2+1D Quantum Field Theories in Large N Limit

2 + 1d Quantum Field Theories in Large N limit

Hamid Omid

M.Sc Theoretical Physics, The University of British Columbia, 2011 B.Sc Theoretical Physics, Isfahan University of Technology, 2009

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF

Doctor of Philosophy

FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Physics)

The University Of British Columbia (Vancouver)

January 2017

c Hamid Omid, 2017 Abstract

In Chapter 1, we present a brief introduction to the tight-binding model of graphene and show that in the low-energy continuum limit, it can be modeled by reduced

QED2+1. We then review renormalization group technique which is used in the next chapters.

In Chapter 2, we consider a quantum field theory in 3 + 1d with the defect of a large number of fermion flavors, N. We study the next-to-leading order contributions to the fermions current-current correlation function h jµ (x) jν (y)i by performing a large N expansion. We find that the next-to-leading order contributions 1/N to the current-current correlation function is significantly suppressed. The suppression is a consequence of a surprising cancellation between the two contributing Feynman diagrams. We calculate the model’s conductivity via the Kubo formula 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 large N expansion to explore the renormalization of the Fermi velocity in the screening dominated regime of charge neutral graphene with a Coulomb interaction. We show that inclusion of the fluctuations of the magnetic field lead to a cancellation of the beta function to the leading order in 1/N. The first non-zero contribution to the beta function turns out to be of order 1/N2.

In Chapter 4, we study the phase structure of a φ 6 theory in large N. The

ii leading order of the large N limit of the O(N) symmetric phi-six theory in three dimensions has a phase which exhibits spontaneous breaking of scale symmetry accompanied by a massless dilaton. In this chapter, we show that this “light dilaton” is actually a tachyon. This indicates an instability of the phase of the theory with spontaneously broken approximate scale invariance. We rule out the existence of Bardeen-Moshe-Bander phase.

In this thesis, we show that Large N expansion is a powerful tool which in regimes that the system is interacting strongly could be used as an alternative to coupling expansion scheme.

iii Preface

This thesis is based on notes written by myself during my PhD program and also publications authored by my collaborators and me. Most of the calculations were done by myself. The ideas were developed during several meetings between my supervisor and myself. Chapter 2 is the study of AC conductivity of 2 + 1d Dirac semi-metal in the large N Limit. A version of this chapter is prepared to be published. Chapter 3 is the study of the beta function of charge neutral 2 + 1d Dirac Semi-metal in the large N. A version of this chapter is also prepared to be published. In Chapter 4, we investigate φ 6 theory in the large N limit. A version of this chapter is accepted to be published in Phys. Rev. D .

iv Table of Contents

Abstract ...... ii

Preface ...... iv

Table of Contents ...... v

List of Tables ...... viii

List of Figures ...... ix

Acknowledgments ...... xii

1 Introduction ...... 1 1.1 Graphene ...... 2 1.2 Renormalization Group ...... 10 1.3 Dimensional Reduction of Electromagnetism ...... 16 1.4 Outline and Results ...... 17

2 AC Conductivity of 2 + 1d Dirac Semi-metal in the large N Limit . . 19 2.1 Introduction ...... 19 2.2 Next-to-Leading Order Contributions to h jµ (x) jν (y)i ...... 28

2.2.1 Π1 ...... 31 2.2.2 Evaluation of Π1B ...... 34

2.2.3 Evaluation of Π1A + Π1B ...... 35 2.3 Evaluation of Π2 ...... 36

v 2.4 Combining Π1 and Π2 ...... 37 2.5 The Current-Current Correlator in Presence of a Condensate . . . 40 2.6 Experimental Results ...... 47 2.7 Conclusion ...... 49

3 The Beta Function of Charge Neutral 2 + 1d Dirac Semi-metal in the Large N ...... 50 3.1 Introduction ...... 50 3.2 Corrections to the Electron Propagator ...... 56 3.3 Electron Self-Energy ...... 59 3.4 Infrared Contributions to the Fermi Velocity Beta Function . . . . 66 3.5 Experimental Results ...... 72 3.6 Conclusion ...... 74

4 φ 6 Theory in the Large N Limit ...... 76 4.1 Introduction ...... 76 4.2 Spontaneous Symmetry Breaking ...... 83 4.3 Effective Action Technique ...... 86 6 4.4 Tachyonic Excitations in φ2+1 ...... 91 4.5 Conclusion ...... 109

Bibliography ...... 110

A Trace over Π1 Dirac Matrices ...... 117

B Recurrence Relations for Is ...... 121

C Calculation of I1 ...... 127

D Instantaneous Limit of Graphene in Large N ...... 129

E Dimensional Regularization of φ 6 in Large N ...... 131 E.1.1 ~j2 Term ...... 132

vi E.1.2 ∆(p) Term ...... 132 E.1.3 ∆(p)∆(−p) Term ...... 133 E.1.4 ∆(p)~j2 Term ...... 134 E.1.5 ∆(p)3 Term ...... 135 E.1.6 All Terms Together ...... 136

vii List of Tables

Table B.1 Table of Recurrence Relations for Is ...... 126

viii List of Figures

Figure 1.1 A honeycomb lattice consists of two triangular sub-lattices (black and white atoms)...... 3 Figure 1.2 Graphene spectrum has two bands with opposite signs. Bands meet at K points and form a Dirac cone...... 5 Figure 1.3 Scattering diagram for φ 4 theory...... 11

Figure 2.1 Partial sum over fermionic loops to get the effective propagator for the screening dominated regime...... 20 Figure 2.2 Fermionic bubble, the elementary ingredient for our partial summation...... 22 Figure 2.3 The Feynman diagram of the expansion of the fermion determinant is depicted. The series is even due to particle-hole and time reversal symmetry. The Feynman integrals for diagrams with more than two legs are ﬁnite...... 27 Figure 2.4 The next-to-leading order Feynman digrams that contribute to current-current correlator to the next-to-leading order. . . . . 29 Figure 2.5 The master diagram for the two-loop calculations of the current- current correlation function ...... 29

ix 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 account a nonlinearity and triangular warping of graphene’s electronic spectrum. The gray area indicates the standard error for the measurements. (Inset) Transmittance of white light as a function of the number of graphene layers...... 48

Figure 3.1 The Feynman diagram of the expansion of the fermion determinant is depicted. The series is even due to particle-hole and time reversal symmetry. The Feynman integrals for diagrams with more than two legs are ﬁnite...... 54 Figure 3.2 The leading contribution to the beta function in the large N limit comes from the Feynman diagram where the dotted line is the relativistic large N propagator and the insertion into the photon propagator is the tree-level classical Coulomb action which is non-relativistic. This diagram is of order 1/N2. . . . 57 Figure 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 light velocity is our measure for velocity), we ﬁnd that the Lorentz symmetry prevents the Fermi velocity from running. The beta function in [55] violates this condition as its Lorentz symmetry is violated by construction but not the presence of the two velocities in the Lagrangian. . . . 65 Figure 3.4 We have plotted the beta function in [55](dashed lines) vs. Eq.(3.19) (solid lines). The orange, purple and green lines respectively correspond to N = 4,10,100 ...... 66 Figure 3.5 By zooming into p q regime, we only check that a given theory with an infrared limit that corresponds to the ultraviolet regime of Eq.(3.21) will be infrared divergent free...... 69

x 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 1 mc ∼ n 2 . The dotted line is the behavior of cyclotron mass derived from the standard value of Fermi velocity. Graphene’s spectrum renormalized due to electron-electron interactions is expected to result in the dependence

shown by the solid curve. (b) Cyclotron mass plotted as a variable of vF . . 73 Figure 3.7 Adapted from [20]. N = 1 to N = 6 LLs’ energy as a function of level number for different values of carrier density and B= 2T. For ﬁxed density the curves are highly linear, resulting in a possible negligible renormalization of the Fermi velocity. (Inset) Residuals from the linear ﬁt showing very good linearity in the LLs...... 75

Figure 4.1 N× the beta function of large N regime of g2(~φ 2)3 theory in 2 three dimensions. The infrared fixed point is gIR = 0 and the 2 ultra-violet fixed point occurs at gUV = 192. The critical coupling where in the infinite N limit scale symmetry breaking occurs is g2 = (4π)2 ≈ 158...... 82 Figure 4.2 Spontaneous breaking of the internal rotation symmetry in φ space. The field φ chooses a ground state that violates the internal U(2) symmetry in the potential V(φ) = φ ∗φ...... 84 Figure 4.3 Connected reducible three-point function in terms of irreducible vertices...... 89 Figure 4.4 Connected reducible four-point function in terms of irreducible vertices...... 89 Figure 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 of second order phase transitions and a line of first g2 ~ 2 3 order phase transitions where the potential is equal to 6 (φ ) . 95

xi Acknowledgments

The writing of this dissertation would not have been possible without the help and encouragement of many people.

