µν =[N16+14(1− 929pi2)+19pi2+ . . .](kδµν − kµkνk)=[N16−0.008+0.011+ . . .](kδµν − kµkνk). (2.73)Or using the Kubo formalism, we can find the conductivityσ =N16+14(1− 929pi2)+19pi2+ . . . (2.74)46After rescaling the gauge field so that our units match the standard units used, wefind that the conductivity of the phase with the condensate is given byσ =e24(1−0.03+0.04)+ . . . . (2.75)As discussed in the introduction, the dependence of the conductivity on the phasecan be used experimentally to distinguish the phases that the Dirac semi-metallives in.2.6 Experimental ResultsGraphene’s optical conductivity has been determined experimentally by measur-ing the transmittance T and reflectance R of suspended graphene sheets [7–9].The experimental results of these studies affirm that graphene’s conductivity isvery close to the conductivity of non-interacting 2+1d fermion gases.In these experiments, the transmittance (T) and reflectance (R) of a sample aredirectly recorded by passing a beam of light through the sample. The conductiv-ity is then derived using the measured T and R. The authors find the conductivityby using the transmittance through the formula T = (1+ 2piσ/c)−2. They findthat the optical conductivity of a single layer graphene is given by (1.0±0.15) e24h¯ ,(1.0± 0.1) e24h¯ , and (1.01± 0.04) e24h¯ [respectively in [7], [8], and [9]]. These re-sults are very close to the optical conductivity of non-interacting two-dimensionalDirac fermions σ = e24h¯ , illustrated in Fig.2.6. The experimental results are inagreement with our theoretical result for graphene’s optical conductivity with-/without the presence of a condensate. In the phase that the condensate is absent,we find that the conductivity of graphene is given by σ = e24h¯ (1−0.035) and in thephase with the condensate, it is given by σ = e24h¯ (1+0.054). Both of the resultsare in the error interval of the experimental results and more accurate experi-mental results are needed for choosing the phase in which graphene lives in. Asdiscussed in [9], surface contamination of graphene membranes by hydrocarbon47decreases the experiments’ accuracy. Measuring conductivity of cleaner samplesof graphene and also using devices with wider ranges of frequencies can providemore accurate measurements of graphene’s conductivity.Figure 2.6: Adapted from [9]. The red line is the transmittance expected for non-interacting two-dimensional Dirac fermions, whereas the green curve takes into ac-count a nonlinearity and triangular warping of graphene’s electronic spectrum. Thegray area indicates the standard error for the measurements. (Inset) Transmittanceof white light as a function of the number of graphene layers.482.7 ConclusionWe investigated a defect quantum field theory in 3+ 1d with large number offermion flavors and studied the next-to-leading order contributions to the current-current correlation function 〈 jµ(x) jν(y)〉. We found that the next-to-leading ordercontributions from the fermion determinant to the effective kinetic term of pho-tons are significantly suppressed. This suppression helps to validate the largeN expansion of the interacting quantum field theory that models graphene. Ourperturbation scheme seems to compete with the coupling expansion scheme forgraphene. The large N expansion developed here however would work better formaterials with larger number of fermion species. We found that this suppression isa result of a cancellation between two contributing Feynman diagrams. Understat-ing the reason behind this miraculous cancellation might be crucial in understatingthe physics of graphene, which is a strongly interacting system. We studied theconductivity of Dirac semi-metals in presence and absence of a condensate whichmight get turned on due to the short range interactions. The derived conductivityof graphene is very close in both phases and both of the results are inside the rangeof the error bars of the experimental results. To infer the phase that the materiallives in by measuring the conductivity, one needs more accurate experimental re-sults.49Chapter 3The Beta Function of ChargeNeutral 2+1d Dirac Semi-metal inthe Large N3.1 IntroductionThere are both experimental [18]-[21] and theoretical [22]-[45] indications thatthe Coulomb interaction renormalizes the Fermi velocity of the relativistic elec-trons in a Dirac semi-metal. Graphene is the best and most studied example.Experimentally, the dependence of the Fermi velocity on both density and energyhave been investigated independently. The experiments find an increase in theFermi velocity by varying the density of the carriers and approaching the chargeneutral point. However, they find less than 1% change in the Fermi velocity in adecade change of the energy scale. As discussed in Section 3.8, the renormaliza-tion of the Fermi velocity can be thought of it acquiring a non-zero beta function,βvF ≡ dd lnΛvF , and becoming an energy scale dependent parameter. The Fermivelocity changes (runs) from its value of vF ∼ c300 to larger values in the lowerenergies. As well as providing a fascinating example of scale invariance breakingby renormalization in a quantum field theory, it has interesting implications for the50electronic properties of Dirac materials. Within an electron volt of the Fermi en-ergy, neutral graphene is modeled by a 2+1-dimensional quantum field theory withfour species of massless Dirac electrons [51], [52]. As the renormalization in theFermi velocity due to the change in the energy scale is negligible, the free particlepicture of graphene gives a remarkably good description of many of its physicalproperties and it is often said that graphene is a weakly interacting system. Anenduring puzzle [54] is what happens to interactions such as the Coulomb inter-action, which is strong before screening is taken into account, and which is notrelativistic, but do not seem to ruin the relativistic nature of the electron spectrum.The parameter which characterizes the strength of the interaction is the graphenefine structure constant, αg = e24pi h¯vF =e24pi h¯ccvF≈ 300137 ≈ 2.2 (we will set h¯= 1). It islarge, therefore coupling constant perturbation theory is not reliable without somere-summation that takes screening into account. In this chapter, we focus on therenormalization in the Fermi velocity due to change in energy scale and will out-line a mechanism whereby this strong interaction is screened in a Dirac material.We will make use of the large N expansion, which has already been applied tographene [55], and which is very similar to the random phase approximation em-ployed and argued to have good convergence in reference [54]. Our essential newobservation will be that, in the large N limit, the retarded nature of the Coulombforce, and fluctuations of the magnetic field, cannot be ignored. When included,they lead to a significant reduction in the magnitude of the beta function βvF whenN is large, leaving the beta function small. Physically this implies that the mate-rial is quantum critical with conformal and Lorentz invariance.In the instantaneous limit, in which the magnetic part of the action is dropped,the leading order contribution to the beta function is of order 1/N. However, weobserve that the beta function in the presence of both electric and magnetic terms,in the non-instantaneous limit, gets suppressed by a factor of N. We find that inthis case, the Lorentz symmetry of the leading contributions to the beta functionmakes it vanish. We find that in the presence of the full Maxwell action, the lead-51ing contribution to the beta function of our model is of order 1/N2.We will idealize Coulomb interacting graphene as an infinite 2+1d defect embed-ded in 3+1d space-time that was explained in the introduction chapter. Grapheneelectrons, with their emergent massless relativistic dispersion relations, are the de-grees of freedom occupying the sheet. They interact by exchanging photons whichare allowed to propagate in the surrounding bulk of space-time. We describe thistheory by the defect quantum field theory with the Euclidean space actionS=∫d3x[ψ¯aγ t(i∂t+At)ψa+ vF ψ¯a~γ · (i~∇+~A)ψa]+εc2e2∫d4x[1c~E2+ c~B2]. (3.1)The integral in the first line is over the 2+ 1d defect with the integrand beingthe Lagrangian density of N species of two-component spinor fermions. It has aglobal U(N) symmetry. For graphene, we should set N = 4 to compare our re-sults with the experimental results for graphene’s Fermi velocity renormalization.There are topological insulators which are also described by defect field theoriesof this kind with various values of N. The second line in Eq.(3.1) is the EuclideanMaxwell action integrated over the bulk of the 3+1d space-time surrounding thedefect.The fields (At ,~A) are the usual Maxwell theory vector potential and the electricand magnetic fields are ~E = ~˙A−~∇At and ~B= ~∇×~A, respectively. The vector po-tential also appears in the defect fermion action in the first line of Eq.(3.1) whereit is minimally coupled to the fermion field. Note that the emergent semi-metal’sspeed of light, the Fermi velocity vF , differs from the vacuum speed of the photonc which appears in the Maxwell action. Here c is the speed of light in the sur-rounding medium which we are assuming is the same on both sides of the defect.It would be straightforward to generalize our consideration to the case where it isdifferent.52In the following, we will be interested in an expansion of the quantum field theorydescribed by the action in Eq.(3.1) in the limit where N→∞. Of course, as statedabove, to describe graphene N should be set equal to 4. One might question thevalidity of expanding about the large N limit. The value of N = 4 is near the strongcoupling regime of the 1N expansion, although it is thought to be at least slightlyabove the value of N where the U(N) symmetry is broken dynamically by stronginteractions. That critical value of N has several estimates, all of which are lessthan 4 [10], [56], [57], [58]. Some topological insulators can be described by thisaction, and they have N = 2, or even N = 1, which means their large N expansionsare in the strongly coupled regime where dynamical symmetry breaking of chiralsymmetry is expected.The photon field, whose action is in the second line, interacts with the electronby minimal coupling. The parameter vF is the Fermi velocity and ε and c are thedielectric constant and the speed of light in the substance which surrounds the de-fect. The Dirac matrices are 2×2 and obey {γµ ,γν}= 2δ µν . To model graphene,as well as taking N = 4, we should set the Fermi velocity to vF ≈ c/300 at thescale of the ultraviolet cut-off, ≈ 1A˚−1.The quantum field theory in Eq.(3.1) has no dimensional parameters and it isscale invariant at the classical level. However, renormalization will introduce ascale. It is also clear that, since the photon field is free field theory in the regionasymptotically far away from the defect, the coefficients of the E2 and B2 termsdo not renormalize. In the end, we shall only require renormalization constantsfor the operators z0ψ¯γ t(i∂t +At)ψ and z1vF ψ¯~γ · (i~∇+~A)ψ . Because the modelis not Lorentz invariant, these terms can have different renormalization constants.Their ratio defines a renormalization of the Fermi velocity, v˜F ≡ z1z0 vF . This renor-malization is logarithmic in the cut-off and the Fermi velocity becomes a scale-dependent running coupling constant. The perturbative beta function, in the limit53Figure 3.1: The Feynman diagram of the expansion of the fermion deter-minant is depicted. The series is even due to particle-hole and timereversal symmetry. The Feynman integrals for diagrams with morethan two legs are finite.where c/vF is very large has been computed to two-loop orderβvF =αg4vF +[N12− 10396+32ln2]α2gvF + . . . ,and for graphene can be approximated as βvF ' 2vF .As discussed previously, in this chapter we use a different perturbation schemeand use the large N expansion that we developed in the previous chapter to studythe running of the Fermi velocity in our Lagrangian. To implement the largeN expansion, we integrate out the fermions to get the effective field theory forthe gauge fields. The fermions contribute terms in the effective action given bythe Feynman diagrams in the series depicted in Fig. 3.1. The model describingour Dirac semi-metal after integrating out the fermions is given by the following54actionSeff =N16∫d3x~E · 1/vF√−~∇2− 1v2F∂ 2t~E+~B vF√−~∇2− 1v2F∂ 2t~B+N∞∑n=4∫dx1...dxnAµ1(x1) . . .Aµn(xn)Γµ1...µn(x1, ...,xn)+εce2∫d3x[~E · 1/c√−~∇2− 1c2∂ 2t~E+~B c√−~∇2− 1c2∂ 2t~B]+λ∫d3x(vF~∇ ·~A+ A˙t)2−∫d3xη¯a 1z0γt(i∂t+At)+z1vF~γ·(i~∇+~A)ηa. (3.2)The first line in Eq.(3.2) is the expansion of the fermion determinant to quadraticorder in the gauge fields. It has a factor of N in front of it. This term containsthe leading order screening of the Coulomb interaction by the relativistic elec-trons, and we will refer to it as the “screening-generated term”. The second linein Eq(3.2) contains the higher order, multi-photon interaction terms coming fromthe fermion determinant. These terms are of order N. To emphasize this, we haveexplicitly extracted a factor of N. The Γµ1...µn(x1, ...,xn) are connected irreduciblemulti-photon correlation functions arising from a single species of electrons, com-puted to one-loop order. Due to charge conjugation (particle-hole) symmetry, thisis an even series, and it begins at order four (with photon-photon scattering). Inthe third line of equation Eq.(3.2) , we have presented the same Maxwell theory ofthe photon as in the second line of equation Eq.(3.1), but dimensionally reduced to2+1d, with the component of the photon field that is perpendicular to the defecteliminated using its linear equation of motion. Note that this action has becomenon-local and resembles the screening-generated term in the first line, albeit witha different speed of light. The fourth line of Eq.(3.2) is a gauge-fixing term. In thelast line of Eq.(3.2), we have introduced anti-commuting sources ηa, η¯a in orderto generate correlation functions with fermion fields.55The first two lines are of order N and they dominate the large N limit. Thevacuum kinetic term for the photon in the third line is ignored to the leading orderin the large N limit. The theory obtained by retaining the first two lines of Eq.(3.2)is a Lorentz invariant conformal field theory whose speed of light is vF . The onlyrenormalization is that of the fermion wave-function and, because of the Lorentzinvariance, the temporal and spatial parts get the same renormalization contribu-tions, z0 = z1. This can be observed from the Feynman diagram in Fig.3.2 wherethe dotted line is the inverse of the suitably gauge fixed quadratic form in the firstline of Eq.(3.2). This contribution could give the fermion operators non-trivialanomalous dimensions. The beta function for vF vanishes at order 1/N as a con-sequence of Lorentz invariance, with vF playing the role of light velocity.Both Lorentz and conformal invariance are broken by turning on the third linein Eq.(3.2). This results is a nonzero beta function βvF . The leading contribution,which is depicted in Fig.3.2, is of order 1/N2. Again, because of the Lorentzinvariance of the remaining order 1/N2 contributions to the fermion self-energy,the other self-energy corrections do not contribute to βvF . Computing the betafunction simply entails computing the ultraviolet divergent part of the contributiondisplayed in Fig.3.2. We will show that the infrared regime does not contribute tothe beta function to the order of 1/N2.3.2 Corrections to the Electron PropagatorThe symmetries of the system, including U(N), parity and time reversal, spatialrotation, and charge conjugation invariance limit the matrix form of the fermionpropagator. If we define〈0|Tψa(x)ψ†b (y)|0〉=∫ d3k(2pi)3e−ik·(x−y)Sab(k), (3.3)56Figure 3.2: The leading contribution to the beta function in the large N limitcomes from the Feynman diagram where the dotted line is the relativis-tic large N propagator and the insertion into the photon propagator isthe tree-level classical Coulomb action which is non-relativistic. Thisdiagram is of order 1/N2.thenS−1(k)ab = δab[Σt(k0, |~k|)γ tk0+Σs(k0, |~k|)vF~γ ·~k]. (3.4)Generally, the electron propagator is highly gauge dependent. However, we expectthe pole in the propagator to be gauge invariant. This suffices to define the speedof the fermions. In fact, the dispersion relation is the solution of the followingequationΣt(iω(k),k) iω(k) = ± Σs(iω(k),k)vFk. (3.5)We rewrite the above formula asvF(k) = vFΣs(iω(k),k)Σt(iω(k),k)in which vF is the bare Fermi velocity and vF(k) = vF + δvF(k). In perturbationtheory, ω(k) = vFk+δvF(k)k and we can write the self-energies asΣt = 1+δ z0+δΣt(k0,k) , Σs = 1+δ z1+δΣs(k0,k). (3.6)57δ z0 and δ z1 are the contributions coming from counter-terms δ z0ψ¯γ t(i∂t +At)ψand δ z1vF ψ¯~γ ·(i~∇+~A)ψ . Then we can plug the above into the dispersion relationand findδvF(k) = δ z1+δΣs(ivFk,k)− [δ z0+δΣt(ivFk,k)] . (3.7)The counter-terms δ z0 and δ z1 must be chosen so as to cancel the logarithmicallyultraviolet divergent terms in Σt and Σs, respectively. Then, the finite part of δ z1−δ z0 can be chosen by the condition that vF is the Fermi momentum at some scalek = µ . ThenδvF(k) =δΣs(ivFk,k)−δΣt(ivFk,k)−δΣs(ivFµ,µ)+δΣt(ivFµ,µ). (3.8)In deriving the renormalization contributions we use a specific gauge, and oneneeds to make sure that the final result that describes the physics is gauge invariant.Fortunately, using the extended BRST symmetry of the original action Eq.(3.1),Nielsen identities [59] can be derived. From the Nielsen identities, in the absenceof infrared divergences, it can be shown that the poles of the fermion self-energyare gauge invariant [60], [61]. Nielsen identities can be written as independenceof the on-shell self-energy from the gauge parameter λ , ∂Σ−1∂λ = 0. Implementingthis condition on our expression for the self-energy shows that the contributionto the Fermi velocity renormalization is gauge invariant. In the presence of in-frared divergences, the gauge invariance of the Fermi velocity is not guaranteedby Nielsen identities. We investigate infrared divergences in the following sec-tions and conclude that they are not present. As we discuss in the section devotedto an investigation of infrared divergences, turning on the Coulomb interactionregulates the infrared divergences and leaves the Fermi velocity gauge invariant.583.3 Electron Self-EnergyWe now start our perturbative analysis of the corrections to the electron prop-agator. The fermion field gets a divergent (and gauge dependent) wave-functionrenormalization. This should be canceled by adding local counter terms δ z0ψ¯γ0(i∂ t+At)ψ and δ z1ψ¯~γ · (~∇+~A)ψ to the original Lagrangian. As we discussed, renor-malization of the speed of light appears because of the different contributions tothe spacial and temporal parts of the electron wave-function. Logarithmic di-vergences in the wave-function coefficients result in infinite renormalization ofgraphene’s speed of light and makes it a scale-dependent parameter with a betafunction. For readers’ convenience, we recopy our effective action Eq.(3.2) hereS=εe2∫d3x~E 1√−~∇2− 1c2∂ 2t~E+~Bc2√−~∇2− 1c2∂ 2t~B+N16vF∫d3x~E · 1√−~∇2− 1v2F∂ 2t~E+~B v2F√−~∇2− 1v2F∂ 2t~B . (3.9)We have dropped the other terms as they only contribute to higher orders in 1/N,and we set the source equal to zero. The gauge term will be discussed in thefollowing sections where we will choose a gauge that makes the computationsmore convenient. The vertex which couples the electron to the photon is Γ =(γ0v,γivF). We use this vertex to get the leading correction to the electron self-energyΣ(p0, p) =∫ d3q(2pi)3∆µν(q)Γµ [γ0(p0−q0)+vFγ · (~p−~q)]Γν(p0−q0)2+v2F(~p−~q)2(3.10)where ∆µν(q) is the photon propagator. ∆µν(q) is clearly gauge dependent. Weneed to fix a gauge to go further; we choose a gauge that makes the term inducedby fermion 1-loop summation diagonal. This can be accomplished by imposingthe gauge condition v2F∇ ·~A+ ∂tA0 = 0. In Appendix D, we derive the propaga-59tor and check its instantaneous limit versus the previously studied instantaneouspropagator. We find that our propagator indeed reduces to the instantaneous prop-agator in the instantaneous limit. After imposing our gauge, we can find that thenew propagator for the gauge field has the following complicated form∆(p0,~p) =16Nc2pP+ξ p20η1p2ξη1p0p1p2ξη1p0p2p2ξη1p0p1p2η4+ξ 2Pp21−ξc2p(p20+c2p22)p2η1η2ξη3η1η2p1p2)p2ξη1p0p2p2ξη3η1η2p1p2p2η4−ξc2p(p20+c2p21)+ξ 2Pp22p2η1η2 , (3.11)η1 = (ξ p+ c2P),η2 = (c2p+ξP),η3 = (c4p+ξP),η4 = c4p2P+ξc2p(p2+P2). (3.12)We have set vF = 1 and defined P2 = p20+ c2−→p 2, p= p20+−→p 2 and ξ = 32εc3e2N .To extract information about the running of the Fermi velocity, vF , due to the in-teractions, we should rewrite the calculated form of the fermion self-energy in theform of Σ= Σt p0γ0+Σs~p.~γ . The leading divergent correction to the fermion self-energy (2-point function) comes from the Feynman diagram depicted in Fig.3.2.This indeed has a logarithmic divergence which should be canceled by the counterterms. However, the leading term of order 1/N in Eq.(3.9) is equivalent to the limitin which 1e2 → 0. This contribution is Lorentz invariant resulting in δΣt = δΣs and,consequently, δvF = 0. To get a non-Lorentz invariant contribution, we must re-store the electrodynamic terms (finite 1e2 ) to the action, where we can consider theadded term as a perturbation. Their leading contribution appears in the diagramsdepicted in Fig.3.2, and is of order 1/N2.60Orthogonality of the γ matrices can be used to check that p0Σt = 12tr(γ0Σ) andpiΣs = 12tr(γiΣ). Ultimately, to find the dispersion relation we need to imposethe on-shell constraint. In the following computations to evaluate δvF , we canmake derivations easier by imposing q2 = 0 and restricting the variables to liveon-shell. As we are computing the first non-zero contributions to the free fieldtheory dispersion relation, we can safely use the zeroth order dispersion relation(q20+q21 = 0) for imposing the on-shell constraint. With a small amount of manip-ulation, the following simplified expressions for the δΣs and δΣt coefficients canbe found. Let’s start with computing δΣs, the spatial part of the self-energyδΣs =− 16Nq1∫R3|−→p |d|−→p |dp0dt(2pi)3{|−→p |(c4p(−pP+ξ |−→p |2)+ξc2p(−P2+ p0(p0−2iq1))+ξ 2P(p20+ |−→p |2−2ip0q1))cos t+q1(c4p2P+ξ 2Pp20+ξc2p(P2+ p20)−ξ (c4p+ξP)|−→p |2 cos2t)} 1p20+−→p 2−2ip0q1−2|−→p |q1 cos t1p2(c2p+ξP)1(ξ p+ c2P)(3.13)in which we have defined p2 = |−→p |sin t and p1 = |−→p |cos t. We first integrate overthe angle variable t, and then scale the internal momentum by q1. The scalingin q1 cancels the 1/q1 factor in Eq.(3.13)(q1 appears in bounds of integral, thehard cut-off regulator). Due to our interest in the ultraviolet limit, we then expandthe result in momenta and keep only the divergent terms. The original integral islinearly divergent, however, the linearly divergent term vanishes and leaves only alogarithmic divergence as the leading term. In other words, the first order terms of61the δΣs Taylor expansion around small q1 are the only divergent terms that surviveδΣs =− 16(2pi)2N∫ ∞0|~p|d|~p|∫ ∞−∞dp0(c4p20√p20+ c2|~p|2+ξ 2(p20+ |~p|2)√p20+ c2|~p|2+ξc2√p20+ |~p|2(2p20+ c2|~p|2))1ξ + c2√p20+c2|~p|2p20+|~p|21c2+ξ√p20+c2|~p|2p20+|~p|21(p20+ |~p|2)3. (3.14)To proceed with the evaluation of the spatial part of the self-energy, we perform achange of variable by defining p0 = x|~p|. The integral can be separated in termsof x and p integrations; the x integral is finite and the p integral is logarithmicallydivergentδΣs =− 16(2pi)2N∫ ∞0d|~p||~p|∫ ∞−∞dx(c4x2√x2+ c2+ξ 2(x2+1)√x+ c2+ξc2√x+1(2x2+ c2)) 1ξ + c2√x2+c2x2+11c2+ξ√x2+c2x2+11(x2+1)3. (3.15)We regulate the integral by introducing a hard cut-off Λ. The result can be sum-marized asδΣs =− 16(2pi)2N[{ξ√ξ 2− (1+ξ 2)c2+ c4(−2ξ 2+ξ (−1+2ξ )c2+2c4)− c2√−ξ 2+ c2(2c4+ξ 2(−2+ c2)) tan−1[√−1+ c2]+2c3(ξ 2− c4)(tan−1[ ξ√−1+ c2c√−ξ 2+ c2 ]− tan−1[c√−1+ c2√−ξ 2+ c2 ])}1ξ 3(−1+ c2) 32√−ξ 2+ c2]ln[Λ/q1]. (3.16)The temporal part of the self-energy, δΣt , can be calculated in the same way asδΣs. As the calculation is fairly similar to the calculation for the spatial part, we62simply report the result of the calculationsδΣt=− 8N(2pi)21ξ 3c2[{4√−ξ 2+ c2(−ξ 2c4+2c8+ξ 4(−1+ c2))tan−1√−1+ c2+4c4((ξ 2−2ξc2)√(−1+ c2)(−ξ 2+ c2)− c(−2c4+ξ 2(1+ c2))(tan−1[ξ√−1+ c2c√−ξ 2+ c2 ]− tanh−1[c√−1+ c2√−ξ 2+ c2 ]))} 1(−1+ c2) 32√−ξ 2+ c2)+4ξ 5{tan−1[ξ√−1+ c2√−ξ 2+ c6 ]− tanh−1[c2√−1+ c2√−ξ 2+ c6 ]}1√(−1+ c2)(−ξ 2+ c6)] ln[Λ/q1].(3.17)Recall from the previous sections, the measurable quantity which needs to begauge invariant is the difference of these two coefficients. The Fermi velocitybeta function, which describes the rate of change of the Fermi velocity with achange of scale can be found by applying its definition βvF =δvFδ lnΛ . We compareour results with the results in [55] by taking the instantaneous Coulomb interactionlimit ( 1c2 → 0) [To compare our results, we restoring vF by the help of dimensionalanalysis.]. The beta function in the instantaneous limit is give by the followingexpressionβ instvF (v) =−vNpi2(8+4pivξ [−1+√v2ξ 2−1+ v2ξ 2 ]+8v2ξ 2ArcCoth[√1− v2ξ 2]√1− v2ξ 2).(3.18)Taking into account that in [55] the author works in a four dimensional represen-tation of the Clifford algebra instead of our two dimensional representation, theresults are in agreement (N f = 2 in [55]). In the last section, we will come backto the beta function in this limit, and whether there will be any contributions fromthe infrared regime.63The beta function can be written in terms of vF by taking c→ 1vF while keepingεce2 fixed, and then setting c = 1. We find that the Fermi velocity beta function isgiven by the following complicated functionβvF (v) =−4v3ξ 3Npi2((−ξ√(ξ 2− v2)(−1+ v2)(−6v2+3ξv2+2ξ 2(−1+ v2))+v√−ξ 2+ v2(−6v2+2ξ 4(−1+ v2)+ξ 2(−1+4v2)) tan−1[√−1+1/v2]−2v2{−3v2+ξ 2(1+2v2)}(tan−1[√1− v2√−ξ 2+ v2 ]− tan−1[ξ√1− v2√−ξ 2+ v2 ]))1(−1+1/v2) 32 v5√−ξ 2+ v2+2ξ 5 tanh−1[√1−v2√1−ξ 2v2 ]− tanh−1[ ξ√1−v2√1−ξ 2v2 ]√(−1+ v2)(−1+ξ 2v2))(3.19)where, as before, ξ = 32εe2N . We have plotted our improved beta function versusthe beta function found in [55] in Fig.3.3. As we expected, they start to deviate atlarger Fermi velocities when the retarded nature of the interactions becomes moreimportant.It is instructive to compare the derived beta function for various values of N,the number of fermions. For three different values of N = 4,10 and 100, we haveplotted the beta function in Fig.3.4. The dashed lines correspond to the instanta-neous approximation results. We observe that the beta function gets smaller forlarge N, and the difference between the instantaneous approximation and the in-clusion of the full Maxwell Lagrangian becomes more significant.Let us look at the large the N expansion of our beta function. We need toexpand the complicated expression above in terms of ξ as it depends on N thoughξ = 32εe2N . We find that, as expected from the Lorentz invariance of the lowest order64Figure 3.3: We have plotted the beta function in [55] (purple) vs. Eq.