I am grateful to my supervisor Gordon. W. Semenoff, who let me follow my interests and helped me by his brightness and wide range of knowledge. I would 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 to develop my background in condensed matter physics. I would like to thank Ian Afﬂeck and Eric Zhitnitsky for their enthusiasm in training new physicist; Ian and Eric always welcomed me and my questions. Special thanks to Ian Aﬂeck, Ryan McKenzie, and Rocky So for proofreading this thesis.

To the University of British Columbia for providing me the funding through the Graduate Entrance Fellowship, International Partial Tuition Scholarship, the PhD Tuition Fee Award and University Four Year Fellowship.

I would like to thank my great friends around the world who made life more interesting. Special thanks to Shahzad Ghanbarian, Omid Nourbakhsh, Ali Na- rimani and Hamid Atighechi. And last but not least, I am deeply grateful to my parents for their endless love and devotion to their children.

xii Chapter 1

Introduction

Quantum field theories have been essential tools for studying various physical systems in the past century. Quantum electrodynamics, the first achievement of the quantum field theory paradigm, is still one of the most successful theories used widely in many fields of studies. Quantum field theories equipped with renormalization group techniques provide such a strong paradigm for studying various systems that they are used in many fields outside of physics including social networks and financial markets. We owe our understanding of the fundamental constituents of matter, the elementary particles, to the Standard Model of particles physics, which is a quantum field theory in 3 + 1 d. On the other hand, quantum field theories can be used to model phenomena such as superconductivity in a very different 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 + 1d quantum field theories that enjoy infinite tower of symmetries and 3 + 1 dimensional 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 experimental discovery of the two-dimensional cousin of graphite, graphene in 2004 by Novoselov et al. [1], was a seminal event in electronic materials science. The

1 novel features of graphene such as its emergent relativistic dispersion relation has attracted a lot of attention. The fundamental ingredients of graphene interact via exchanging photons that live in 3+1 d. The interactions turn out to be strong and as a consequence of the low density of electrons in graphene, the usual methods of expansion in terms of particle density seems invalid. Most of our knowledge of quantum ﬁeld theories come from perturbative solutions. In contrast to weakly coupled systems, where one can use the coupling as a perturbation parameter, in strongly coupled ﬁeld theories there is no obvious candidate to be used as the perturbative parameter, a parameter that should be adequately small.

In this chapter, we provide an introduction to the tight-binding model of graphene and its continuum limit as a quantum ﬁeld theory in 2 + 1d. We then present a brief introduction to the renormalization group technique that turns out to be the oil of our machinery. We ﬁnish the introduction by presenting the outline of the remainder of this thesis and a brief introduction to the following chapters.

1.1 Graphene Graphene is made of a honeycomb lattice with two distinct type of atoms. A honeycomb lattice can be described in terms of two triangular sub-lattices, A and B, 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 a2 are given by √ 1 3 a1 = a( , ), 2 2√ 1 3 a = a( ,− ). (1.1) 2 2 2

We deﬁne the following δ vectors such that they connect different sites to each-

2 Figure 1.1: A honeycomb lattice consists of two triangular sub-lattices (black and white atoms).

other.

1 δ = (a − a ) 1 3 1 2 1 δ = (a + 2a ) 2 3 1 2 1 δ = (−2a − a ) (1.2) 3 3 1 2

The sp2 hybridization of the atomic s orbital with two p orbitals leads to the trigonal 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 and the robustness of graphene. The remaining p orbitals (pz) are perpendicular to the plane of this planar structure and have a weak overlap. The resulting covalent bond between pz orbitals of the neighbouring atoms lead to the formation of π band. Therefore, we model our material, graphene, using a tight-binding model

3 for the π orbitals. The Hamiltonian would be given by the following operator

† H = −t ∑ ai,sbi+δ,s + h.c., (1.3) i,δ,s where t ' 2.8eV is the nearest neighbour hopping parameter, ai,s and bi+δ,s are the 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. In the momentum representation the Hamiltonian reads

Z 2 ! d k † 0 ϕ(k) H = ∑ 2 ψs (k) ψs(k) s B.Z. (2π) ϕ(k) 0 Z 2 d k † = ∑ 2 ψs (k)H ψs(k), (1.4) s B.Z. (2π) where

ϕ(k) = −t ∑eik·δ δ ik a ik a k a = −t exp( √y ) + 2exp(− √y )cos x . (1.5) 3 2 3 2

The function ϕ(k) can be decomposed into real and imaginary parts corresponding to its magnitude and phase. As H 2 = ϕϕ∗1, the magnitude of ϕ(k) is indeed the absolute value of the energy eigenvalues and is given by

s √ k a k a 3kya ε(k) = t 1 + 4cos2 x + 4cos x cos . (1.6) 2 2 2

Then the eigenvalues of the Hamiltonian are simply

E±(k) = ±ε(k). (1.7)

4 Figure 1.2: Graphene spectrum has two bands with opposite signs. Bands meet at K points and form a Dirac cone.

We plot this spectrum in Fig.1.2. As the lattice is a bipartite lattice, we observe that there are two energy bands. Discrete symmetries of the original Hamiltonian could indeed be used to show that for any band, there is another band with the opposite sign in the energy. As it can be observed from Fig.1.2, the valence and conduction band meet at six points, the K points. Two of these points are independent and we choose them to be the following K± points

4π K = ± (1,0) (1.8) ± 3a

The linearized dispersion around these points get the form of the dispersion relation for a Dirac cone. The Dirac cone turns out to be massless as a consequence of the discrete symmetries of the Hamiltonian including the spatial inversion symmetry (P : (x,y) → (x,−y)). These discrete symmetries protect the dispersion relation from getting gapped.

5 Let us linearize the Hamiltonian around K points and study the low energy excitations of our model around those points. As graphene is half-ﬁlled, these points are naturally located on the zero-chemical potential line. We can approximate ϕ and linearize it as ϕ(K± + k) = ±vF (k1 ∓ ik2). As the name of the coefﬁcient of the 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 as

Z 2 d k h † † i H = ∑ 2 ψs (K+ + k)HK+ ψs(K+ + k) + ψs (K− + k)HK− ψs(K− + k) , s (2π) (1.9) where the Hamiltonian densities are given by

HK+ = 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 in a more familiar form of 4 × 4 matrices, forming a reducible version of the Dirac Hamiltonian. The irreducible representation of the Dirac algebra in 2+1d is given by 2 × 2 matrices and there are two such irreducible representations. In above Hamiltonian, we are using both of these representations. We can always combine two irreducible representations and make them into a reducible representation. We choose our new basis such that the fermionic ﬁeld is given by ! ΨK+,s(k) Ψs(k) = ΨK−,s(k)   as(K+ + k)    bs(K+ + k)  =  . (1.11)  b (K + k)   s −  as(K− + k)

6 In this basis the Hamiltonian gets the more conventional form of the Dirac Hamil- tonian given by the following Hamiltonian   0 k1 − ik2 0 0 Z 2   d k †  k1 − ik2 0 0 0  H = Ψ (k) Ψs(k) ∑ (2π)2 s  0 0 0 −k + ik  s  1 2  0 0 −k1 − ik2 0 Z 2 d k † = ∑ 2 Ψs (k)H (k)Ψs(k), (1.12) s (2π) where H can be written in terms of the α matrices

1 2 H (k) = α k1 + α k2, (1.13) where α matrices are given by

1 2 α = (α ,α ) = τ3 × (τ1,τ2) ! τ 0 = . (1.14) 0 −τ

This representation is the original representation of the Dirac Hamiltonian that was found by Paul Dirac in a search of a relativistic version of the Schrdinger equation that was linear in momenta.