(3.19) (orange).As one expects in the limit of v→ 1 (here we have chosen the units such that lightvelocity is our measure for velocity), we find that the Lorentz symmetry preventsthe Fermi velocity from running. The beta function in [55] violates this conditionas its Lorentz symmetry is violated by construction but not the presence of the twovelocities in the Lagrangian.term in 1/N, the first non-zero contribution is of order 1/N2βvF (v) =−1N232εe2pi2[(5−2v2)+(5−16v2+8v4) tan−1[√1− v2/v]v√1− v2]. (3.20)Although this expansion is more accurate than the expansion in [55] for large N,for graphene N is only four, resulting in the above term to be of order 10, and as aresult the expansion in [55] works better.650.2 0.4 0.6 0.8 1.0v-0.0015-0.0010-0.0005ΒFigure 3.4: We have plotted the beta function in [55](dashed lines) vs. Eq.(3.19) (solidlines). The orange, purple and green lines respectively correspond to N = 4,10,100.3.4 Infrared Contributions to the Fermi VelocityBeta FunctionIn this section, we investigate the model with the instantaneous Coulomb inter-action ( 1c2 → 0) with more caution. We will see that the fermion self-energy, toleading order, suffers from infrared (IR) divergences. These infrared divergencesget regulated by the presence of the Coulomb interaction automatically, and thefinal result is infrared divergence free. Let’s start by writing down the action inthis limit. By looking at the coefficient of the magnetic kinetic term, it can beseen that in the instantaneous limit (1c → 0) , the fluctuations of the spatial com-ponent of the gauge field get suppressed and only the temporal part survives. The66instantaneous action can then be written asSinst =∫d3x 1g2~E1√−~∇2~E+N32vF~E 1√−~∇2− 1v2F∂ 2t~E (3.21)where we have defined 1g2 =εe2 . We are interested in studying the infrared behaviorof the following integral, setting vF = 1 we find that the fermion self-energy isgiven byΣ=−∫ d3q(2pi)3γ0(/p−/q)γ0(p0−q0)2+(p−q)21N16q2√q20+q2+ 2qg2=−∫ d3q(2pi)3(p0−q0)γ0− (~p−~q).~γ(p0−q0)2+(p−q)21N16q2√q20+q2+ 2qg2. (3.22)We have used Feynman slash notation defined by /p = γµ pµ , and for the rest ofthis chapter we redefine p2 as p2 = p21+ p22 so that it only has the spatial compo-nents of the momenta in it. In the infrared regime, we expect q0 and q terms in thenumerator not to contribute (at least for the most divergent term). Consequently,the coefficients of γ0 and ~γ are just different in sign. We checked this by explicitintegration of the angle variable using the complex coordinates and looking at theregime where q0 and q are both small. Before getting into the full calculation ofthe self-energy in the presence of the Coulomb interaction, let us do an approx-imate calculation. In the infrared regime, q p, the fermion self-energy can beapproximated by a simplified integral given byδΣIR =−2pi∫ ∞0∫ ∞−∞dqdq0(2pi)3[1p20+ p2 +p20−3p2(p20+ p2)3q20]p0γ0−~p.~γN16q2√q20+q2+ 2qg2, (3.23)suggesting that in the limit of 1g2 → 0, the fermion self-energy is infrared diver-gent. As we discussed before, vF renormalization depends on δΣt − δΣs. As a67result of the above observation, the running of vF is proportional to both δΣs andδΣt (δΣt=-δΣs). By turning off the Coulomb interaction, 1g → 0, both δΣt andδΣs suffer from infrared divergences, therefore vF will also suffer from the samedivergences. We will see that the Coulomb interaction smooths out these diver-gences.Now, it is time for a rigorous treatment of the presence of possible infrareddivergences. We start by integrating over the angle variable. We choose the ex-ternal momenta to have no component in the y direction, (p0, p,0). Using thissimplification, we can simplify δΣt and δΣs a bit. δΣt is given byδΣt .p0 =∫ ∞0∫ ∞−∞∫ 2pi0qdqdq0dφ(2pi)3−1N16q2√q20+q2+ 2qg2(p0−q0)(p0−q0)2+ p2+q2−2pqcosφ=∫ ∞0∫ ∞−∞qdqdq0(2pi)3∮ dziz1N16q2√q20+q2+ 2qg2(q0− p0)(p0−q0)2+ p2+q2− pq(z+ z−1)=∫ ∞0∫ ∞−∞dq2dq0(2pi)31N16q2√q20+q2+ 2qg2pi(q0− p0)√((p−q)2+(p0−q0)2)((p+q)2+(p0−q0)2).(3.24)We can use the same trick to integrate the angle variable in the spatial contributionto the self-energy. δΣs is then given by68δΣs.p=∫ ∞0∫ ∞−∞∫ 2pi0qdqdq0dφ(2pi)31N16q2√q20+q2+ 2qg2(p−qcosφ)(p0−q0)2+ p2+q2−2pqcosφ=∫ ∞0∫ ∞−∞qdqdq0(2pi)3∮ dziz1N16q2√q20+q2+ 2qg2(p−2q(z+ z−1))(p0−q0)2+ p2+q2− pq(z+ z−1)(3.25)=∫ ∞0∫ ∞−∞dqdq0(2pi)31N16q√q20+q2+ 2g2[ pip (p2−q2− (p0−q0)2)√((p−q)2+(p0−q0)2)((p+q)2+(p0−q0)2)+1].By expansion of these equations in the region where p q, we are able to verifythe results in [55]. As is mentioned in [55] , there is no infrared divergence inthat regime; however, this does not prove that the theory is infrared divergencefree. By zooming into this particular regime, we only check that a given theorywith an infrared limit that corresponds to the ultraviolet regime of Eq.(3.21) willbe infrared divergent free, as illustrated in Fig.3.5.Figure 3.5: By zooming into p q regime, we only check that a given theory with aninfrared limit that corresponds to the ultraviolet regime of Eq.(3.21) will be infrareddivergent free.The integrals could be expanded in terms of 2g2 , but the expansion would notbe well defined. As the integrals are infrared divergent at 2g2 → 0, the perturba-69tive expansion would be invalid and non-perturbative terms would show up. Byexploring the most dominant terms in the strict infrared limit, we see that loga-rithmic terms in g2 show up. As a result, we can expand the self-energies in thefollowing manner and look at their behavior in the infrared regime.δΣIRt = αt ln[g−2]+O(g−2), (3.26)δΣIRs = αs ln[g−2]+O(g−2). (3.27)αt and αs can be found by taking the logarithmic derivative of Σt and Σs respec-tively.αt =∂Σt∂ ln(g−2)=− 1p0lim1g2→0∫ ∞0∫ ∞−∞dqdq0(2pi)34pi(p0−q0)g−2√(p−q)2+(p0−q0)2(N16q√q20+q2+ 2g2)−2√(p+q)2+(p0−q0)2=− 1p0∫ ∞0∫ ∞−∞dqdq0(2pi)32pi(p0−q0)√(p−q)2+(p0−q0)2δ ( Nq16√q20+q2)√(p+q)2+(p0−q0)2(3.28)αt =− 1p0∫ ∞0∫ ∞−∞dqdq0(2pi)32pi(p0−q0)√(p−q)2+(p0−q0)216√q20+q2√(p+q)2+(p0−q0)21Nδ (q)=− 1p0∫ ∞−∞dq0(2pi)32pi(p0−q0)p2+(p0−q0)216|q0|N. (3.29)Above, we have used the following representation of the Dirac delta distributionto push the calculations forwardδ (x) = lima→0a(x+a)2.As the computation for the spatial part is similar to the temporal part, we report70only the results here. The spatial part can be found in the same way, and we copythe resultαs =∫ ∞−∞dq0(2pi)32pip2+(p0−q0)216|q0|N. (3.30)The remaining integrals can be evaluated explicitly as well,∫ dx (a− x)xb2+(a− x)2 =−x−b tan−1[a− xb]− a2ln[b2+(a− x)2],∫ dx xb2+(a− x)2 =−abtan−1[a− xb]+12ln[b2+(a− x)2],and we find that the difference between the α coefficients is given byαt−αs =− 4pi2Np20+ p2p0p(tan−1[Λ− p0p]+2tan−1[p0p]− tan−1[Λ+ p0p]),(3.31)where Λ is the ultraviolet cutoff. Although each of αs and αt are ultraviolet di-vergent, the subtraction is ultraviolet finite. Then, at the limit that the ultravioletcut-off goes to infinity, we have αt −αs = − 8pi2N 1p0p(p20 + p2) tan−1(p0p ). If wecast this result into our expansion, and restore the vF by dimensional analysis, wewould find that the infrared contribution to the Fermi velocity renormalization isgiven byδ IRvF =− 8pi2Np20+ v2F p2p0ptan−1[1vFp0p] ln[g−2vF ]. (3.32)The above equation should be evaluated on-shell, for which we need to use thedefinition of the renormalized Fermi velocity p20 +(vF + δvF)2p2 = 0. We findthat the infrared contribution to the renormalization of the Fermi velocity is finite,and is independent of the infrared cut-off. However, it is worth noticing that the71infrared contribution to the renormalization equation of vF starts with a higherorder in 1/N when comparing to the ultraviolet contributions. We observe that inthe instantaneous model the contributions from the ultraviolet divergences are ofthe order 1/N, δUVvF =− 8pi2N vF δΛΛ , however, δ IRvF is of order O(lnN/N2) .As the infrared contribution to the self-energy turned out to be finite and in-dependent of the infrared cut-off, it does not contribute to the beta function. Thislack of contribution can be observed by the use of Eq.(3.8) and observing that theinfrared terms cancel out due to their independence of the scale.3.5 Experimental ResultsThe Fermi velocity in graphene has been measured through measurements oftransmittance T and reflectance R of suspended graphene sheets [7], scanning tun-neling spectroscopy [19, 20], angle-resolved photo-emission spectroscopy mea-surement of the Dirac cones [21] and effective cyclotron mass [18] . Experimen-tally, the dependence of the Fermi velocity on both density and energy have beeninvestigated independently. The experiments find an increase in the Fermi veloc-ity by varying the density of the carriers and approaching the charge neutral point.The most prominent renormalization of the Fermi velocity is seen in its de-pendence on density. In [18], the authors used the cyclotron mass to measurethe Fermi velocity. A significant renormalization of the Fermi velocity is ob-served in this experiment, an enhancement by a factor of 3 at the lowest density,n' 109cm−2, of electrons that was reached by [18]. As discussed in the introduc-tion, in theory presented above, we are interested in not the change of the Fermivelocity by changing density but instead for fixed density as a function of energy,our calculations were performed assuming zero density. Although, the Fermi ve-locity shows substantial renormalization by changing the density, its behavior un-der change of energy scale at fixed density can be quite different. As such, we need72to compare our results with the experiments that investigate the renormalizationof the Fermi velocity by changing the energy scale of the charge-neutral graphenein which the density is minimal (n 1015cm−2) and fixed. In this measurementhowever, they do not investigate the renormalization of the Fermi velocity at con-stant densities and their results can not be used as an experimental check for ourresults.Figure 3.6: Adapted from [18]. (a) Cyclotron mass as a function of Fermi wave-vector.The dashed curves are the best linear fits with assumption that mc ∼ n 12 . The dot-ted line is the behavior of cyclotron mass derived from the standard value of Fermivelocity. Graphene’s spectrum renormalized due to electron-electron interactions isexpected to result in the dependence shown by the solid curve. (b) Cyclotron massplotted as a variable of vF .In [20], the authors studied the renormalization of the Fermi velocity in grapheneusing scanning tunneling spectroscopy and measuring the dispersion relation ofthe excitations as a function of their energies. In graphene, the linear dispersionyields the Landau level (LL) spectra that is given by EN =E0+vF sgn(N)√2eh¯NB,where E0 is the energy for the zeroth level, N is the landau level number and Bis the magnetic field. This dependence allows us to study the dispersion of Fermivelocity at fixed density and magnetic field by studying the energy bands for dif-73ferent LLs. Chae et al. investigated the LLs of graphene and found that at fixeddensity and magnetic field the dispersion of the excitations is highly linear in√N,illustrated in Fig.3.7, and the Fermi velocity can be extracted consistently fromEN =E0+vFsgn(N)√2eh¯NB . As a result of the linearity, we can infer that chang-ing the energy scale does not renormalize the Fermi velocity substantially and theexcitations do not see the effect of the interactions in their dispersion. Chae et al.extended their study by investigating the Fermi velocity dependence on density.They found that by changing the density and fixing the energy scale, the Fermivelocity of the excitations gets renormalized. They confirmed the previous resultsthat suggested the renormalization of the Fermi velocity due to changes in density.From the inset in Fig.3.7, we observe that the Fermi velocity has less than 1%renormalization in a decade change of the energy scale. Our calculations show arenormalization of 0.2% in a decade, which is consistent with the experimentalresults.3.6 ConclusionWe conclude this chapter by summarizing our main results. It was shown that2+ 1d fermions in large N interacting via 3+ 1d electrodynamics acquire a betafunction for their Fermi velocity. Due to Lorentz invariance, the beta functionstarts at second order in 1/N, in contrast to models with an instantaneous Coulombinteraction that acquire a beta function of order 1/N. We made a careful inves-tigation of the infrared regime. It was argued that possible infrared divergencesget regulated by the presence of the Coulomb interaction, and the infrared regimeis divergence free. The gauge invariance of the Fermi velocity in presence of theinteractions was discussed, and it was argued that, based on the absence of theinfrared divergences, the Fermi velocity is gauge invariant.74Figure 3.7: Adapted from [20]. N = 1 to N = 6 LLs’ energy as a function of level numberfor different values of carrier density and B= 2T . For fixed density the curves arehighly linear, resulting in a possible negligible renormalization of the Fermi velocity.