The more familiar description of Dirac particles in quantum ﬁeld theory is given by the Dirac representation of the Hamiltonian. We deﬁne β as τ1 × 1. The well-known gamma matrices that the Dirac Hamiltonian can be written in terms of and form a Clifford algebra are related to alpha matrices through the following

7 relations

γ0 = β, γ = βα. (1.15)

† If we deﬁne Ψ¯ ≡ Ψ γ0, the Hamiltonian now can be written as

Z 2 d k ¯ 1 2 H = vF ∑ 2 Ψs(k) γ1k + γ2k Ψs(k) s (2π) Z 2 ¯ 1 2 = ivF ∑ d xΨs(x,t) γ1∂ + γ2∂ Ψs(x,t). (1.16) s

The Lagrangian formalism is a more natural gear for quantum field theories. We can infer the Dirac Lagrangian from our Hamiltonian. It can be found simply by applying a Legendre transformation to the Hamiltonian. We introduce a new field that is coupled to our fermionic field through minimal coupling to the fermionic field. The added gauge field can be implemented through Peierls substitution in the original tight-binding Hamiltonian [2]. Our final Lagrangian for the emergent excitations in graphene coupled to a U(1) gauge field, A(x,t), is given by the following well-known Lagrangian

Z h i 2 ¯ 0 0 ~ ~ L = ∑ d x i Ψs(x,t) γ0(∂ − i eA(x,t) ) + vF~γ · (∂ − i eA(x,t)) Ψs(x,t). s (1.17)

The Lagrangian density of Eq.1.17 in a more formal notation using Feynman slash 0 notation, A/ ≡ γ0A + vF~γ ·~A, can be written as

¯ L = ∑ i Ψs(x,t) ∂/ − i eA/(x,t) Ψs(x,t). (1.18) s

To consider the dynamics of the gauge ﬁelds, we need to add their kinetic term to

8 the Lagrangian. The interacting action for the excitations is given by the following

Z h i εc Z 1 S = d3x ψ¯ γt(i∂ + A )ψ + v ψ¯ ~γ · (i~∇ +~A)ψ + d4x ~E2 + c~B2 . a t t a F a a 2e2 c (1.19)

We can examine the relative strength of the above terms and investigate whether we can treat the Coulomb interaction as a perturbation. The ratio between the potential term and the kinetic term is given by

V e2 α = ' , (1.20) K hv¯ F where we used the linear dispersion of the kinetic term and the usual Coulomb potential. In usual many-body systems, this ratio depends on n (the density of the particles) such that the interactions are suppressed in the high density limit. For example, in 2 + 1d materials with a quadratic dispersion relation, this ratio is proportional to n−1/2. However, this ratio for graphene is independent of n and is close to 2.2, a value which is not small. Consequently, the Coulomb interaction term can not be investigated by using the usual coupling expansion scheme.

In the next two chapters, we focus on the effects of electromagnetic interactions in Dirac semi-metals. In graphene, our favorite Dirac semi-metal, the Coulomb interaction term is the dominant interaction and based on the above es- timation, its strength is around few electron volts. Other interactions, including phonon-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. We extend 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 to develop a well-deﬁned perturbative method. In our calculations, this perturbative

9 method takes the place of the usual coupling expansion method.

1.2 Renormalization Group The pioneers of quantum field theory were enormously puzzled by the divergent integrals that they often encountered in their calculations of various physical properties of quantum field theories. They spent the 1930’s and 1940’s struggling with these infinities. As a result of these seemingly inconsistencies in quantum field theories, many advocated abandoning quantum field theory altogether. Eventu- ally, a so called renormalization procedure was developed where the infinities were removed and finite physical results were obtained. However for many years many physicists looked at renormalization theory suspiciously as a sleight of hand. Eventually, starting in the 1960’s a better understanding of quantum field theory was developed through the efforts of Leo Kadanoff and Ken Wilson and many others. Renormalization group was used to infer observable consequences of the presence of these singularities. Field theorists gradually came to realize that divergences in quantum field theory imply deep physical consequences that could not be explained before the invention of renormalization group. Later, many different renormalization schemes where developed that each were more convenient than the others in different scenarios.

The initial treatment of divergences in quantum field theories was based on the fact that only observables need to be finite. Observable quantities in the laboratories are not the same as the bare coupling constants in Lagrangians but are related to them. The main idea was to change the bare coupling in a way that the observable quantities turn out finite. This redefinition of the coupling constants generally makes coupling constants a function of momenta. For example, a coupling constant expansion of the scattering amplitude, as shown in Fig.1.3, in a φ 4 theory in terms of the cut-off and the external momenta can be written as the

10 Figure 1.3: Scattering diagram for φ 4 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-momenta as

2 s = (p1 + p2) , 2 t = (p1 − p3) , 2 u = (p1 − p4) . (1.22)

Λ2 V(x) turn out to be divergent and is given by C ln x where C is a ﬁnite constant and Λ is the hard cut-off. If we assume that the scattering amplitude at a given momentum that corresponds to s0, t0 and u0 is λR, then at any other momenta it can be found to be 2 s t u 3 M = −λR −CλR ln + ln + ln + O(λR) (1.23) s0 t0 u0 and now the scattering amplitude becomes ﬁnite for any momenta. Later, a more

11 systematic way of renormalization was suggested that would make calculations easier for higher order calculations. To do so, one would need to add a finite number of infinite terms called counter-terms that would make the observables finite. As these counter-terms are not observable, one can remove infinities of renormalizable theories consistently without putting the physicists in danger.

In the limit where the Mandelstam ratio becomes large or small, we observe that Eq.(1.23) becomes an invalid approximation. Our perturbative analysis breaks down as the coupling constant, the perturbative parameter, becomes large. To deal with this issue, one can use the renormalization group equation that is known as the Callan-Symanzik equation. This equation is based on the observation that the bare couplings in our Lagrangian are independent of the renormalization criteria which we impose. Callan-Symanzik showed that the renormalized Green’s functions of our theory would satisfy the following differential equation ∂ ∂ (n) µ + β + nγ(λ) G (x ,...,xn; µ,λ) = 0, (1.24) ∂ µ ∂λ 1 where µ is the renormalization scale, β and γ are functions of the coupling constant only dictated by the Lagrangian and n corresponds to the number of ﬁelds in the Green function.

Using the Callan-Symanzik equation, one can ﬁnd that the accurate version of Eq.(1.23) is given by the following expansion in the coupling constant

λR 3 M = − h i + O(λR), (1.25) 1 −Cλ 2 ln s + ln t + ln u R s0 t0 u0 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-

12 tive low energy theory, valid up to some energy scale. This view was suggested by Wilson after the work that was done by Kadanoff to explain the scaling behavior of systems at the critical points. In this scheme, one integrates out the fast modes (UV) and leaves out the slow modes (IR). The question of interest is to ﬁnd a function for the renormalized couplings for the slow modes by starting the analysis with a more fundamental model that consists of both fast and slow modes.

Here we briefly discuss the Wilson renormalization scheme. We follow the path integral treatment discussed in [6]. The scheme has three stages. We first eliminate fast modes by integrating them out. In other words, we divide the modes into two categories, fast and slow. Let’s assume our field representation in momentum space is φ(k) then

φ< = 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 them

S(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 have Z S0(φ<) S0(φ>)+Sint(φ<,φ>) exp(Seff(φ<)) = e [dφ>(k)]e . (1.28)

The beauty of the path integral formalism becomes more evident here as the task of eliminating the fast modes in path integral language is the most natural.

Although Seff provides a good description of the slow mode physics, the renormalization group transformation has two more steps besides the above mode elimination. We need to rescale the momenta so that comparing our renormalized

13 couplings with the previous couplings makes sense. The previous theory was integrated up to Λ while the effective theory is only integrated to Λ/s. We then define k0 = sk which run over the same range as k did before the elimination of the fast modes. However, theories that their actions are only different by a rescaling of the constituting fields are equivalent. To make sure that we incorporate this observation in our analysis, we rescale the field such that a certain coupling in the quadratic part of the action has a fixed coefficient. Explicitly, we rescale the field by a factor ξ and write the effective action in terms of renormalized fields φ 0(k0) given by

0 0 −1 0 φ (k ) = ξ φ<(k/s ). (1.29)

The new couplings that describe the theory made of φ 0(k0)s are our effective coupling constants. In renormalizable field theories, we can use the same trick that we used in Eq.(1.25) and eliminate the dependencies on the cut-off, Λ. We are left with a consistent theory that is not sensitive to the cut-off and works well for slow degrees of freedom. The flow in the coupling constants are typically described using their beta function. For a generic coupling constant, g, its beta function is defined as

dg β(g) = (1.30) d lns

In general, the beta function for a coupling constant at a given scale could behave in 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 this class, if the sign of the beta function persists to stay positive, the coupling be-

14 comes larger after every step of reducing the cut-off and the perturbative analysis breaks down at a given scale. The operators that are associated with these coupling constants are called relevant.