(Inset) Residuals from the linear fit showing very good linearity in the LLs.75Chapter 4φ 6 Theory in the Large N Limit4.1 IntroductionIn this chapter, we will consider scalar field theories in 2+ 1d and study theirbehavior in large N limit. We first review the relevance of these model for study-ing fermionic theories that have a fermionic condensate by using the Hubbard-Stratonovich transformation and adding a real scalar field to the original La-grangian. The scenario where a quantum field theory can have a parametricallysmall beta function resulting in an approximate scale invariance has attracted at-tention, particularly when the approximate scale symmetry can be spontaneouslybroken, generating a pseudo-Goldstone boson in the form of a light dilaton. Forexample, the notion that the tree level scale invariance of the SU(2)×U(1) elec-troweak theory is softly broken by the Higgs potential or dynamically broken bysome physical mechanism beyond the Standard Model leaves the Higgs boson it-self as the dilaton, with some testable physical consequences [62]-[75]. Walkingtechni-color [65]-[71], confining and chiral symmetry breaking gauge field theo-ries with approximate infrared conformal symmetry [72] are also scenarios wherespontaneous breaking of approximate scale invariance could play an importantrole.76One of the prototypical examples of spontaneously broken scale symmetry oc-curs in the large N limit of the tri-critical O(N) symmetric g2(~φ2)3-theory in threespace-time dimensions which is an interesting quantum field theory in its ownright. The phase diagram of g2(~φ2)3-theory in three space-time dimensions is ofimportance in studying graphene. The importance is a consequence of the renor-malization of the coupling constant under renormalization group.Let us consider the effect of short-range interactions by adding a Hubbard termto our Lagrangian of graphene. Suggested by the seminal work of Hertz [76],Hubbard interactions could be studied using the Hubbard-Stratonovich transfor-mation. Let’s consider the half-filled Hubbard Hamiltonian given by the followingon-site interaction term in the Hamiltonian,HHubbard =U∑i(ni,↑− 12)(ni,↓−12). (4.1)In the above Hamiltonian, U is the Hubbard parameter which implies the impor-tance of the interaction term in comparison to the other terms in the full Hamilto-nian, the sum is over the sites and ni,↑ (ni,↓) is the number operator for the spin-up(spin-down) electrons in the site i. We can write the above interaction Hamilto-nian in terms of the spin operator and then use the following identity to do theHubbard-Stratonovich transformation and decouple the spin-up/spin-down opera-tors,ea2/2 =∫ ∞−∞dx√pie−x2/2−ax. (4.2)The decoupled Hubbard Hamiltonian then would be given by the following for-mula in which S is the spin operator.HHubbard =∑i(U4− 2U3S2i). (4.3)77We then transform the Hubbard interaction Hamiltonian using the Hubbard-Stratonovichtransformation and add a real scalar field to our Hamiltonian. In path integrallanguage, the original Hubbard term gets substituted by a new scalar field anda coupling term that couples this added scalar field to the spin operator of thefermionseHHubbard = e∑i(U4 − 2U3 S2i)=∫[dφi]e∑i(U4 − 3U8 φ2i +φi.Si). (4.4)The coupling between the scalar field and the fermions spin is given by∑iφi.Si =∑iφic†isσss′cis′, (4.5)which in our Lagrangian would correspond to a φ ·ψσψ term. Following therenormalization group scheme, the Lagrangian for the Hubbard-Stratonovich field,φ , would have all of the interaction terms that are relevant or marginal. This is aconsequence of renormalization of the coupling constants under renormalizationgroup. Here, φ6 being the marginal interaction in 2+1d, makes the investigationof g2(~φ2)3-theory phase structure in three space-time dimensions an importantquestion for a graphene material that has Hubbard interactions in its Lagrangian.The introduced Hubbard-Stratonovich field φ , has three components and enjoysan O(3) symmetry. To study the phase diagram of the effective theory describingthe φ field, we study the O(N) symmetric version of the theory and use N as anexpansion parameter.As N here is small, one might be skeptical about the results that are derived bystudying the large N limit theories. In the real world, N is three so an expansionin powers of l/N may not seem like such a good idea. However, it is possiblethat the l/N expansion might actually be applied to infer valuable informationsabout quantum field theories with a finite and even small N. The actual expansion78parameter would be proportional to 1/N and that coefficient might play an impor-tant role in the validity of the 1/N expansion. This can be observed by remindingourself about the well known coupling expansion for quantum electrodynamicsin 3+ 1d. Although e ' 0.3, the actual expansion parameter α , fine-structureconstant, is given by the following formulaα =14piε0e2h¯c' 1137. (4.6)We observe that although the coupling constant in the Lagrangian, e, is not verysmall, the actual expansion parameter, the fine-structure constant turns out to besmall. At the same-time, as we observed in Chapter 2, contributions with the sameorder in 1/N might cancel the other contributions and leave a small contributionafter these cancellations.The Hubbard-Stratonovich transformation of a fermionic system transforms it intoa fermionic system that is coupled to bosons. Recently bosonic fields coupled tothe fermionic fields that enjoy a non-abelian symmetry with a large rank havereceived more attentions [77]. In Ref.[77], in search of non-Fermi liquid theystart with a fermionic field with a four-point interaction and after introduction ofa scalar field, who plays the role of Hubbard-Stratonovich field, they continuetheir investigation without fully integrating out the fermions. They employ theWilsonian mechanism to study the coupled fermions with bosons and enhancethe symmetry of their Lagrangian by tuning one of the parameters of the La-grangian to zero. The Lagrangian used in Ref.[77] model after the introduction of79a Hubbard-Stratonovich field to the original Lagrangian density is given byL =Lψ +Lφ +Lψ,φ ,Lψ = ψσ (∂0+µ− iv∇)ψσ +λψψσψσψσ ′ψσ ′,Lφ = (∂0φ)2+ c2(∇φ)2+m2φφ2+λφ4!φ4,Lψ,φ = λψ,φψσψσ ′φ , (4.7)where ψ is the fermionic field, φ is the Hubbard-Stratonovich field, v and c are thevelocities of the fermionic and bosonic field respectively, µ is the chemical poten-tial, λψ,φ , λψ and λφ are the coupling constants. In Ref.[77], the authors promotethe model to have N fermion species and an N×N complex scalar field that be-longs to the adjoint representation of the resulting fermions’ SU(N) symmetrygroup. The SU(N) invariant model then can be written asLψ = ψ i(∂0+µ− iv∇)ψi+λψψ iψiψ jψ j,Lφ = tr{(∂0φ)2+ c2(∇φ)2+m2φφ2}+λ (1)φ4!tr(φ4)+λ (2)φ4!(tr(φ2))2,Lψ,φ = λψ,φψ iψ jφji . (4.8)To study this model one might try to study its more symmetric version and gain in-sight through studying that more simple model. The model in Eq.4.8 would havean enhanced symmetry for λ (1)φ = 0. The scalar field would enjoy an enhancedSO(N2) symmetry. The interaction term, λψ,φ , breaks this enhanced symmetryhowever, it is an approximate symmetry of the Lagrangian. In 2+1d, this modelcould generate more terms, including phi-six terms that are marginal in that di-mension. To study this model in 2+ 1d then one needs to study the effect ofother terms in the Lagrangian including the following term in the bosonic fieldLagrangianLφ6 = g2(tr(φ2))3. (4.9)80In this chapter, we will study a critical model defined by the following LagrangianL=12∂µ~φ∂µ~φ +g2N2(tr(φ2))3−~j ·~φ , (4.10)where the φ field is a vector representation of SO(N) group. The g2(~φ2)3 inter-action is scale invariant at the classical level. Its beta function is of order 1/Nand, it is therefore suppressed at large N. As a result, g2(~φ2)3 remains exactlymarginal at the leading order in the large N expansion and, at the next-to-leadingorder, it becomes a marginally irrelevant slowly running coupling. The theory isapproximately scale invariant. The beta function, depicted in Fig. 4.1, has a trivialinfrared fixed point at g2 = 0. In addition, as was argued long ago [12]-[15], itexhibits a nontrivial ultraviolet fixed point at g2 = 192. The ultraviolet fixed pointrenders the field theory asymptotically safe in that the ultraviolet cut-off can beremoved without forcing triviality [14].However, Bardeen, Moshe and Bander [16] showed that, with sufficientlystrong coupling, the infinite N limit of g2(~φ2)3 theory has a quantum phase tran-sition to a phase where an O(N) singlet composite operator, 1N~φ2, gets an expec-tation value. This operator has a non-zero scaling dimension and its expectationvalue breaks the scale symmetry. The phase transition occurs at a critical valueof the coupling, g∗2 = (4pi)2 ≈ 158 that is somewhat smaller than the ultravioletfixed point, g2UV = 192. They also showed that the ultra-violet cut-off can be re-moved only when g2 is tuned to their fixed point, g2→ g∗2. The condensate breaksthe exact scale symmetry of that limit and they showed that the spectrum of thetheory contains a massless dilaton. David, Kessler and Neuberger [78, 79] did acareful analysis of the phase diagram of the infinite N model. They also conjec-tured that, “for all finite N the BMB (Bardeen-Moshe-Bander) phenomenon doesnot survive.” The latter is something that we shall demonstrate to be so in thefollowing. In a subsequent lattice study, Kessler and Neuberger demonstrated thatthe infinite N model with nearest neighbour lattice interactions does not exhibit812 4 6 8 10 12 14 g200400600800Β NFigure 4.1: N× the beta function of large N regime of g2(~φ2)3 theory inthree dimensions. The infrared fixed point is g2IR = 0 and the ultra-violet fixed point occurs at g2UV = 192. The critical coupling where inthe infinite N limit scale symmetry breaking occurs is g2 = (4pi)2 ≈158.the Bardeen, Moshe and Bander phenomena [80]. Their conclusion suggests thatthe BMB phase is regulation scheme dependent and this dependency is a hint ofinstability of the BMB phase.It was suggested in the original work of Bardeen, Moshe and Bander [16]that if one considers the large but not infinite N limit, where the scale symme-try becomes approximate, in their strong coupling phase, the latter is representedas a spontaneously broken approximate symmetry. The dilaton becomes a quasi-Goldstone boson, acquiring a mass of order 1/N. In this chapter, we will examinethis issue by studying the tri-critical O(N) vector model in the leading and next-to-leading order of the large N limit. Of particular interest will be the interplaybetween the loss of tune-ability of the coupling constant when it has a renormal-ization group flow and dynamical breaking of scale invariance which is driven bystrong coupling dynamics and occurs at a specific fixed point. Our central con-82clusion will be that the “light dilaton” of this theory is actually a tachyon. Thisindicates an instability of the phase of the theory with spontaneously broken ap-proximate scale invariance. Our computation is in complete perturbative control,at least in the context of a renormalization group improved large N expansion,when N is large enough. We note that, potential instability, based on the fact thatthis is ultimately a theory with a cubic potential was pointed out by Gudminds-dottir et. al. [81].The composite operator effective action for the same model was originallycomputed by Townsend [12] and our results are in agreement with his where theyoverlap. The main difference is that he studied O(N) symmetry breaking whereaswe study the the massive phase which occurs near the tri-critical point.4.2 Spontaneous Symmetry BreakingSpontaneous symmetry breaking occurs when the ground state is not invariantunder the symmetry transformations of a Lagrangian like the cartoon in Fig.4.2.Spontaneous symmetry breaking surprisingly provides a large amount of infor-mation about the spectrum of a theory. The phenomena of spontaneous symmetrybreaking plays an important role in our understanding of various phenomenas innature. Most simple phases of matter and phase transitions, like crystals, magnets,and conventional superconductors can be simply understood from the viewpointof spontaneous symmetry breaking. The mechanism of spontaneous broken sym-metry plays an important role in the context of the strong interactions, specificallychiral symmetry breaking. In the Standard Model of particle physics, sponta-neous symmetry breaking of the SU(2) × U(1) gauge symmetry associated withthe electroweak force generates masses for several particles, and separates theelectromagnetic and weak forces.It is a well known fact that spontaneous breaking of a symmetry results inemergent massless particles, called Goldstone modes. The existence of Goldstone83Figure 4.2: Spontaneous breaking of the internal rotation symmetry in φspace. The field φ chooses a ground state that violates the internalU(2) symmetry in the potential V (φ) = φ∗φ .modes can be proved in classical limit by investigating the classical Lagrangian.In the next section, we review the “effective action” technique that makes it pos-sible to extend the following arguments to full quantum level. Consider a theoryconsisting of generic fields φi(x) given by a Lagrangian of the formL (φi,∂φi) = (kinetic terms)−V (φi), (4.11)where we have assumed our Lagrangian is local and can be separated in kineticand potential terms. We assume that there exist φ0 such that it minimizes V (φ)and as a result is a stationary point as well∂V∂φi∣∣∣∣∣φ(x)=φ0= 0. (4.12)84Let us expand V around its minimum, we find thatV (φi) =V (φ0i )+12(φ −φ0)i(∂ 2V∂φi∂φ j)φ(x)=φ0(φ −φ0) j+ . . . . (4.13)The term m2i j=∂V∂φi∂φ jis symmetric and its eigenvalues give the mass of fields. Theeigenvalues are by definition positive around the minimum. Goldstone’s theoremstates that every continuous symmetry of the Lagrangian that is not a symmetryof minimum solution, φ0, gives rise to a massless particle corresponding to a zeroeigenvalue of m2i j.Let us write the general form of a continuous symmetry transformation for thefield φ(x), assuming that α is an infinitesimal parameterφ → φ +α∆(φ). (4.14)We consider constant fields. We drop the derivative terms in the Lagrangian anduse the invariance of the Lagrangian under the symmetry transformation to find∆(φ)∂V∂φi= 0. (4.15)Differentiating with respect to φ and imposing φ = φ0 we find that∆(φ)∂ 2V∂φi∂φ j∣∣∣∣∣φ=φ0= 0. (4.16)We finally find that for any transformation that respects the ground state sym-metries and consequently leaves φ0 unchanged, the above equation is satisfiedtrivially, ∆(φ) = 0. For any spontaneously broken symmetry ∆(φ) 6= 0, ∆(φ) isthe eigenstate with zero eigenvalue corresponding to a massless particle.85Here, we started with the assumption that the original Lagrangian enjoys anexact continuous symmetry. In the next section, the effective potential Veff(φ)plays the role of a classical potential V (φ). In case of having the original symme-try present at Veff(φ) after including the loop corrections to classical potential, wemight end up with spontaneous symmetry breaking scenario. For example, we canobserve spontaneous breaking of O(N) symmetry in O(N) model and compute themass of the Goldstone bosons explicitly. As O(N) is an internal symmetry of themodel, it survives the quantum corrections. Indeed the effective potential turnsout to be O(N) symmetric, resulting in an exact O(N) symmetry and a masslessGoldstone boson. Later in this chapter, we study the dilaton, the Goldstone bosonresponsible for breaking an approximate conformal symmetry. Although we startwith a classical conformal symmetry and find that to the leading order in our per-turbation theory it stays exact, resulting in a massless dilaton, we find that highercorrections make the dilaton massive. The emergent mass of such dilaton stemsform the fact that renormalization of fields introduces a scale in the effective po-tential and reduces our exact symmetry to an approximate symmetry. As it can beexpected, the dilaton mass turn out to be proportional to the term that breaks thescale invariance.4.3 Effective Action TechniqueThe effective potential method is useful in studying spontaneous symmetry break-ing in field theories. Effective potentials are generating functionals for single-particle irreducible Green functions of a field theory, and were introduced firstby Euler, Heisenberg and Schwinger. Their usefulness was first pointed out byJona-Lasinio [82] and was used extensively by several authors [83]-[87]. In thissection, we review the effective potential and the technique suggested by DeWittand Jackiw [86] for calculating the effective potential Γ, using functional methodsin the multi-loop approximation.Consider a field theory given by a Lagrangian L (φi,∂φi). Here φi represents86all of the fields and for simplicity we assume that only the first derivative of φi ispresent in the Lagrangian. By adding a source term J(x) that couples to the fieldsin the Lagrangian, we can make a generating functional such that its functionalderivatives result in single-particle reducible Green functions. In the presence ofa source, the action in d+1 dimensions is given by,S [ϕ] =∫dd+1x{L (φi,∂φi)+ Ji(x)φi(x)}. (4.17)Next, one can define a functional W [J] in terms of the probability amplitude forthe vacuum state in the far past to go into the vacuum state in the far future in thepresence of the external sourcee−W [J] =〈0+∣∣0−〉 . (4.18)We chose to work in Euclidean space and as a result we have a negative signinstead of i. The term W [J] is the generating functional for the connected Greenfunctions; that is, we can writeW [J] =∑{n}1n1! . . .nk!∫dd+1x1 . . .dd+1xn1 . . .dd+1wnkGn1...np(x1 . . .xp)J1(x1). . .J1(xn1) . . .Jk(wnk), (4.19)in which Gn1...np(x1 . . .xp) is the sum of all connected Feynman diagrams with n1external lines of type 1, n2 external lines of type 2, etc. A classical field can bedefined as a 1-point correlator of the field (expectation value of field) and can befound using W [J],ϕi(x) =〈0+ |φ(x)i |0−〉〈0+ |0−〉=δW [J]δJi(x). (4.20)The term W [J] and the classical field ϕi(x) which are functionals of source J can87be used to perform a Legendre transformation and obtain the effective action,Γ[ϕ], a functional of the classical fieldΓ[ϕ] =W [J]−∫dd+1x ϕiJi(x). (4.21)Here, Γ[ϕ] is a generating functional for connected 1-particle irreducible Greenfunctions. It can be expanded in classical fieldsΓ[ϕ] =∑{n}1n1! . . .nk!∫dd+1x1 . . .dd+1xn1 . . .dd+1wnkΓn1...np(x1 . . .xp)ϕ1(x1). . .ϕ1(xn1) . . .ϕk(wnk). (4.22)The Γn1...np(x1 . . .xp) are the 1-particle irreducible Green functions, defined as thesum of all connected Feynman diagrams which cannot be disconnected by cut-ting a single internal line; these are evaluated without propagators on the externallines. It can be verified by the following observations.For simplicity, we assume that we only have one species of field φ . The defi-nition of the classical field Eq.(4.20), can be used to deduce that G(x,x′) = δϕ(x)δJ(x′) .The expansion for the effective action, W [J], Eq.(4.19,4.22), and the effective ac-tion definition Eq.(4.21) can be used to relate Γp(x1 . . .xp) and Gp(x1 . . .xp). Forexample, for 3-point functions, we find that,G3(x1,x2,x3) =∫dd+1w1 . . .dd+1w3 G(x1,w1)G(x2,w2)G(x3,w3)Γ3(x1,x2,x3),(4.23)or diagrammatically for 3-point and 4-point functions, we can show the results asFig.4.3 and Fig.4.4. In other words, the connected 3-point and 4-point functionscan be constructed as a sum of irreducible 3-point and 4-point vertices with exactpropagators in the external line. This procedure can be extended to an arbitraryn-point function and one may convince oneself that the one-particle structure was88Figure 4.3: Connected reducible three-point function in terms of irreduciblevertices.Figure 4.4: Connected reducible four-point function in terms of irreduciblevertices.fully analyzed such that Γ vertices are indeed irreducible.As we are interested in spontaneous symmetry breaking of our symmetricalLagrangian, we are interested in solutions of Eq.(4.20) with non-zero ϕ in the ab-sence of source, J(x) = 0. We will focus our investigations on spatial and temporal89constant solution, ϕ(x) = ϕ0. In this case, one can define an effective potential asΓ(ϕ0) =−Veff(ϕ0)∫dd+1x. (4.24)Expansion around a constant classical field configuration is equivalent to expand-ing Γ in powers of the external momenta about the point where all external mo-menta are zero; effective potential is the zeroth-order term of such an expansion,Γ=∫dd+1x{−V (ϕ0)+ 12Z(ϕ0)∂ϕ.∂ϕ+ . . .}. (4.25)By comparing the expansion in Eq.(4.22) and Eq.(4.25), it is easy to notice thatnth derivative of the effective potential with respect to the classical field is the sumof 1-particle irreducible graphs with n vanishing external momenta.To approximate the effective potential, we use the “background field” methoddeveloped by Jackiw and DeWitt [85], [86]. Their result can be written asΓ[ϕ] = S[ϕ]+12lndetD−1+ . . . , (4.26)where D−1(ϕ) = δ2S[φ ]δφ2∣∣∣φ=ϕ. In deriving the above, authors assume that thereexists a solution φ0, that satisfies the classical equations of motion.δS[φ ]δφ∣∣∣φ=φ0= 0. (4.27)In this method, essentially we expand the action around the stationary point of theaction but later on we treat the field corresponding with the stationary point asthe classical field. In other words we expand the action around the classical fieldand drop the linear-terms (tadpole). The power of this methods comes from thefact that in a compact way, it sums up all of 1-loop graphs which contribute tothe effective potential. A task which is quite computationally hard and vulnerableagainst errors if one chooses to use the usual Feynman graph summation and take90into account the combinatorial factors. It is suggested that some mechanisms ofspontaneous symmetry breaking only can be obtained by including an infinitesubset of loop diagrams. As a result, it is crucial to employ a method that at leastbenefits from summing up an infinite subset of loop diagrams[83], [84].4.4 Tachyonic Excitations in φ 62+1In this section, we find the effective potential for φ6 model in 2+1 d. The O(N)vector field will be denoted ~φ(x). We will find it convenient to describe the theoryby two variables, the composite field χ(x) = 1N~φ2 and an auxiliary field M(x),whose expectation value is proportional to the ~φ -field mass. Both χ(x) and M(x)have classical dimension one and M(x)χ(x) is dimensionless. Whenever 〈M(x)〉 isnot zero, the ~φ -field is massive and it does not obtain an expectation value. Wewill find that, at the leading and next-to leading order in the 1/N expansion, therenormalized background field effective action [86], isS= N∫d3x{χ3(x)6(g2(M(x))−g∗2(M(x)χ(x)))++∂M(x) ·∂M(x)96pi|M(x)| + . . .}, (4.28)where g2(M) is the running coupling at scale M and g∗2(x), where x = Mχ , is thescale invariant part of the non-derivative terms in the effective action, containingcontributions of order one and of order 1N . The ellipses denote contributions of or-der 1N2 or higher of any type and terms with more than two derivatives. Althoughχ is nominally a positive operator, an infinite normal ordering constant has beensubtracted from it so that it can now be either positive or negative. The couplingshave been tuned so that terms proportional to (~φ2)2 or ~φ2 are absent.To use the background field effective action (4.28), we should first solve the91equations which determine its extrema,δSδχ(x)= 0 ,δSδM(x)= 0. (4.29)Solutions of these equations are the classical fields which we shall denote by M0and χ0. If there are more than one solution (there will not be in our example), weshould choose the solution where S, when evaluated on the solution, has the small-est real part. The expansion of the action in (4.28) in derivatives assumes that M0and χ0 are non-zero and that they are slowly varying functions, sufficiently so thatthe expansion in their derivatives is accurate. (M0 and χ0 are usually constants.)Then, in order to compute a one-particle irreducible correlation function of thefields χ(x) and M(x), we take functional derivatives of the background field ac-tion S by the variables χ(x) and M(x), and we subsequently evaluate the resultingfunctions “on-shell” by setting χ(x) and M(x) to χ0 and M0, respectively. Thisyields the renormalized, connected, one-particle-irreducible multi-point correla-tion functions of the quantum fields χ(x) and M(x). For example, the connectedtwo-point correlation functions are found by inverting the one-particle irreducibletwo-point functions which are obtained as functional second derivatives of theeffective action. They are thus given by[〈χχ〉−〈χ〉〈χ〉 〈χM〉−〈χ〉〈M〉〈Mχ〉−〈M〉〈χ〉 〈MM〉−〈M〉〈M〉]==[δ 2Sδχ2δ 2Sδχ∂Mδ 2Sδχ∂Mδ 2SδM2]−1∣∣∣∣∣∣M,χ=M0,χ0.For example, we obtain the composite operator correlation function〈 1Nφ2(x) 1Nφ2(y)〉−〈 1Nφ2(x)〉〈 1Nφ2(y)〉=1N∫ d3p(2pi)3eip(x−y)48piχ20/M0p2+ 24piχ30M0β (g2(M0)). (4.30)92Let us review a few interesting features of our results:1. We are putatively working in the leading and next-to-leading orders of thelarge N expansion. The quantities in brackets in (4.28) are of order one andof order 1N . The running coupling constant, g2(M), on the other hand, isthe solution of the renormalization group equation using the beta functionwhich is of order 1N . If expanded in1N , it contains all orders of1N , multi-plied by powers of logarithms of the mass scale ratio. This “sum of leadinglogarithms” is needed in order to accommodate possible very small or verylarge values of the condensate, M ∼ µ exp(N · . . .).2. At this order in the large N expansion, the only renormalization group func-tion entering the affective action (4.28) is the running coupling constantg2(M) which is to be evaluated at the scale determined by the condensate.3. The generic features of the result in equation (4.30) do not depend much onthe details of the function g∗2(x) in equation (4.28). It relies only on thefact that its leading contribution at large N is independent of N and the factthat it is scale invariant, that is, it is a function of only the dimensionlessratio Mχ and g2. Validity of the derivative expansion also requires non-zeroχ0 and M0 as the classical solutions. When evaluated on the solutions of theequations of motion (4.29), g∗2 = (4pi)2+O( 1N). At leading order in largeN, g∗ = 4pi , the value of the coupling at the Bardeen-Moshe-Bander fixedpoint.4. When N goes to infinity, the beta function vanishes and g2(M) becomesM-independent and tunable. Consequently, at this limit, the action (4.28)has an extremum only when g2 = g∗2 where the potential is flat and doesnot