In the second class, the coupling constant decreases by reducing the cut-off. In this case, if the coupling constant stays monotonic, the coupling becomes smaller after every step of the cut-off reduction and loses its importance gradually. The operators that are associated with these coupling constants are called irrelevant as we can ignore them in the strict infrared regime.

In the third class, the coupling does not run (get renormalized) and is independent of the scale. The operators that are associated with these coupling constants are called marginal.

Of great importance are the points in which the beta function vanishes. At these points, the theory becomes scale independent. These points are called the fixed points of the theory and describe the critical behavior of the corresponding Lagrangian. At these points, the correlation length ξ either becomes zero or infinite, corresponding to a trivial fixed point or critical fixed point respectively. Phase transitions happen when the correlation length tends to infinity. The fact that the theory becomes scale invariant at the critical point can be used to explain the observed scaling behavior of critical systems in experiments.

In the next two chapters, we will use the renormalization group technique to study low energy excitations of graphene. It turns out that renormalization group manages to somehow suppress the interactions in graphene. In the last chapter of this thesis, we use renormalization group to study the stability of a conjectured phase in φ 6 theory. Investigating this model and its relevance to graphene would be argued in the last chapter.

15 1.3 Dimensional Reduction of Electromagnetism In this section, we consider a gauge field that lives in 3 + 1d and is coupled to a field that lives in one lower dimension, 2 + 1d. We assume that the gauge field is coupled to the current of the lower dimensional field through minimal coupling and its kinetic term is described by the Maxwell Lagrangian. This model describes the coupling of a 2 + 1d Dirac semi-metal, for example graphene, to the electromagnetic field. The Euclidean action of the gauge field and its coupling to the current is given by the following equation " # Z 1 S = d3+1x F F + A j , (1.32) 4e2 µν µν µ µ where Aµ is the gauge field, Fµν is the electromagnetic field tensor and jµ is the current 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 to live 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 it always carries a delta function, we observe that every interaction vertex for them would carry a delta function. In Fourier space, the delta function results in an integration 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 following expression

Z ∞ dp 1 D(pa pa) = 2 −∞ 2π pa pa + pz 1 1 = √ , (1.33) 2 pa pa where a runs over {0,1,2}.

Now we can use the above propagator to ﬁnd the reduced Lagrangian in 2+1d.

16 It can be written in the following form,

Z " # 3+1 1 1 Sreduced = d x Fab √ Fab + Aa ja . (1.34) 2e2 −∂ 2

1.4 Outline and Results In Chapter 2, we consider a 2+1d defect in quantum field theory living in a 3+1d space-time. We assume that our defect theory has a large number of fermionic flavors, N. The corresponding defect quantum field theory is of interest as it has shown 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 ignored. We investigate the linear response of our model to the electromagnetic fields using a large N perturbative scheme. This investigation is motivated by the observed experimental results which suggest that our model, despite being strongly-interacting can be approximated well by free quantum field theories. We study the next-to-leading order contributions to our model’s current-current correlation function h jµ (x) jν (y)i by performing a large N expansion. We find that the next-to-leading order contributions 1/N to the current-current correlation function is significantly suppressed. The suppression is a consequence of a surprising cancellation between the two contributing Feynman diagrams and is not explicitly dependent on N being large. Through the computed current-current correlator in this chapter, we study the optical conductivity of our Dirac semi-metal system. We calculate the model’s conductivity via the Kubo formula and compare our results with the observed conductivity for graphene. The results show a good agreement with 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 large N expansion to explore the renormalization of the Fermi velocity in the screening dominated regime of the charge neutral graphene with a Coulomb interaction.

17 We find that in contrast to the instantaneous model in which the speed of light is assumed to be infinite and acquires a beta function of order of 1/N for the Fermi velocity, our model acquires a beta function of order 1/N2. We show that inclusion of magnetic fluctuations restores the Lorentz symmetry to the leading contribution and results in a cancellation between electric and magnetic parts. As the next-to-leading order is of order 1/N2, the first non-zero contribution to the beta function turn out to be of the order 1/N2. Consequently, the beta function would much smaller in large N than the values quoted in the current literature and the Fermi velocity would renormalize significantly slower. We prove that the Fermi velocity is gauge invariant given that the infrared regime is divergence-free. We perform a careful analysis of the possible infrared divergences and show that the 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 in three dimensions has a phase which exhibits spontaneous breaking of scale symmetry accompanied by a massless dilaton which is a Goldstone boson. At the next-to-leading order in large N, the phi-six coupling has a beta function of order 1/N and it is expected that the dilaton acquires a small mass, proportional to the beta function and the condensate. However, the stability of the phase is not guaranteed and needs further investigations. In this chapter, we show that this “light dilaton” is actually a tachyon. This indicates an instability of the phase of the theory with spontaneously broken approximate scale invariance. We rule out the existence of the Bardeen-Moshe-Bander phase by showing that the vacuum is unstable in that phase. Our analysis suggests that for a phi-six vector model the only non-trivial critical behavior is present at the Wilson-Fisher point.

18 Chapter 2

AC Conductivity of 2 + 1d Dirac Semi-metal in the large N Limit

2.1 Introduction In this chapter, we use the large N technique to study the current-current correlator of a 2 + 1d defect theory. We study the AC conductivity of a two-dimensional Dirac semi-metal and model the latter using a defect quantum ﬁeld theory consisting of N species of Dirac fermions occupying an inﬁnite planar 2 + 1d defect embedded in 3 + 1d Minkowski space-time. The conductivity of materials is an important quantity in probing their underlying physics and is relatively easy to measure in laboratories. Experimental measurements of graphene’s conductivity show that despite the strong electromagnetic interactions in graphene, the AC conductivity of this material is very close to the AC conductivity of non-interacting 2 + 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 the screening dominated regime of the 2 + 1d defect by doing a partial sum over the fermion loops shown in Fig.2.1 and then study the next-to-leading order contributions to the current-current correlator. The contributions from the electromagnet interactions are suppressed by a factor of N, the number of fermions, and would

19 Figure 2.1: Partial sum over fermionic loops to get the effective propagator for the screening dominated regime. not contribute to our calculations in this chapter. This is due to the fact that we are working in the screening dominated regime in which the induced kinetic term for the gauge ﬁelds would be of order N while the Maxwell Lagrangian is independent of N. This hierarchy in N can be observed by considering the following fermion partial summation and observing that each fermionic species contributes independently 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 integral for a mass-less fermion.

Z d3k 1 1 Πµν = tr γ µ γν fermion-loop (2π)3 /k + /p /k Z d3k /k + /p /k = tr γ µ γν . (2.1) (2π)3 (p + k)2 k2

The functional form of the polarization tensor is dictated by the Ward identity µ µν k Πfermion-loop = 0. The only tensors that can appear in the polarization tensor are the metric η µν and kµ kν . This implies that the polarization tensor is proportional to the projection operator p2η µν − pµ pν . We can ﬁnd the exact form of the polarization tensor by performing the integral. We use dimensional regularization as it makes the calculations easier here

20 Z dD+1k /k + /p /k Πµν (p) = tr γ µ γν fermion-loop (2π)D+1 (p + k)2 k2 Z dD+1k kµ (k + p)ν + kν (k + p)µ − η µν k · (k + p) = 2 . (2.2) (2π)D+1 k2(k + p)2

We use our projector to simplify the calculations. Projecting the polarization tensor using a transverse projector results in

Z dD+1k −2p · k p · (k + p) (p2η − p p )Πµν = µν µ ν fermion-loop (2π)D+1 k2(k + p)2 Z dD+1k 1 = (p2)2 (2π)D+1 k2(k + p)2 3 D+ → p ==1 3 . (2.3) 8

Then the polarization tensor is given by the following ﬁnite tensor

1 Πµν (p) = pη µν − pµ pν /p . (2.4) fermion-loop 16

1 We observe here that the polarization tensor is proportional to p in contrast to the 1 usual p2 behavior. As we observe in the next chapter, the propagator for the gauge ﬁeld after the dimensional reduction to 2 + 1d acquires the same form and has a pole proportional to p instead of p2.