determine the scale. The fluctuations of the scale, M or χ , is in a flatdirection and forms a massless dilaton. This is the Bardeen-Moshe-Bandersolution.5. The signature of the dilaton is the presence of the pole in the correlation93function of〈φ2(x)φ2(y)〉in equation (4.30). Note that the mass of this poleis proportional to the beta function, β (g2(M)). The latter vanishes at infiniteN, leaving the dilaton massless in that limit.6. For large but finite N, the pole in the two-point function (4.30) occurs at−p2 = 24piχ30M0β (g2(M0)).The beta function is positive, β (g2(M0)) > 0. However, the values χ0 andM0 which solve the equations of motion turn out to have opposite signs.That opposite sign results in the mass squared in the pole in the propagatorin (4.30) having a negative sign and the dilaton has become a tachyon. Thetachyonic mass indicates that the phase that we are describing is unstable tofluctuations.In the following, we will present our derivation of equation (4.28) and the simplecomputation leading from equation (4.28) to (4.30). We consider the Euclideanquantum field theory which has N real scalar fields ~φ = (φ1,φ2, . . . ,φN) and O(N)symmetry in three space-time dimensions. The classical Landau-Ginzburg poten-tial is given byV (~φ2) =r2~φ2+u4N(~φ2)2+g26N2(~φ2)3, (4.31)where~φ2 ≡N∑a=1φaφa.When u> 0, there is a line of second order phase transitions at r = 0 as depictedin Fig. 4.5. When u< 0 there is a line of first order phase transitions. These linesof transitions terminate at the tri-critical point O where u = r = 0. At the classi-cal level this phase structure persists for all positive values of g2 and the g2(~φ2)3coupling is exactly marginal.94Figure 4.5: The phase diagram of the Landau-Ginzburg potential in equation(4.31). The tri-critical point O appears at the intersection of a line ofsecond order phase transitions and a line of first order phase transitionswhere the potential is equal to g26 (~φ2)3.To examine fluctuations, we consider the Euclidean functional integralZ[ j] =∫[d~φ ] e−∫d3xL[~φ ,~j], (4.32)with the Lagrangian densityL=12∂µ~φ∂µ~φ +NV (~φ2/N)−~j ·~φ , (4.33)where µ = 1,2,3 and repeated indices are summed and we have introduced asource ~j(x) in order to use the functional integral as a generating functional forcorrelators of ~φ(x). In order to study the large N limit, we introduce two auxiliary95fields by inserting1 =∫ ∞−∞[dχ(x)]δ (χ(x)−~φ2/N) (4.34)=∫ ∞−∞[dχ(x)]∫ i∞−i∞[dm2(x)]e∫ 12m2(Nχ−~φ2) (4.35)into the functional integral (4.32). This introduces two new fields χ(x) and m2(x)and it will allow us to integrate out the scalar field ~φ(x). We must be careful tonote that the integration for m2 is on the imaginary axis. We will find out andexplicit form for the scale M2 that was present in (4.28) as a function of m2. Withthese additional fields, the Lagrangian becomesL=12∂µ~φ∂µ~φ +m22~φ2−Nm22χ+NV (χ)−~j ·~φ . (4.36)The ~φ -fields now appear in a quadratic form and we integrate them exactly to getan effective actionS[m2,χ,~j] =N2Trln(−∂ 2+m2)+∫ (NV (χ)−Nm22χ− 12~j1−∂ 2+m2~j). (4.37)To find the partition function, it remains to integrate χ and m2,Z[ j] =∫[dm2dχ] e−S[m2,χ,~j] (4.38). This would yield a generating functional where functional derivatives with re-spect to j give the correlation functions of the ~φ -fields.We will study the region of the phase diagram where the O(N) symmetry is notspontaneously broken. Instead, there will be a condensates 〈χ(x)〉 and 〈m2(x)〉which will result in a mass gap for the ~φ -field. To begin, it is instructive to put the96source j(x) to zero and to write the effective action in (4.37) in an expansion inderivatives of the variable χ(x) and m2(x) [88, 89],SN=∫ {Λm24pi2− |m|312pi+V (χ)− m2χ2+∂m.∂m96pi|m| + . . .}, (4.39)where Λ is the ultra-violet cut-off and the ellipses represent terms with more thantwo derivatives of m. We have dropped a constant term that is m2 and χ inde-pendent. The effective action in (4.39) has an ultra-violet divergent Λ-dependentterm which must be removed by renormalization. We can renormalize the expres-sion by introducing counter-terms. This is accomplished by replacing V (χ) byV(χ− Λ2pi2). Then, after a field translation, χ(x) = χ˜(x)+ Λ2pi2 , the cut-off depen-dent term cancels from (4.39). Although χ(x) was originally a positive field, χ˜(x)can be positive or negative. We hereafter drop the tilde from χ˜ . The second, thirdand fourth terms in (4.39) are the effective potential for m and χ at the leadingorder in the large N expansion. In the remainder of this chapter, we will choosethe potential V (χ) to be the specific dimension-three operatorV (χ) =g26χ3(x)+ counterterms, (4.40)where the counterterms will be needed to cancel divergences at higher orders in1N . With this choice, the field theory is scale invariant at the classical level and,since there are no logarithms in (4.39), it remains scale invariant at the quantumlevel in the leading order in the large N expansion.In the large N limit, we can use the saddle-point technique to evaluate theremaining functional integral (4.38). The saddle points are field configurationswhich solve the equations of motion derived from the effective action (4.37). Wewill use the notation χ0 and m0 to denote fields which satisfy the equations ofmotion. When the fields are constant, the saddle points are extrema of the renor-malized effective potential obtained from (4.39), The potential in (4.39) has a line97of extrema, located at |m0|=−4piχ0 and these extrema exist only when the cou-pling constant g is set to the Bardeen-Moshe-Bander fixed point at g→ g∗ = 4pi .To see this, consider the equation of motion for m. This equation does not involvethe coupling constant. It has the solution|m0|={−4piχ χ < 00 χ > 0 .A massive solution exists only when χ is negative. Let us assume this is so. Wecan plug this solution into the effective action to get (We use Sˆ to distinguish thispartially on-shell action from S defined elsewhere.)Sˆ= N∫ {−|χ|36(g2− (4pi)2)+ . . .} , (4.41)and ask whether there is now a solution for χ . When g > g∗ this expression hasno extrema and for g < g∗ there is no spontaneous symmetry breaking( χ = 0).However at g= g∗, the potential is flat and any (negative) constant χ0 is a solution.To find the effective action to the next-to-leading order in large N, we use thebackground field technique. To implement this technique, we do the substitutionχ → χ+δχ , m2→ m2+ iδm2 (4.42)and, following the recipe in [86], we drop the linear terms in δχ and δm2. Then,the action expanded to quadratic order isS=N2Trln(−∂ 2+m2)−∫NΛm24pi2+∫ (NV (χ)−Nm22χ− 12~j1−∂ 2+m2~j)+N2∫[δχ,δm2][V˜ ′′(χ) −i/2−i/2 ∆/2+J [~j]][δχδm2]+ . . . , (4.43)98where∆(x,y) = 〈x| 1−∂ 2+m2 |y〉〈y|1−∂ 2+m2 |x〉 (4.44)J [x,y; j] =1N∫dwdz ja(w) ja(z)·· 〈w| 1−∂ 2+m2 |x〉〈x|1−∂ 2+m2 |y〉〈y|1−∂ 2+m2 |z〉 . (4.45)When m2 is a constant,∆(x,y) =∫ d3p(2pi)3 eip(x−y)∆(p)∆(p) = 14pi p arctanp2|m| . (4.46)Before we proceed, we can use the action (4.43) to study the spectrum of fluc-tuations in the infinite N limit. For this purpose, we invert the quadratic form in(4.43) and find the propagator〈1N~φ2 1N~φ2〉−〈1N~φ2〉〈1N~φ2〉= 〈δχ δχ〉=2N∆(p)1+2V ′′(χ)∆(p)=2N14pi p arctanp2|m|1− 2mp arctan p2m≈ 3mNpi1p2, (4.47)where, in the last equality, we have put the condensate on shell and the couplingconstant equal to the fixed point value, 4pi . The last expression reproduces thesum of bubble diagrams which would be expected from studying the Feynmandiagrams for this correlation function. The massless pole is due to the dilatonwhich is a Goldstone boson for spontaneous breaking of the scale symmetry whichis exact at this order in the large N expansion. We can see that this massless poleis the only pole by studying the denominator of (4.47).1− 2mparctanp2m=∫ 10dxx24m2p2 + x2(4.48)99in the complex −p2-plane. It is easiest to see from the integral representation ofthe function that the only zero is at −p2 = 0. There is also a cut singularity on thepositive −p2-axis beginning at 4m2 due to intermediate φ -particle pairs.To study the next order in the large N expansion, we do the Gaussian integralover the fluctuations in (4.43) to get the effective actionS=N2Trln(−∂ 2+m2)+∫ (NV (χ)−Nm22χ− 12j1−∂ 2+m2 j)+12lndet[V˜ ′′(χ) −i/2−i/2 ∆/2+J]+ . . . , (4.49)where the ellipses stand for corrections of order 1/N and higher. When we assumethat the source j and the classical fields m2 and χ are constants, we obtain theeffective action evaluated on constant fields,S= N∫ {− 112pi[m2] 32 +V (χ)− m22χ−~j2/N2m2+12N∫ d3p(2pi)3ln[1+2V ′′(χ)(∆(p)+2~j2/Nm4(p2+m2))]+ . . .} . (4.50)Corrections represented by the ellipses in the last line of (4.50) are functions of1/N2 or higher order with m2,χ and j and terms with derivatives of m2,χ and j.The first line in (4.50) is the leading order in large N and the second line isthe next-to-leading order. The integral in the next-to-leading order is ultra-violetdivergent and renormalization is required. The linear term in the effective action100in a Taylor expansion in ~j2/N is−~j22N[1m2− 4Ng2χm4∫ d3p(2pi)31p2+m211+2g2χ∆(p)]=−~j22N[1m2− 4Ng2χm4∫ d3p(2pi)31−2g2χ∆(p)(p2+m2)− 4Ng2χm4∫ ′ d3p(2pi)31p2+m211+2g2χ∆(p)]≡−~j22N1M2, (4.51)where the parameter M is proportional to the renormalized mass of the ~φ -field.At this order in the large N expansion, the ~φ -field wave-function renormalizationis finite. The prime on the integration in the third line means that the first twodivergent terms which are written before it have been subtracted, resulting in afinite integral. (These divergent terms are the first two terms in a Taylor expansionof the integral in g2.) Keeping M finite as the ultra-violet cut-off is scaled toinfinity requires that we take m2 to be a divergent function of M2,m2 =M2− 4Ng2χ(Λ2pi2− M4pi)+g4χ22pi2NlnΛξ1M− 4Ng2χ∫ ′ d3p(2pi)31p2+M211+2g2χ∆(p), (4.52)where ξ1 parameterizes the finite part of the logarithmically divergent integral.101The effective action isS= N∫ {− 112piM3+V (χ)−M22χ−~j2/N2M2+(χ2+M8pi)[4Ng2χ(Λ2pi2− M4pi)− g4χ22pi2NlnΛξ1M+4Ng2χ∫ ′ d3p(2pi)31p2+M211+2g2χ∆(p)]+12N∫ ′ d3p(2pi)3ln[1+2g2χ(∆(p)+2~j2/NM4(p2+M2))]+1N∫ d3p(2pi)3[V ′′∆− (V ′′∆)2+ 43(V ′′∆)3]+ . . .}. (4.53)As before, the prime on the integral in the fourth line indicates that the term oforder ~j2/N and the divergent terms which are written in the fifth line (these arethe first, second and third order terms in a Taylor expansion in g2) have beensubtracted to render the integral finite. The terms that have been introduced bythe mass renormalization are proportional to(χ2 +M8pi)which vanishes on-shell.Here, we will first renormalize the effective action off-shell and then later on wewill put the variables on-shell. We will find that the action is both on-shell andoff-shell renormalziable. The divergent terms in the fifth line are1N∫ [V ′′∆− (V ′′∆)2+ 43(V ′′∆)3]=g2χN(Λ2pi2− M4pi)2−g4χ2Npi2Λ−4M ln ΛMξ226pi3+g6χ33 ·28pi2 lnΛMξ3,where ξ2 and ξ3 are constants which parameterize the finite parts of divergent in-tegrals.Putting these in the effective action, we find a miraculous cancellation. All di-vergent terms with a power of M in the numerator cancel. The remaining divergent102terms can be canceled by counter-terms added to V (χ) alone. The counter-termsintroduce the scale µ in the action. What remains isS= N∫ {− 112piM3+g26χ3−M22χ−~j2/N2M2−(χ+M4pi)g2χM2piN+g2χM216pi2N− g4χ2M16pi3Nlnξ2ξ1+(χ+M4pi)g2χMpi2N∫ ′dpp2p2+111+ g2χMarctan p/22pi p+M34pi2N∫ ′dpp2 ln[1+g2χM(arctan p/22pi p+4~j2/NM6(p2+1))]− g4χ34pi2Nlnµξ1M+g6χ33 ·28pi2N lnµMξ3+ . . .}. (4.54)We shall set j2 = 0 and seek solutions of the equations of motionδSδM= 0 (4.55)δSδχ= 0. (4.56)Here we used hard cut-off in order to regulate our integrals. In Appendix.E, weuse dimensional regularization to regulate all of the diverging integrals and verifyour results’ independence from the choice of the regulator.There are three important lessons to be learned from the form of the effectiveaction (4.54).1. First of all, to this order in 1/N the theory is off-shell renormalizable. Theeffective action that we have computed can be used to find the renormal-ized correlation functions of φ ,χ, im2-fields where all external lines havevanishing momenta.1032. The second lesson is that scale invariance is indeed violated at next-to-leading order in large N, by the last two, logarithmic terms in (4.54). Fromthose terms we can find the beta function for the g26N2(~φ2)3 interaction. Theeffective action is a physical quantity, the volume times the energy of thetheory when the fields are constrained to have certain expectation values.As such, it should not depend on the renormalization scale µ . This is so if gdepends on µ in such a way that the action does not depend on µ explicitly.This yields∂∂µ(g2(µ)− 1N[3g4(µ)2pi2− g6(µ)27pi2]lnµM)= 0,β (g) = µddµg2(µ) =1N(3g42pi2− g627pi2)+ . . . , (4.57)where the ellipses denote contributions of order 1/N and higher. This resultmatches the large N limit of the known perturbative beta function [13–15].3. The third important feature of the effective action in (4.54) is that the ar-gument of the logarithms in the µ-dependent terms contains only M and µ ,and not χ . Moreover, its coefficient contains only χ3 and does not dependon M. As a result of this structure, the equation of motion for M, (4.55),does not depend on µ , and it is therefore scale invariant. If we set j2 = 0,δδMS is a homogeneous function of χ and M and it is therefore solved byM = αχ whence it gives an equation for α . That equation is solved byα =−4pi+δα where δα ∼ 1N .We can write the effective action in the formSN=∫ χ36[g2(µ)−g∗2(Mχ ,g)−β (g(µ)) lnµM+ . . .], (4.58)104whereg∗2 (x,g) =12pix3+3x2+O(1N), (4.59)and the g-dependence is only in the higher orders in 1/N and can be substitutedfor its leading order g= 4pi . Also, on-shell,g∗2 (x0,g) = (4pi)2+O(1N). (4.60)In −χ36 g∗2, we have gathered all of the terms in the effective action (4.54) exceptthose proportional to ln µM in the last line and theg36 χ3 term in the first line. In thelast equality, we have used the leading order solution of the equation δS/δM = 0,which is , Mχ0 =−4pi+O(1/N) in g∗2.Then minimum of (4.58) occurs atM = µ exp(g∗2−g2β (g)− 13), (4.61)where we use equation (4.60) for g∗. This solution is non-perturbative, both inthe sense that, since β ∼ 1N , it does not have a Taylor expansion in 1/N, and in thesense that, when it is substituted into the effective action, the logarithm produces afactor of 1β ∼N which invalidates the large N expansion. In higher orders, powersof ln Mµ will produce factors of N which can cancel their large N suppression. Thisis similar to the phenomenon in the scalar field theory example in Coleman andWeinberg’s work [87] on dynamical symmetry breaking. There, they used therenormalization group to re-sum higher order logarithmic terms to obtain a moreaccurate result. When they did, the minimum went away - there was no longer asymmetry breaking solution. In the present case, we will be more fortunate. Thisallows us to find a solution is the presence of g∗ in the action. To begin, we will usethe renormalization group to sum the leading logarithms of perturbation theory to105all orders. In this particular case, it is very simple. We replace the combinationwhich occurs in the effective action,g2(µ)−β (g(µ)) ln µM, (4.62)by the running coupling at scale M, g2(M), which is defined by integrating thebeta function∫ g2(M)g2(µ)dg2β (g)= lnMµ. (4.63)The result of the integral, g2(M), has a 1/N expansion and the leading terms repro-duce (4.62). The corrections have higher orders in 1N lnµM . The renormalizationgroup improved potential energy of the effective action is then the one given inequation (4.28), which we recopy here for the reader’s convenience,S= N∫d3x{χ3(x)6(g2(M(x))−g∗2(M(x)χ(x)))++∂M(x) ·∂M(x)96pi|M(x)| + . . .}.We will now study the states of the theory using this effective action. The equa-tions of motion are,0 =δSδχ(x)=χ202(g2(M0)−g∗2)+χ0M06g∗2′ (4.64)0 =δSδM(x)=χ306(1M0β − 1χ0g∗2′), (4.65)where g∗2′ is a derivative of g∗2 by its argument M/χ . M0 and χ0 are the solutionsof these equations. We have dropped the derivative terms since we assume thatthe solutions will be constant fields.106Equations (4.64) and (4.65) implyg∗2′ =χ0M0β (g2(M0)), (4.66)g2(M0)−g∗2 = − 13β (g2(M0)). (4.67)Equation (4.66) is an algebraic equation containing terms of order one and of or-der 1N and the variablesM0χ0 and g2. g2 appears only in the terms of order 1N andit can therefore be regarded as a constant, and set to 4pi . In the leading order,equation (4.66) has the solution M0χ0 = −4pi +O(1/N) and the order1N terms areeasily computable.We recall that g∗2 has a similar structure to equation (4.66), it contains termsof order one and of order 1N and the variablesM0χ0 and g2, and g2 appears only inthe terms of order 1N . We can then plug the solution forM0χ0 which we discussed inthe above paragraph into g∗ to obtain a 1N corrected expression for it.Then we use the corrected g∗2 in equation (4.67). The solution of equation(4.67) is a mass scale, that is, the value of the mass scale where the running cou-pling solves the equation. This yields the value of the condensate M0 and theabove considerations then determine χ0 = − 14piM0 +O( 1N ). Due to the order 1Nviolation of scale invariance, and unlike the scale invariant infinite N limit, the val-ues of these condensates are no longer arbitrary, but they are fixed by the value ofthe running coupling constant at some reference scale. We substitute the solutioninto the effective action and then obtainSon−shell = N∫ { M3018(4pi)3β (g(M0))+ . . .}, (4.68)where the ellipses are terms of order 1N2and higher.To examine the fate of the dilaton, we return to the action (4.28) and we con-107sider the fluctuation matrix about the solution that we have found,1Nδ 2Sδχ2=χ03β − M206χ0g∗2′′, (4.69)1Nδ 2Sδχ∂M=χ206Mβ +M06g∗2′′, (4.70)1Nδ 2SδM2=− χ306M20β − χ06g∗2′′+p248pi|M0| , (4.71)where we have used equations (4.66) and (4.67) to simplify the right-hand-sides.We can determine determinant of the fluctuation matrix and find,1N2det[δ 2Sδχ2δ 2Sδχ∂Mδ 2Sδχ∂Mδ 2SδM2]=M2032pi2β − p212, (4.72)where we have used the fact thatg∗2 =12piM3χ3+3M2χ2+ . . .= (4pi)2+ . . . (4.73)and,g∗2′′ =3piMχ+6+ . . .=−6+ . . . . (4.74)The determinant of the the fluctuation matrix is proportional to the inversepropagator of the χ- and M- fields. The beta function is positive over the inter-esting range of g2 (see Fig. 4.1). Clearly, from equation (4.72), we see that theseexcitations are tachyonic with mass given by,m2dilaton =−3M208pi2β . (4.75)1084.5 ConclusionWe conclude this chapter by summarizing our results. We found that, althoughat leading order in 1/N, phi-six theory in three dimensions exhibits spontaneousscale symmetry breaking accompanied by a massless dilaton (4.47), at the next-to-leading order in 1/N, dilaton acquires a tachyonic mass (4.75) and the spon-taneously broken phase is therefore unstable. We found that BMB phase is notstable. Our background field technique found the perturbative beta function of theO(N) symmetric g2(φ2)3 theory. The result agreed with the beta function origi-nally found in [13–15]. Of course, our analysis applies only if N is not infinite,but if it is large enough that our large N expansion is accurate. We found that thephase in which the interactions induce a condensate is unstable in finite N.109Bibliography[1] A. K. Geim, K. S. Novoselov, Nature Materials 6, 183-191 (2007).[2] R. Peierls, Z. Phys. 80. 1933[3] Jens Christian Johannsen, Sren Ulstrup, Marco Bianchi, Richard Hatch,Dandan Guan, Federico Mazzola, Liv Hornekr, Felix Fromm, ChristianRaidel, Thomas Seyller, Philip Hofmann, J. Phys. Condens. Matter 25,094001 (2013)[4] Hongki Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, Leonard Kleinman, andA. H. MacDonald Phys. Rev. B 74, 165310 (2006)[5] Michael E. Peskin, Daniel V. Schroeder, “An Introduction To Quantum FieldTheory”, Westview Press, 1995.[6] R. Shankar, Rev. Mod. Phys. 66, 129 (1994).[7] Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L.Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).[8] Kin Fai Mak, Matthew Y. Sfeir, Yang Wu, Chun Hung Lui, James A. Mis-ewich, and Tony F. Heinz Phys. Rev. Lett. 101, 196405 (2008).[9] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T.Stauber, N. M. R. Peres, A. K. Geim , Science 06 (2008).110[10] T. W. Appelquist, M. J. Bowick, D. Karabali and L. C. R. Wijewardhana,Phys. Rev. D 33, 3704 (1986).[11] T. Appelquist, D. Nash and L. C. R. Wijewardhana, Phys. Rev. Lett. 60, 2575(1988).[12] P. K. Townsend, Phys. Rev. D 12, 2269 (1975).[13] P. K. Townsend, Nucl. Phys. B 118, 199 (1977).[14] R. D. Pisarski, Phys. Rev. Lett. 48, 574 (1982).[15] T. Appelquist and U. W. Heinz, Phys. Rev. D 25, 2620 (1982).[16] W. A. Bardeen, M. Moshe and M. Bander, Phys. Rev. Lett. 52, 1188 (1984).[17] H. Omid and G. W. Semenoff, L.C.R. Wijewardhana, arxiv:hep-ph/1605.00750.[18] D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Mo- rozov, A. A. Zhukov,P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea,and A. K. Geim, Nat. Phys. 7, 701 (2011).[19] G. Li, A. Luican, and E. Y. Andrei, Phys. Rev. Lett. 102, 176804 (2009).[20] J. Chae, S. Jung, A. F. Young, C. R. Dean, L. Wang, Y. Gao, K. Watanabe,T. Taniguchi, J. Hone, K. L. Shep- ard, P. Kim, N. B. Zhitenev, and J. A.Stroscio, Phys. Rev. Lett. 109, 116802 (2012).[21] D.A. Siegel, C.-H. Park, C. Hwang, J. Deslippe, A.V. Fedorov, S.G. Louie,A. Lanzara, PNAS 108, 11365 (2011).[22] J. Gonzalez, F. Guinea, M. A. H. Vozmediano, Nucl. Phys. B424, 595(1994).[23] J. Gonzalez, F. Guinea, Phys. Rev. B59, R2474 (1999); Phys. Rev. B 63,134421 (2001).111[24] E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy,Phys. Rev. B66, 045108 (2002)[25] H. Leal, D. V. Khveshchenko, Nucl. Phys. B687, 323 (2004).[26] D. V. Khveshchenko, Phys. Rev. B74, 161402 (2006).[27] K. Novoselov, Nature Materials 6, 720 - 721 (2007).[28] S.Y. Zhou, G.-H. Gweon, A.V. Fedorov, P.N. First, W.A. de Heer, D.-H. Lee,F. Guinea, A.H. Castro Neto, A. Lanzara Nature Materials 6, 770 (2007).[29] M. I. Katsnelson, Materials Today 10, 20 (2007).[30] E. G. Mishchenko, Phys. Rev. Lett. 98, 216801 (2007).[31] D. E. Sheehy, J. Schmalian, Phys. Rev. Lett. 99, 226803 (2007).[32] O. Vafek, M. J. Case, Phys. Rev. B77, 033410 (2008).[33] M. S. Foster and I. L. Aleiner, Phys. Rev. B 77, 195413 (2008).[34] J. E. Drut and D. T. Son, Phys. Rev. B 77, 075115 (2008).[35] D. V. Khveshchenko, Phys. Rev. Lett. 87, 246802 (2001).[36] O. Vafek, Phys. Rev. Lett. 98, 216401 (2007).[37] Joshua L. Davis, Hamid Omid, Gordon W. Semenoff, JHEP 1109 (2011).[38] Hamid Omid, Gordon W. Semenoff, Phys. Rev. D 88, 026006 (2013).[39] G. L. Yu, R. Jalil, Branson Belle, Alexander S. Mayorov, Peter Blake, Fred-erick Schedin, Sergey V. Morozov, Leonid A. Ponomarenko, F. Chiappini,S. Wiedmann, Uli Zeitler, Mikhail I. Katsnelson, A. K. Geim, Kostya S.Novoselov, and Daniel C. Elias PNAS 110 3282-3286 ( 2013).[40] Hamid Omid, Gordon W. Semenoff, in preparation112[41] M. A. H. Vozmediano, Nat Phys 7(9) 671-672 (2011).[42] I. F. Herbut, V. Juricic, O. Vafek, Phys. Rev. Lett. 100, 046403 (2008).[43] Julia M. Link, Peter P. Orth, Daniel E. Sheehy, Jrg Schmalian, Phys. Rev. B93, 235447 (2016).[44] I. F. Herbut, Phys. Rev. Lett. 97, 146401 (2006).[45] E. Barnes, E. H. Hwang, R. Throckmorton, S. Das Sarma, Phys. Rev. B 89,235431(2014).[46] Andreas W. W. Ludwig, Matthew P. A. Fisher, R. Shankar, G. Grinstein,Phys. Rev. B 50, 7526 (1994).[47] E. G. Mishchenko, EPL 83 , 17005 (2008).[48] D. E. Sheehy and J. Schmalian, Phys. Rev. B 80 , 193411 (2009).[49] V. Juricic, O. Vafek, and I. F. Herbut, Phys. Rev. B 82, 235402 (2010).[50] B. Rosenstein, M. Lewkowicz, and T. Maniv, Phys. Rev. Lett. 110, 066602(2013).[51] P. R. Wallace, Phys. Rev. 71, 622 (1947).[52] G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).[53] V. A. Smirnov, “Evaluating Feynman Integrals”, Springer Tracts in ModernPhysics, 2005.[54] J. Hofmann, E. Barnes, S. Das Sarma, Phys. Rev. Lett. 113, 105502 (2014).[55] D. T. Son, Phys. Rev. B75, 235423 (2007).[56] C. Strouthos and J. B. Kogut, arXhiv:0804:0300[hep-lat]113[57] A. Kocic, S. Hands, J. B. Kogut and E. Dagotto, Nucl. Phys. B 347, 217(1990).[58] T. Appelquist and L. C. R. Wijewardhana, arXiv:hep-ph/0403250.[59] N.K. Nielsen, Nucl. Phys. B101, 173 (1975).[60] J.C. Breckenridge, M.J. Lavelle, T.G. Steele, Z.Phys. C65, 155 (1995).[61] Ashok K. Das, R. R. Francisco, and J. Frenkel, Phys. Rev. D 88, 085012(2013).[62] W. D. Goldberger, B. Grinstein and W. Skiba, Phys. Rev. Lett. 100, 111802(2008).[63] B. Bellazzini, C. Csaki, J. Hubisz, J. Serra and J. Terning, Eur. Phys. J. C 73,no. 2, 2333 (2013).[64] C. Csaki, C. Grojean and J. Terning, arXiv:1512.00468 [hep-ph].[65] B. Holdom, Phys. Lett. B 150, 301 (1985).[66] T. Akiba and T. Yanagida, Phys. Lett. B 169, 432 (1986).[67] K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335(1986).[68] T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett.57, 957 (1986).[69] T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D 36, 568 (1987).[70] A. G. Cohen and H. Georgi, Nucl. Phys. B 314, 7 (1989).[71] D. Dietrich, F. Sannino, Kimmo Tuominen , Phys.Rev. D, 72, 055001(2005).114[72] T. Appelquist and Y. Bai, Phys. Rev. D 82, 071701 (2010).[73] M. Hashimoto and K. Yamawaki, Phys. Rev. D 83, 015008 (2011).[74] B. Grinstein and P. Uttayarat, JHEP 1107, 038 (2011).[75] O. Antipin, M. Mojaza and F. Sannino, Phys. Lett. B 712, 119 (2012).[76] J. . Hertz Phys. Rev. B 14, 1165 (1976).[77] A. Liam Fitzpatrick, Shamit Kachru, Jared Kaplan, S. Raghu, Phys. Rev. B88, 125116 (2013).[78] F. David, D. A. Kessler and H. Neuberger, Phys. Rev. Lett. 53, 2071 (1984).[79] F. David, D. A. Kessler and H. Neuberger, Nucl. Phys. B 257, 695 (1985).[80] David A. Kessler and Herbert Neuberger, Phys.Lett. B157 (1985) .[81] R. Gudmundsdottir, G. Rydnell and P. Salomonson, Phys. Rev. Lett. 53,2529 (1984).[82] G. Jona-Lasinio, Nuovo Cimento 34, 1790 (1964).[83] R. Jackiw and K. Johnson Phys. Rev. D 8, 2386 (1973).[84] J. M. Cornwall and R. E. Norton, Annals of Physics, 8, 3338 (1973).[85] B. DeWitt, “in Magic Without Magic: John Archibald Wheeler, a Collectionof Essays in Honor of His 60th Birthday” , Freeman, 1972).[86] R. Jackiw, Phys. Rev. D 9, 1686 (1974).[87] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).[88] Lai-Him Chan, Phys. Rev. Lett 54,1222 (1985).