Now we can do the geometric series in Fig.2.1 and focus on the regime that the screening is dominant which corresponds to the regime in which the number of fermions is large. The polarization tensor gets a contribution from any fermion species and it can be written as

N Πµν (p) = pη µν − pµ pν /p , (2.5) leading-order 16

21 Figure 2.2: Fermionic bubble, the elementary ingredient for our partial summation.

in which N is the number of the fermionic species. As we will see soon, this partial sum of fermion loop can be done systematically by investigating the large N expansion of our theory.

The large N expansion is used to be an alternative to a coupling constant expansion in theories that the underlying degrees of freedom interact strongly. In these theories, the coupling constant expansion becomes inaccurate and loses its validity [10]-[17]. Graphene is like a perfect laboratory for examining such large N expansion methods as the coupling constant in graphene, α, is around 2.2 [18]. The experimental results suggest that graphene, despite having strongly interacting underlying degrees of freedom, can be approximated by a non-interacting model [18]-[21] .

There has been a number of studies using both large N and coupling constant expansion methods to investigate various properties of graphene [22]-[45] . The results seem to agree with graphene flowing to a Fermi liquid fixed point with an approximate Lorentz symmetry. in this chapter, we investigate the theoretical validity of the large N expansion by computing the next-to-leading order contributions to the current-current correlation function h jµ (x) jν (y)i. We use several loop calculation techniques developed in quantum field theory to evaluate the two contributing Feynman diagrams to the effective photon kinetic term.

The current-current correlator corresponds to the AC conductivity of the ma-

22 terial through the well known Kubo formalism. The AC conductivity of graphene has been measured in several experiments [9], and was found to be very close to e2 the value of 4h¯ , 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 Kubo formalism and then use it to derive the conductivity for a Dirac semi-metal using the large N limit. Let us review the Kubo formula derivation for the conductivity first. We would like to find the conductivity of a given system. Assume that the external field couples to the current in the following manner

Z δS[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 the variables of j and A. We need to find the vacuum expectation value (VEV) of the current, h ji. If the external field can be treated as a perturbation (probe field), we expect the VEV to be linear in the external field and the proportionality constant would be called the conductivity. Using the fact that h ji can be generated via the action in the following way, we can proceed further and find a formula for the conductivity

1 Z h j(1)i = Dψe−S[A, j;ψ] j(1) Z δ lnZ[A, j] = − δA(1) Z δ 2 lnZ[A, j] ' − | A(2). (2.7) δA(1)δA(2) A=0

In the first line, ψ is the collection of the fields and Dψ is the path integral measure of this collection; j(1) means that we are computing the field configuration at the point 1 of the integration space and the same for A(1). In last line, we used perturbative expansion of Z, and the integration is over the coordinates of A(2).

23 We ﬁnd that the linear response coefﬁcient is given by the following expression

δ 2 lnZ[A, j] K(1,2) = − | . (2.8) δA(1)δA(2) A=0

The functional derivatives would produce two terms. Assuming that there is no current in the absence of the external field, the term that is proportional to VEV of the current would vanish in absence of an external field and the response coefficient of our material, K, can be written as

1 δ 2Z[A, j] K(1,2) = − | . (2.9) Z δA(1)δA(2) A=0

We can write down the Kubo formula in terms of the current-current correlator by doing the functional derivatives. Using the deﬁnition for the current, we ﬁnd that

1 δ Z K(1,2) = − Dψe−S[ j,A;ψ] j(2;A) Z δA(1) A=0

= −hδ(1 − 2)∂A j(A)|A=0i + h j(1;A = 0) j(2;A = 0)i. (2.10)

The first term in the above formula is known as the diamagnetic term and the minimally coupled fermionic field turns out to be proportional to the current expectation value. In this chapter, we focus on the second term that is known as the paramagnetic term as we are interested in linear response in presence of a probe field and the diamagnetic term is not present in this limit.

Th actual conductivity is a retarded response and one needs to be more careful as in above we used the time-ordered correlation functions. To take this observation into account, we first need to employ the general formalism to derive the imaginary-time response and then analytically continue to real frequencies. We also notice that the actual optical conductivity is the linear response to the electric field, which is the time derivative of the gauge field considered above. The usual optical conductivity is given by the following formula

24 1 σ(ω) = − lim K(q) . (2.11) ~q→0 ω iω→ω+i0

The conductivity for graphene has been calculated using coupling constant expansion formalism [42],[43]. The conductivity in the regime that the coupling constant expansion is accurate gets the following form

2 e 2 σ = 1 + C α + O(α ) , (2.12) 4 in which α is the fine structure for graphene and C is a constant. There has been a controversy on the exact value of C . A few calculations find C to be 19−6π 22−6π C = 12 ' 0.01 [47, 48] while other authors find C = 12 ' 0.26 [49, 50]. It is suggested [43] that this controversy stems from the inaccurate use of the reg- 19−6π ularizations and the accurate result is given by C = 12 . As the fine structure constant for graphene is around 2.2, the coupling constant expansion seems to be less reliable and for the latter value for C the next-to-leading order contribution makes the expansion converge poorly.

In this chapter, we will model graphene as an infinite 2 + 1d defect embedded in 3 + 1d space-time. Graphene electrons are the degrees of freedom that are occupying the the 2+1d sheet [51, 52]. As we discussed, the tight-binding model of graphene results in emergent massless electrons with an emergent relativistic dispersion relations. The Lorentz symmetry of this emergent dispersion relation will simplify the future calculations significantly. The electrons interact by exchanging 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 Euclidean

25 space action

Z h i εc Z 1 S = d3x ψ¯ γt(i∂ + A )ψ + v ψ¯ ~γ · (i~∇ +~A)ψ + d4x ~E2 + c~B2 . a t t a F a a 2e2 c (2.13)

The ﬁrst integral is over the 2+1d defect with the integrand being the Lagrangian density of N species of two-component spinor fermions. The Lagrangian has an extra global U(N) symmetry. For graphene, N = 4 which corresponds to the two valleys and the spin degree of freedom. In the following, we test the validity of the large 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 and consider their effect on the conductivity. Short range interactions can be studied using the usual Hubbard-Stratonovich transformation. We present a detailed review of this transformation in Chapter 4. Hubbard-Stratonovich transformation introduces a new scaler field in the Lagrangian that is coupled to the fermions through a Yukawa interaction term, gφψψ¯ . 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 be distinguished 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 the effective ﬁeld theory for the gauge ﬁelds. The fermions contribute terms in the effective action which are given by the Feynman diagrams in the series depicted

26 Figure 2.3: The Feynman diagram of the expansion of the fermion determinant is depicted. The series is even due to particle-hole and time reversal symmetry. The Feynman integrals for diagrams with more than two legs are ﬁnite.

in Fig. 2.3. We obtain the following effective action for our Dirac semi-metal   Z N 3 1/vF vF Seff = d x~E · r ~E +~B r ~B. ~ 2 1 2 ~ 2 1 2 16 −∇ − 2 ∂t −∇ − 2 ∂t vF vF ∞ Z ...... + N ∑ dx1 dxnAµ1 (x1)...Aµn (xn)Γµ1...µn (x1, ,xn) n=4 Z " # εc 3 1/c c + d x ~E · q ~E +~B q ~B (2.14) e2 −~∇2− 1 ∂ 2 −~∇2− 1 ∂ 2 c2 t c2 t

The ﬁrst line in Eq.(2.14) is the expansion of the fermion determinant to the quadratic order in the gauge ﬁelds. It has a factor of N in front of it. This term contains the leading order screening of the Coulomb interaction by the relativistic electrons. The second line in Eq.(2.14) contains the higher order, multi-photon interaction terms coming from the fermion determinant. These terms are of order N.

... To emphasize this, we have explicitly extracted a factor of N. The Γµ1...µn (x1, ,xn) are connected irreducible multi-photon correlation functions arising from a single 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 (with photon-photon scattering). In the third line of Eq.(2.14) , we have presented the

27 same Maxwell theory of the photon as in the second line of Eq.(2.13), but dimensionally reduced to 2 + 1d, with the component of the photon ﬁeld that is perpendicular 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 leading order in the large N limit. The theory obtained by retaining the first two lines of Eq.(2.14), which is a Lorentz invariant conformal field theory whose speed of light is vF . In the following, we are interested in the evaluation of the second line up to three-loops. To have a sensible expansion, the new contributions needs to be suppressed in comparison to the leading order contributions.

2.2 Next-to-Leading Order Contributions to h jµ(x) jν(y)i In this section, we shall compute the next-to-leading order contributions to the current-current correlation function h jµ (x) jν (y)i. We would need to compute the contributions 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 computation 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 can not be reduced to a one-loop calculation. Indeed the corresponding topology of Fig.2.5 is the most general topology in the two-loops calculations of our theory and all other diagrams are equivalent to this topology. Our main task would be ﬁnding 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, let D E us deﬁne the following convention. For the rest of this chapter, we deﬁne ...

28 Figure 2.4: The next-to-leading order Feynman digrams that contribute to current-current correlator to the next-to-leading order.

as the following integral

D E Z d3 p d3q ... ≡ .... (2.15) R6 (2π)3 (2π)3

The next-to-leading order contribution to the current-current correlation function µν has a complicated form and can be written as a sum of two distinct terms, Π1 (k) µν and Π2 (k). Each corresponds to a distinct Feynman diagram in Fig.2.4. We can write down the algebraic form of each Feynman diagram following the

Figure 2.5: The master diagram for the two-loop calculations of the current- current correlation function usual Feynman recipe. Due to the presence of several vertices and propagators,

29 the integrals get a relatively complicated form

µν µν µν Π (k) = Π1 (k) + Π2 (k), (2.16) Tr(/q + /p)γ µ (/q + /p + /k)γρ (/q + /k)γν /qγρ Πµν (k) = −16 , (2.17) 1 (q + p)2(q + p + k)2(q + k)2q2 ps Tr/qγ µ (/q + /k)γν /qγρ (/q + /p)γρ Πµν (k) = −32 . (2.18) 2 (q + p)2(q + k)2[q2]2 ps

In the above expression, we used Πleading-order that we found previously and is defined as 16 kρ kσ Πρσ (k) = (δ ρσ + κ ). (2.19) leading-order N k2 The minus sign comes from the presence of a fermionic loop in our Feynman diagrams. The combinatoric factor is one for each diagram. The additional factor of two in Π2 is due to the fact that there are two diagrams. We are working in the Feynman gauge and have regulated the integrals by changing the photon propa- 16 δ ρσ 16 δ ρσ gator from N p to a new propagator given by N ps . This will be sufficient to define the integrals properly and regulate the ultraviolet divergences that would be present in our integrals.

As a consequence of the Ward identities, Lorentz invariance and dimensional analysis, the vacuum polarization gets a particular form. The vacuum polarization generally has the form given by the following expression

k k Π (k) = Π(k) δ − µ ν , (2.20) µν µν k2 which makes the computations simpler; as a result, we will compute Π(k) from the following trace,

1 Π(k) = Π (k) = 16Π + 16Π . (2.21) 2 µµ 1 2

We perform our calculations in Euclidean space and can use the following deﬁni-

30 tions for the Clifford Algebra in a space with arbitrary dimensions and Euclidean signature. We use the Feynman slash notation deﬁned by /p = γµ pµ .

{γ µ ,γν } = 2δ µν , {/p,/q} = 2/p/q. (2.22)

With a small amount of manipulation, the above Clifford Algebra could be used to show that in 2 + 1d, the following identities hold. These identities need to get corrected in other dimensions

µ γ /pγµ = −/p, tr(γ µ γν γρ γσ ) = 2(δ µν δ ρσ − δ µρ δ νσ + δ µσ δ 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 reported in the appendices instead.

2.2.1 Π1

We begin with Π1, which is the more complicated contribution to Π. This contribution is a faithful two-loop calculation and could not be decomposed into multi- ple one-loop calculations.

1 Tr(/q + /p)γ µ (/q + /p + /k)γρ (/q + /k)γ µ /qγρ Π = − (2.24) 1 2 (q + p)2(q + p + k)2(q + k)2q2 ps

In Appendix A, we computed the numerator in equation Eq.(A.7) and simpliﬁed it using the symmetries of the integral that it belongs to. We would need to use the identities that follow the Clifford Algebra and were listed previously. We copy

31 the results here

µ ρ µ ρ Num1 =Tr(/q + /p)γ (/q + /p + /k)γ (/q + /k)γ /qγ = + (q + p + k)2 8k2 − 4(q + k)2 + 2q2 + 8p2 − 4(q + p)2 − 2k4 − 2p4 − 5k2 p2. (2.25)

Now, we insert the numerator into the Feynman integral and obtain the following integral

1D 2k4 + 2p4 + 5k2 p2 Π (k) = − − + 1 2 (q + p)2(q + p + k)2(q + k)2q2 ps (q + p + k)2 8k2 − 4(q + k)2 + 2q2 + 8p2 − 4(q + p)2E (q + p)2(q + p + k)2(q + k)2q2 ps

≡ Π1A + Π1B. (2.26)

We have divided Π1 into two separate terms Π1A and Π1B. We shall compute each term in the following subsections.

Π1A

Consider the ﬁrst contribution to Π1A,

1 8k2 − 4(q + k)2 + 2q2 + 8p2 − 4(q + p)2 Π = − 1A 2 (q + p)2(q + k)2q2 ps D −4k2 2 = + + (q + p)2(q + k)2q2 ps (q + p)2q2 ps 1 4p2 − − (q + p)2(q + k)2 ps (q + p)2(q + k)2q2 ps 2 E + . (2.27) (q + k)2q2 ps

The second and the last term have no dependence on external momenta, k, and consequently must contribute zero to Π1A. Then we are left with the following

32 three terms

D −4k2 1 4p2 E Π = − − . 1A (q + p)2(q + k)2q2 ps (q + p)2(q + k)2 ps (q + p)2(q + k)2q2 ps (2.28)

The p integral can be done using standard one-loop integrals [53]. Let us copy the formula here

Z d3 p 1 1 1 Γ[A + B − 3 ]Γ[ 3 − A]Γ[ 3 − B] = 2 2 2 . 3 2 A 2 B 3 2 A+B−3/2 (2π) [p + q) ] [p ] (4π) 2 [q ] 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. The new 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 deﬁnes Gamma functions, Γ[A] = (A−1)Γ[A−1], we arrive at the following simple result.

s 1 1 3 s *" 2 # 1 Γ[ 2 − 2 ]Γ[ 2 ]Γ[ 2 − 2 ] −4k 1 Π1A = + 3 s s s 1 − s 1 2 2 + 2 2 − (4π) 2 Γ[ 2 ]Γ[2 − 2 ] (q + k) [q ] 2 2 (q + k) [q ] 2 2 s 3 1 5 s + 1 Γ[ 2 − 2 ]Γ[ 2 ]Γ[ 2 − 2 ] −4 + 3 s s s 1 2 2 − (4π) 2 Γ[ 2 − 1]Γ[3 − 2 ] (q + k) [q ] 2 2 " −1 Γ[ s − 1 ]Γ[ 1 ]Γ[ 3 − s ] Γ[ s ]Γ[ 1 ]Γ[1 − s ] = 2 2 2 2 2 2 2 2 16 3ks−2 Γ[ s ]Γ[2 − s ] s 1 3 s π 2 2 Γ[ 2 + 2 ]Γ[ 2 − 2 ] Γ[ s − 1 ]Γ[ 1 ]Γ[ 3 − s ] Γ[ s − 1]Γ[ 1 ]Γ[2 − s ] + 2 2 2 2 2 2 2 2 4Γ[ s ]Γ[2 − s ] s 1 5 s 2 2 Γ[ 2 − 2 ]Γ[ 2 − 2 ] # Γ[ s − 3 ]Γ[ 1 ]Γ[ 5 − s ] Γ[ s − 1]Γ[ 1 ]Γ[2 − s ] + 2 2 2 2 2 2 2 2 (2.30) Γ[ s − 1]Γ[3 − s ] s 1 5 s 2 2 Γ[ 2 − 2 ]Γ[ 2 − 2 ]