[89] Josef A. Zuk, Phys. Rev. D 32, 2653 (1985).115[90] H. Cheng, T . T. Wu, “Expanding Protons: Scattering at High Energies”,MIT Press, 1987.116Appendix ATrace over Π1 Dirac MatricesIn this appendix, we compute the trace over the Dirac matrices which appear inthe numerator of Π1 integrand. The numerator of the integral is given by thefollowingNum1 =Tr(/q+ /p)γµ(/q+ /p+/k)γρ(/q+/k)γµ/qγρ=+2Tr(/q+ /p)γρ(/q+/k)(/q+ /p+/k)/qγρ−Tr(/q+ /p)(/q+ /p+/k)γµγρ(/q+/k)γµ/qγρ , (A.1)Num1 =−2Tr(/q+ /p)(/q+/k)(/q+ /p+/k)/q+2Tr(/q+ /p)(/q+ /p+/k)(/q+/k)/q−Tr(/q+ /p)(/q+ /p+/k)γρ(/q+/k)/qγρ , (A.2)Num1 =−2Tr(/q+ /p)(/q+/k)(/q+ /p+/k)/q+Tr(/q+ /p)(/q+ /p+/k)(/q+/k)/q−2Tr(/q+ /p)(/q+ /p+/k)/q(/q+/k), (A.3)117Num1 = 2[(q+ p) · (q+ p+ k)(q+ k) ·q(2+1−2)+(q+ p) · (q+ k)(q+ p+ k) ·q(−2−1−2)+(q+ p) ·q(q+ p+ k) · (q+ k)(−2+1+2)]= 2[(q+ p) · (q+ p+ k)(q+ k) ·q−5(q+ p) · (q+ k)(q+ p+ k) ·q+(q+ p) ·q(q+ p+ k) · (q+ k)]. (A.4)Now, using the following identities2(q+ p) · (q+ p+ k) =−k2+(q+ p)2+(q+ p+ k)2,2(q+ k) ·q=−k2+(q+ k)2+q2,2(q+ p) ·q=−p2+(q+ p)2+q2,2(q+ p+ k) · (q+ k) =−p2+(q+ p+ k)2+(q+ k)2,2(q+ p) · (q+ k) =−p2+q2+(q+ p+ k)2− k2,2(q+ p+ k) ·q=−p2+(q+ p)2+(q+ k)2− k2, (A.5)118we can simplify the numerator, Num1, and get more simple form for it. Thenumerator then can be simplified to the following expressionNum1 =12[(−k2+(q+ p)2+(q+ p+ k)2){−k2+(q+ k)2+q2}+(−p2+(q+ p)2+q2){−p2+(q+ p+ k)2+(q+ k)2}−5{−p2+q2+q+p+ k)2− k2}(−p2+(q+ p)2+(q+ k)2− k2)]=12[(q+ p+ k)2(− k2+(q+ k)2+q2+−p2+(q+ p)2+q2−5{−p2+(q+ p)2+(q+ k)2− k2})+(q+ p)2(− k2+(q+ k)2+q2− p2+(q+ k)2−5(−p2+q2− k2))+(q+ k)2(− k2− p2+q2−5(−p2+q2− k2))− k2(−k2+q2)− p2(−p2+q2)−5(−p2+q2− k2)(−p2− k2)]=12[(q+ p+ k)2(4k2−4(q+ k)2+2q2+4p2−4(q+ p)2)+(q+ p)2(4k2+2(q+ k)2−4q2+4p2)+(q+ k)2(4k2+4p2−4q2)+q2(4k2+4p2)−4k4−4p4−10k2p2]. (A.6)Now, this expression would be integrated using a measure which has the followingdenominator, (q+ p)2(q+ p+ k)2(q+ k)2q2ps. We can use some symmetries ofthis denominator to simplify the above expression. For example, q→−q−k then119p→−p yields(q+ p)→−(q+ p+ k),(q+ p+ k)→−(q+ p),(q+ k)→−q,q→−(q+ k),p→−p,simply permutes the factors. Similarly, q→ q− p and then p→−p does(q+ p)→−q,(q+ p+ k)→ (q+ k),(q+ k)→ (q+ p+ k),q→ (q+ p),p→−p,also simply permutes the factors. Finally, q→−q− k− p does(q+ p)→−(q+ k),(q+ p+ k)→−q,(q+ k)→−(q+ p),q→−(q+ p+ k),p→ p,is a symmetry. We use these symmetries to write the numerator asNum1 =+(q+ p+ k)2(8k2−4(q+ k)2+2q2+8p2−4(q+ p)2)−2k4−2p4−5k2p2. (A.7)120Appendix BRecurrence Relations for IsConsider the integralIs =〈1(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉. (B.1)By dimensional analysis, it is of order k−s−2 and it therefore obeys the identityk · ddkIs+(s+2)Is = 0, (B.2)or(s+2)Is =〈2k · (q+ p+ k)(q+ p+ k)2+2k · (q+ k)(q+ k)2〉. (B.3)We do the transformation q→ q− p then p→−p in the first term to get(s+2)Is = 2〈2k · (q+ k)(q+ k)2〉. (B.4)Then we use 2k · (q+ k) =−q2+ k2+(q+ k)2 to gets2Is =〈k2−q2(q+ k)2〉. (B.5)121Now, we can also use0 =〈3− 2p · (q+ p)(q+ p)2− 2p · (q+ p+ k)(q+ p+ k)2− s〉=〈∂∂ pp(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉. (B.6)Again, in the second-last term, we transform q→−q− p− k to get(3− s)Is+2〈2p · (q+ k)(q+ k)2〉= 0. (B.7)Now, using 2p · (q+ k) = (q+ p+ k)2− p2− (q+ k)2, we have(1− s)2Is =−〈(q+ p+ k)2− p2(q+ k)2〉. (B.8)Finally,0 =〈1− 2q · (q+ p)(q+ p)2− 2q · (q+ p+ k)(q+ p+ k)2− 2q · (q+ k)(q+ k)2〉=〈∂∂qq(q+ p)2(q+ p+ k)2(q+ k)2q2ps〉, (B.9)Is2=〈q · (q+ k)+(q+ p) · (q+ k)+(q+ k+ p) · (q+ k)(q+ k)2〉. (B.10)After a bit of simplifying Is we find that12Is =−〈(q+ p+ k)2+q2− p2− k2(q+ k)2〉. (B.11)122Now, we assemble the results:s2Is =〈k2−q2(q+ k)2〉, (B.12)1− s2Is =−〈(q+ p+ k)2− p2(q+ k)2〉, (B.13)12Is =−〈(q+ p+ k)2+q2− p2− k2(q+ k)2〉. (B.14)Not all of these equations are independent, indeed (B.12)+(B.13)=(B.14). Using(B.12),s−22Is−2 =〈k2p2−q2p2(q+ k)2〉=k2〈−(q+ p+ k)2+ p2(q+ k)2〉+〈k2(q+ p+ k)2−q2p2(q+ k)2〉=k21− s2Is+〈k2(q+ p+ k)2−q2p2(q+ k)2〉, (B.15)orIs−2 =−k2 1− s2− sIs−22− s〈k2(q+ p+ k)2−q2p2(q+ k)2〉. (B.16)123Now, using our integral formula〈k2(q+ p+ k)2(q+ k)2〉= k2〈1[(q+ k)2]2(q+ p)2q2ps〉= k2〈1[(q+ k)2]2q2〉〈1(q+ p)2[p2]s2〉=k2(4pi)32Γ[ s2 − 12 ]Γ[12 ]Γ[32 − s2 ]Γ[ s2 ]Γ[2− s2 ]〈1[q+ k)2]2[q2]s2+12〉=k2(4pi)3[k2]s2+1Γ[ s2 − 12 ]Γ[12 ]Γ[32 − s2 ]Γ[ s2 ]Γ[2− s2 ]Γ[ s2 +1]Γ[−12 ]Γ[1− s2 ]Γ[2]Γ[ s2 +12 ]Γ[12 − s2 ]=132pi2kss2− s (B.17)and 〈q2p2(q+ k)2〉=〈1[(q+ k)2]2(q+ p+ k)2(q+ p)2ps−2〉=〈1(q+ k)2q2〉〈1[(q+ p+ k)2]2[p2]s2−1〉=1(4pi)32Γ[ s2 − 12 ]Γ[−12 ]Γ[52 − s2 ]Γ[2]Γ[ s2 −1]Γ[2− s2 ]〈1[(q+ k)2]s2+12q2〉=1(4pi)3ksΓ[ s2 − 12 ]Γ[−12 ]Γ[52 − s2 ]Γ[2]Γ[ s2 −1]Γ[2− s2 ]Γ[ s2 ]Γ[12 ]Γ[1− s2 ]Γ[ s2 +12 ]Γ[32 − s2 ]=− 132pi2ks3− s1− s . (B.18)Now we can plug in the above result in (B.16 to find an explicit recurrence relationfor Is−2.Is−2 =−k2 1− s2− sIs−22− s132pi2ks[s2− s +3− s1− s]. (B.19)124In evaluation of two-loop integrals, we will need to have a recurrence relation forIs−4. We can work it out using our relation for Is−2 and the result isIs−4 =k4(3− s)(1− s)(4− s)(2− s)Is+116pi2ks−214− s[s(3− s)(2− s)2 +(3− s)2(2− s)(1− s)− 5− s3− s +2− s4− s]. (B.20)125Recurrence RelationsIs−2 =−k2 1− s2− sIs−22− s132pi2ks[s2− s +3− s1− s]Is−4 = k4(3− s)(1− s)(4− s)(2− s)Is+116pi2ks−214− s[s(3− s)(2− s)2 +(3− s)2(2− s)(1− s)−5− s3− s +2− s4− s]Table B.1: Table of Recurrence Relations for Is126Appendix CCalculation of I1In this appendix we evaluate I1 using Schwinger parameters. Lets start by writinga more general form of the integral I1. Using our bracket notation, a more generalform of I1 is given byI(a1, . . . ,a5;d) =〈1((q+ p)2)a1((q+ p+ k)2)a2((q+ k)2))a3(q2)a4(p2)a5〉.(C.1)If we define a = ∑al , Alpha Representation could be used to put this integral inthe following form [53].I(a1, . . . ,a5;d) =(4pi)−d(k2)a−dΓ(a−d)∏l Γ(al)∫R5>0dα1 . . .dα5 δ(1−∑αl)U a−3d/2∏l αal−1lV a−d,(C.2)in which U and V are given byU = (α1+α2+α3+α4)α5+(α1+α2)(α3+α4),V = (α1+α2)α3α4+α1α2(α3+α4)+α5(α1+α3)(α4+α2). (C.3)127We use the Cheng-Wu theorem [90] to change the delta function to δ (α5 = 1) andfocus on I1 by imposing the following constraints a1 = a2 = a3 = a4 = 1, a5 = 1/2and d = 3. We are then left with the following integral for I1I1(k) =1128pi3 k3∫R5>0dα1 . . .dα51V (α1,α2,α3,α4,1)=1128pi3 k3∫R4>0dα1 . . .dα41(α1+α2)α3α4+α1α2(α3+α4)+(α1+α3)(α4+α2). (C.4)Fortunately, the above integral can be evaluated explicitly. The integration can bedone by integrating α1 and α3 first and then the remaining integrals. We couldevaluate I1 and it has a simple formI1(k) =164k3. (C.5)128Appendix DInstantaneous Limit of Graphene inLarge NIn this section, we derive the propagator that we used in calculating the graphene’sbeta function in the large N. We then take the limit in which the light velocity goesto infinity and check that our propagator reduces to the more studied instantaneouslimit propagator.The kinetic terms of the Lagrangian can be decomposed into fermion and pho-ton parts, we fix our gauge such that the fermion part becomes diagonal. Thekinetic terms are given by the following matricesLfermion =Np16 0 00 Np16 00 0 Np16 , (D.1)129Lphoton =2c(p12+p22)εe2P −2cp0p1εe2P −2cp0p2εe2P−2cp0p1εe2P2c(p02+c2p22)εe2P −2c3p1p2εe2P−2cp0p2εe2P −2c3p1p2εe2P2c(p02+c2p12)εe2P . (D.2)We now find the inverse of the sum of the above terms and derive the propa-gator that we used in previous sections to calculate the graphene’s beta function.The propagator has the following form∆(p0,~p) =16Nc2pP+ξ p20η1p2ξη1p0p1p2ξη1p0p2p2ξη1p0p1p2η4+ξ 2Pp21−ξc2p(p20+c2p22)p2η1η2ξη3η1η2p1p2p2ξη1p0p2p2ξη3η1η2p1p2p2η4−ξc2p(p20+c2p21)+ξ 2Pp22p2η1η2 .(D.3)To investigate the instantaneous limit, we need to take the c→ ∞ limit. It is anon-trivial task to take that limit in the propagator however, it can be easily donefor the Lagrangian. Lets expand our Lagrangian around infinite light speed, thefollowing Lagrangian needs to get contracted with AµLkinetic =2~p2εce2P +Np16 −2p0p1εce2P −2p0p2εce2P−2p0p1εce2P2(p22c2+p20)εce2P +Np16 −2p1p2εc3e2P−2p0p2εce2P −2p1p2εc3e2P2(p12c2+p20)εce2P +Np16 . (D.4)As the spacial part of Aµ contributes a term of order c3 to the Lagrangianand the temporal has a contribution of order c, the spacial part (magnetic) getssuppressed. We can then only keep L 00kinetic in our Lagrangian. We then get backthe instantaneous Lagrangian.130Appendix EDimensional Regularization of φ 6 inLarge NIn this section, we analyze φ6 theory in the large N limit using dimensional regu-larization. We need to introduce a scale µ in our Lagrangian so that the couplingconstant has the correct dimensions in any dimension of space-time. The correctedLagrangian is given byL=12∂µ~φ∂µ~φ +m22~φ2−Nm22χ+Ng26µ−2(2ω−3)χ30 −~j ·~φ . (E.1)Assuming that the the source ~j and the classical fields m2 and χ are constants,after performing the same steps to derive the fluctuations’ determinant, we obtainthe following effective potential in arbitrary dimensionsVeffNV=12∫ d2ω p(2pi)2ωln(p2+m2)+g26µ−2(2ω−3)χ30 −m22χ0−~j2/N2m2+12N∫ d2ω p(2pi)2ωln[1+2g2χ0µ−2(2ω−3)(∆(p)+2~j2/Nm4(p2+m2))]. (E.2)Lets expand the log term resulting from the determinant. To simplify the re-131sult, we use the identity ∆(p) = ∆(−p)∫ d2ω p(2pi)2ωln[1+2g2χµ−2(2ω−3)(∆(p)+2~j2/Nm4(p2+m2))]'∫ d2ω p(2pi)2ω{(2∆(p)+4~j2/Nm4p2+m2)g2χµ−2(2ω−3)−(2∆(p)∆(−p)+8∆(−p)~j2/Nm4p2+m2)(g2χµ−2(2ω−3))2+83∆(p)3(g2χµ−2(2ω−3))3}. (E.3)We only kept the terms that are not finite in dimensional regularization. Weevaluate the above term by term in the following sections.E.1.1 ~j2 TermIt is well known that this term is finite however we reconfirm the result by explicitcalculations. This term is given by the following∫ d2ω p(2pi)2ω1p2+m2=Ω[2ω]12(1m2)1−ωpiCsc [ωpi]2ω→3−−−→−2|m|pi2. (E.4)E.1.2 ∆(p) TermWe do not expect this term to be divergent either however, lets confirm our guess.We observe that ∆(p) term is given by the following∫ d2ω p(2pi)2ωd2ωq(2pi)2ω1p2+m21(p+q)2+m2=(∫ d2ω p(2pi)2ω1p2+m2)2, (E.5)132which is finite due to our previous calculation.E.1.3 ∆(p)∆(−p) TermThis term is indeed divergent and contributes to the beta function. Lets start eval-uating it, it is given by the following∫ d2ω p(2pi)2ω∆(p)∆(−p) =∫ 4∏i=1d2ω pi(2pi)2ω1p2i +m2(2pi)2ωδ (Σpi)=∫ ∞−∞dx∫ 4∏i=1d2ω pi(2pi)2ω1p2i +m2exp(ipi.x). (E.6)To evaluate the above equation, we need to find the solution to the Poisson equa-tion in general dimensions. Fortunately, it can be done in a close form given by∫ d2ω p(2pi)2ω1p2+m2eip.x =∫ ∞0∫dλd2ω p(2pi)2ωeip.x−λ (p2+m2)= |2mx|ω−1piω∫ ∞0dλλωe−|mx|2 (λ+λ−1)= (2pi)−ω∣∣∣ xm∣∣∣1−ω Kω−1 (|mx|) . (E.7)133Using the result of our integral and putting it back into Eq.(E.1.3), we find that∆(p)∆(−p) term can be written in terms of hyper-geometric functions,∫ d2ω p(2pi)2ω∆(p)∆(−p) =∫ ∞0dx((2pi)−ω | xm|1−ωKω−1 (|mx|))4x2ω−1Ω[2ω] =1Γ[ω]2−5−4ω∣∣∣∣ 1m∣∣∣∣−4ω |m|−8+2ωpi1−3ω√piCsc[piω](−4ω (E.8)Γ[4−3ω]Γ[3−2ω]Γ[−1+ω]2F1[4−3ω, 32 −ω, 72 −2ω,1]Γ[72 −2ω]− 16Γ[2−2ω]Γ[2−ω]Hyp[{12 ,1,3−2ω},{52 −ω,ω},1]Γ[52 −ω]+Γ[1−ω]Γ[2−ω]Hyp[{1,2−ω,−12 +ω},{32 ,−1+2ω},1]2−5+2ω). (E.9)We expand the result around 2ω = 3 and keeping only the poles. We find that∆(p)∆(−p) term is simply given by∫ d2ω p(2pi)2ω∆(p)∆(−p)∼ |m|32pi3(2ω−3) . (E.10)E.1.4 ∆(p)~j2 TermWe follow the same strategy as our strategy for evaluation of ∆(p)∆(−p) term. Weagain find that ∆(p)~j2 term can be written in terms of hyper-geometric functions,134∫ d2ω p(2pi)2ω∆(−p) 1p2+m2=∫ 3∏i=1d2ω pi(2pi)2ω1p2i +m2(2pi)2ωδ (Σpi) =∫ ∞−∞dx∫ 3∏i=1d2ω pi(2pi)2ω1p2i +m2exp(ipi.x), (E.11)∫ d2ω p(2pi)2ω∆(−p) 1p2+m2=∫ ∞0dx((2pi)−ω | xm|1−ωKω−1 (|mx|))3x2ω−1Ω[2ω]=− 2−3−2ωΓ[52 −ω]Γ[ω]∣∣∣∣ 1m∣∣∣∣−3ωm−6+ωpi 32−2ωCsc[piω]Γ[3−2ω](2F1[1,3−2ω, 52 −ω,14]+32F1[1,2−ω,−12 ,14]−2(−1+ω)2F1[1,2−ω, 12 ,14]). (E.12)Expanding the result around 2+1d, we find the following pole for ∆(p)~j2 term∫ d2ω p(2pi)2ω∆(−p) 1p2+m2=− 132pi2(2ω−3) . (E.13)E.1.5 ∆(p)3 TermWe approximate ∆(p) by its exact value at zero mass∆(p) =∫ d2ωq(2pi)2ω1q2(p−q)2=Γ[2−ω]Γ2[ω−1](4pi)ωΓ[2ω−2] [p2]ω−2. (E.14)135Due to dimensional analysis non-zero mass can not contribute to the divergentterms of ∆(p)3. The integral is originally logarithmic divergent at m = 0 and thecorrection for the non-zero mass can not be divergent. As a result, we regulatethe infrared divergences by a mass term, M2. As the UV part is not sensitive tothis regularization, this regularization is valid. The IR regularization consequentlywill not alter the UV divergent part.∫ d2ω p(2pi)2ω∆3(p)'∫ d2ω p(2pi)2ω(Γ[2−ω]Γ2[ω−1](4pi)ωΓ[2ω−2])3[p2+M2]3ω−62ω→3−−−→− 1(64pi)2(2ω−3) . (E.15)E.1.6 All Terms TogetherNow, we have all of the parts of the puzzle and we can put them together to findthe effective potential. The fluctuations’ determinant is given by∫ d2ω p(2pi)2ωln[1+2g2χµ−2(2ω−3)(∆(p)+2~j2/Nm4(p2+m2))]'(− 1(2ω−3)( |m|16pi3− 14pi2~j2Nm4)(g4χ2 µ−2(2ω−3))µ−2(2ω−3)− g6χ3 µ−2(2ω−3)6(16pi)2(2ω−3)µ−4(2ω−3))=(( |m|8pi3− 12pi2~j2Nm4)g4χ2+23(16pi)2g6χ3)lnµ. (E.16)Here, we isolated g6χ3 µ−2(2ω−3) and g4χ2 µ−2(2ω−3) as the original interactionshave a similar dependence on µ , g2χ3 µ−2(2ω−3). To add the corrections to theoriginal interaction terms, we need to put them in a similar form.Now, we incorporate the newly calculated contributions to our effective po-136tential and find the renormalized mass, MVeffNV=−(m2)3212pi+g26χ3− m22χ−~j2/N2m2− g4N(−|m|χ216pi3+χ24pi2~j2Nm4− g2χ33(16pi)2)lnµ=−(m2)3212pi+g26χ3− m22χ−~j2/2Nm2− 1N g4χ22pi2 lnµ+g4N( |m|χ216pi3+g2χ33(16pi)2)lnµ.(E.17)We find that M2 = m2− 1N g4χ22pi2 lnµm or equivalently m2 = M2 + 1Ng4χ22pi2 lnµM . Werewrite our potential in terms of the renormalized mass, MVeffNV=− M312pi− 1NMg4χ216pi3lnµM+g26χ3−M22χ− 1Ng44pi2lnµMχ3−~j2/2NM2+g4χ2N(M16pi3+g2χ3(16pi)2)lnµM=− M312pi−M22χ−~j2/2NM2+g26χ3+g44Npi2lnµM(−1+ g23 .26)χ3. (E.18)Our final result for the effective potential is in agreement with our previous resultin which we had used the hard cut-off regulator.137


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