33 " # 1 4 1 4 Π = + + 1A 16π2ks−2 (2 − s)(1 − s) (2 − s)(3 − s) (4 − s)(3 − s) 1 3(20 − 15s + 3s2) = . (2.31) 16π2ks−2 (4 − s)(3 − s)(2 − s)(1 − s)

2.2.2 Evaluation of Π1B

Now, consider the other contribution to Π1 given by the following integral

1 2k4 + 2p4 + 5k2 p2 Π = 1B 2 (q + p)2(q + p + k)2(q + k)2q2 ps 5 = k4I + k2I + I , (2.32) s 2 s−2 s−4 where we have deﬁned Is as

1 I = . (2.33) s (q + p)2(q + p + k)2(q + k)2q2 ps

In Appendix C, we evaluate I1 using alpha parameter representation. Here, I1 is an 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-known Triangle Relations [53]. Despite the complicated form of the integrals and the tedious calculation that is needed to compute I1, the result turns out to be given by a simple expression. Here we quote the result for I1

1 I = . (2.34) 1 64ks+2

In the Appendix B, we derive identities that are similar to the Triangle identities and use the recurrence relations derived to ﬁnd an expression for Is−2 and Is−4.

34 Moreover, Is−2 and Is−4 are given by the following equation 2 1 − s 2 1 s 3 − s I = − k Is − + (2.35) s−2 2 − s 2 − s 32π2ks 2 − s 1 − s " 4 (3 − s)(1 − s) 1 1 I =k Is + s−4 (4 − s)(2 − s) 16π2ks−2 4 − s # s(3 − s) (3 − s)2 5 − s 2 − s + − + . (2.36) (2 − s)2 (2 − s)(1 − s) 3 − s 4 − s

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 an explicit result for Π1B. Let us report the results for Π1B here, in terms of Is, Is−2 and Is−4.

5 Π =k4I + k2I + I 1B s 2 s−2 s−4 ( 5 1 − s (3 − s)(1 − s) 4 1 = 1 − + k Is + 2 2 − s (4 − s)(2 − s) 32π2ks−2 " # 2 s(3 − s) (3 − s)2 5 − s 2 − s + − + 4 − s (2 − s)2 (2 − s)(1 − s) 3 − s 4 − s " #) 5 s 3 − s − + . (2.37) 2 − s 2 − s 1 − s

2.2.3 Evaluation of Π1A + Π1B

In the previous subsections we found Π1A and Π1B. We combine them here to ﬁnd an implicit formal for one of the Feynman diagrams self energies, Π1. Although the results seem complex, in the next sections we see that combination of both diagrams cancels most of that complexity and leaves a simple result. Let us ﬁnd

Π1 by combining our results for Π1A and Π1B.

35 Π1 = Π1A + Π1B

( 5 1 − s (3 − s)(1 − s) 4 1 Π = + 1 − + k Is + 1 2 2 − s (4 − s)(2 − s) 32π2ks−2 " # 2 s(3 − s) (3 − s)2 5 − s 2 − s + − + 4 − s (2 − s)2 (2 − s)(1 − s) 3 − s 4 − s ) " 5 s 3 − s 1 4 − + + 2 − s 2 − s 1 − s 16π2ks−2 (2 − s)(1 − s) # 1 4 + + . (2.38) (2 − s)(3 − s) (4 − s)(3 − s)

2.3 Evaluation of Π2

In 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 convenience

DTr/qγ µ (/q + /k)γν /qγρ (/q + /p)γρ E Πµν (k) = −2 . (2.39) 2 (q + p)2(q + k)2[q2]2 ps

µν µν We ﬁnd the trace of Π2 (k) and use it for calculating Π2 (k) in the last step. The µν trace of Π2 (k) is given by the following expression

Tr/q(/q + /k)/q(/q + /p) Π (k) = − 2 (q + p)2(q + k)2[q2]2 ps 2q · (q + k)q · (q + p) − q2(q + k) · (q + p) = − 2 . (2.40) (q + p)2(q + k)2[q2]2 ps

36 First by introducing the Feynman parameter, we do the integral over p and then perform the integration on the Feynman parameter

s * Γ[ + 1] Z 1 s 2 2 −1 Π2(k) = −2 s dα(1 − α) Γ[ 2 ] 0 + 2q · (q + k)q · (q(1 − α) + p) − q2(q + k) · (q(1 − α) + p) s +1 (q + k)2[q2]2[p2 + α(1 − α)q2] 2

s * q·(q+k) + Γ[ + 1] Z 1 s (q+k)2 = −2 2 dα(1 − α) 2 s 2 2 2 s +1 Γ[ 2 ] 0 q [p + α(1 − α)q ] 2 s s 1 * + 2Γ[ + 1]Γ[ − ] Z 1 s q · (q + k) 2 2 2 2 = − 3 dα(1 − α) s 1 s 1 . s 2 s 0 2 2 2 + 2 2 − 2 Γ[ 2 ](4π) Γ[ 2 + 1] (q + k) [q ] [α(1 − α)] (2.41)

Now, the only variable that is left out to be integrated over is q. After a few steps of algebra, we arrive at a relatively simple answer for Π2 which we use in the next section to evaluate the ﬁnal form of Π, the full vacuum polarization tensor

s s 1 3 3 s Z 3 Γ[ 2 + 1] Γ[ 2 − 2 ] Γ[ 2 ]Γ[ 2 − 2 ] d q q · (q + k) Π2(k) = −2 s 3 s 3 s 1 Γ[ ] 2 s Γ[3 − ] (2π) 2 2 2 + 2 2 (4π) Γ[ 2 + 1] 2 (q + k) [q ] s 1 3 3 s Z 3 2 2 2 1 Γ[ 2 − 2 ] Γ[ 2 ]Γ[ 2 − 2 ] d q k − q − (q + k) = 3 s s s 1 3 2 2 + (4π) 2 Γ[ 2 ] Γ[3 − 2 ] (2π) (q + k) [q ] 2 2 " # 1 Γ[ s − 1 ]Γ[ 3 − s ] Γ[ s ]Γ[1 − s ] Γ[ s − 1]Γ[2 − s ] = 2 2 2 2 2 2 − 2 2 128π2ks−2 Γ[ s ]Γ[3 − s ] s 1 3 s s 1 5 s 2 2 Γ[ 2 + 2 ]Γ[ 2 − 2 ] Γ[ 2 − 2 ]Γ[ 2 − 2 ] 1 1 = − . (2.42) 8π2ks−2 (4 − s)(3 − s)(2 − s)(1 − s)

2.4 Combining Π1 and Π2 Finally, we have all of the pieces to put together and ﬁnd the next-to-leading order correction to the current-current correlator. We simply add Π1(k) and Π2(k)

37 derived in the previous sections and deduce the next-to-leading order contribution we were seeking

Π = Π1 + Π2 ( 2 5 − 5s (3 − s)(1 − s) 4 1 2 h 3s − s = 1 − + k Is + 4 − 2s (4 − s)(2 − s) 32π2ks−2 4 − s (2 − s)2 (3 − s)2 5 − s 2 − si 5 s 3 − s h 8 + − + − + + (2 − s)(1 − s) 3 − s 4 − s 2 − s 2 − s 1 − s (2 − s)(1 − s) ) 2 8 i 4 + + − . (2.43) (2 − s)(3 − s) (4 − s)(3 − s) (4 − s)(3 − s)(2 − s)(1 − s)

The residue of the pole at s → 1 cancels out and leaves a ﬁnite result for the polarization tensor. We can therefore combine the terms with the pole in order to cancel the singularity and obtain the following ﬁnite result at the region that s is close to one ( h 5 1 − s (3 − s)(1 − s)i 4 1 Π = 1 − + k Is + 2 2 − s (4 − s)(2 − s) 32π2ks−2 2 hs(3 − s) 5 − s 2 − si 5 h s i h − + − + 4 − s (2 − s)2 3 − s 4 − s 2 − s 2 − s ) 2 8 i −34 + 21s − 3s2 + + . (2.44) (2 − s)(3 − s) (4 − s)(3 − s) (4 − s)(3 − s)(2 − s)

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 of I = 1 1 64ks+2 , we ﬁnally arrive at an explicit result for Π, which has a surprisingly simple form and is given by the following expression

k 92 Π = 1 − . (2.45) 64 9π2

38 16 Restoring the overall factors of N , the polarization tensor and N, we can ﬁnd Πµν by using our explicit form for Π. The equation for Πµν is given by the following

k 92 k k Π = 1 − δ − µ ν . (2.46) µν 4 9π2 µν k2

Now we can add our next-to-leading order result to the leading-order result and ﬁnd the corrected polarization tensor " # N 1 92 k k π = + 1 − + ... kδ − µ ν . (2.47) µν 16 4 9π2 µν k

For graphene, there are four fermion species in its Lagrangian, N = 4, and the numeric 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. This small numeric factor for the next-to-leading order is a result of a miraculous cancellation that we observed between the two Feynman diagrams which contribute to the next-to-leading order.

Using the derived Kubo formalism, we can ﬁnd the conductivity of the phase that does not have a condensate. It is simply given by the following equation for a Dirac semi-metal with N fermion species

N 1 92 σ = + 1 − + .... (2.48) 16 4 9π2

To compare our results with experimental results and the previously found results through theoretical works, we rescale the gauge ﬁeld so that our units match the standard units used by them, A → eA. For graphene, we ﬁnd that the conductivity is given by

e2 σ = (1 − 0.03) + .... (2.49) 4

39 2.5 The Current-Current Correlator in Presence of a Condensate In this section, we investigate the effect of a scaler ﬁeld that is coupled to the fermions through a Yukawa term, gφψψ¯ . We will ﬁnd the new current-current correlator of the fermions and deduce the conductivity of this phase. As discussed in the introduction, this model describes the phase of our system in which the short range interactions introduce a condensate in the Lagrangian. After rescaling the φ and replacing it with φ/g, the action that describes this phase is given by

Z h i εc Z 1 S = + d3x ψ¯ γt(i∂ + A )ψ + v ψ¯ ~γ · (i~∇ +~A)ψ + d4x ~E2 + c~B2 a t t a F a a 2e2 c Z + d3xφψψ¯ + ..., (2.50) where the kinetic term and the higher order interaction terms of φ field are dropped as they contribute to the higher orders of N. The fermions can be integrated out in the same way that we integrated them out and found Eq.(2.14). The propagator for the scaler field (after the same partial re-summation that we did for the vector field) would be given with a similar functional form but with a different numerical coefficient,

Z d3k 1 1 πleading-order = −N tr (2π)3 /k + /p /k Z d3k 2k · (k + p) = −N (2π)3 (k + p)2k2 Z d3k (k + p)2 + k2 − p2 = −N . (2.51) (2π)3 (k + p)2k2

40 The calculation of πleading-order is straightforward, we ﬁnd that

Z d3k 1 1 π = N p2 leading-order (2π)3 (k + p)2 k2 1 1 1 N Γ[ 2 ]Γ[ 2 ]Γ[ 2 ] = 3 p (4π) 2 Γ[1] N = p. (2.52) 8

The next-to-leading contribution to the current-current correlator can be found by using our results for the leading-order result above. The re-summed propagator 8 1 for the scaler ﬁeld is given by N p . Using the usual Feynman rules, we ﬁnd that the next-to-leading contributions are given by

µν µν µν π (k) = π1 (k) + π2 (k), (2.53) Tr(/q + /p)γ µ (/q + /p + /k)(/q + /k)γν /q π µν (k) = −8 , (2.54) 1 (q + p)2(q + p + k)2(q + k)2q2 ps Tr/qγ µ (/q + /k)γν /q(/q + /p) π µν (k) = −16 . (2.55) 2 (q + p)2(q + k)2[q2]2 ps

As a consequence of the Ward identities, Lorentz invariance, and dimensional analysis, the vacuum polarization acquires a particular form. The vacuum polarization generally has the form given by the following expression

k k π (k) = π(k) δ − µ ν , µν µν k2 which makes the computations simpler. As a result, we will compute π(k) from the following trace 1 π(k) = π (k) = 8π + 8π , 2 µµ 1 2

41 and then recover πµν (k). The traces of π1 and π2 are

1 Tr(/q + /p)γ µ (/q + /p + /k)(/q + /k)γ µ /q π (k) = − , (2.56) 1 2 (q + p)2(q + p + k)2(q + k)2q2 ps Tr/qγ µ (/q + /k)γ µ /q(/q + /p) π (k) = − . (2.57) 2 (q + p)2(q + k)2[q2]2 ps

We use the trace identities to simplify the traces and write them in terms of inner products of the momenta,

µ µ Num1 =Tr(/q + /p)γ (/q + /p + /k)(/q + /k)γ /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 π1 can be written as

2 2 2 2 Num1 = − (q + p + k) (−q − 2(q + p) + 2p ) − (q + k)2(−2q2 − (q + p)2 + 2p2) − p2(2q2 + 2(p + q)2 − 2p2 − k2) . (2.59)

42 Using the symmetries of the integral, we can simplify the numerator of π1 and ﬁnd

2 2 2 2 Num1 = − (q + p + k) (−2q − 4(q + p) + 8p ) + 2p4 + p2k2 . (2.60)

As before, we divide the terms into two groups, π1A and π1B, and evaluate each individually,

π1(k) = π1A + π1B, in which

1 −2q2 − 4(q + p)2 + 8p2 π = + 1A 2 (q + p)2(q + k)2q2 ps D 1 2 4p2 E = − + − . (2.61) (q + p)2(q + k)2 ps (q + k)2q2 ps (q + p)2(q + k)2q2 ps

The second term in the second line vanishes and we ﬁnd that

D −1 4p2 E π = + . (2.62) 1A (q + p)2(q + k)2 ps (q + p)2(q + k)2q2 ps

43 The evaluation of the remaining parts of π1A is straightforward. The integrals can be decomposed into 1-loop integrals, we ﬁnd that

* s 1 1 3 s 1 Γ[ 2 − 2 ]Γ[ 2 ]Γ[ 2 − 2 ] 1 π1A = − 3 s s s 1 2 2 − (4π) 2 Γ[ 2 ]Γ[2 − 2 ] (q + k) [q ] 2 2 s 3 1 5 s + 1 Γ[ 2 − 2 ]Γ[ 2 ]Γ[ 2 − 2 ] −4 + 3 s s s 1 2 2 − (4π) 2 Γ[ 2 − 1]Γ[3 − 2 ] (q + k) [q ] 2 2 " 1 Γ[ s − 1 ]Γ[ 1 ]Γ[ 3 − s ] Γ[ s − 1]Γ[ 1 ]Γ[2 − s ] = − + 2 2 2 2 2 2 2 2 16π3ks−2 4Γ[ s ]Γ[2 − s ] s 1 5 s 2 2 Γ[ 2 − 2 ]Γ[ 2 − 2 ] # Γ[ s − 3 ]Γ[ 1 ]Γ[ 5 − s ] Γ[ s − 1]Γ[ 1 ]Γ[2 − s ] − 2 2 2 2 2 2 2 2 Γ[ s − 1]Γ[3 − s ] s 1 5 s 2 2 Γ[ 2 − 2 ]Γ[ 2 − 2 ] " # 1 1 4 = − 16π2ks−2 (2 − s)(3 − s) (4 − s)(3 − s) 1 3s − 4 = . (2.63) 16π2ks−2 (4 − s)(3 − s)(2 − s)

We need to take the limit in which s → 1, around s = 1 it is given by

k π = − . (2.64) 1A 96π2

Now, we consider the other part of π1 which is given by the following integral

1 2p4 + p2k2 π = − 1B 2 (q + p)2(q + p + k)2(q + k)2q2 ps 1 = − k2I − I . (2.65) 2 s−2 s−4

44 To evaluate π1B, we use our results in Appendices B and C. Using the recurrence relations that we found, π1B can be written as 4 1 − s 1 1 s 3 − s 4 (3 − s)(1 − s) π == + k Is + + − k Is 1B 4 − 2s 2 − s 32π2ks−2 2 − s 1 − s (4 − s)(2 − s) " # 1 1 s(3 − s) (3 − s)2 5 − s 2 − s − + − + . (2.66) 16π2ks−2 4 − s (2 − s)2 (2 − s)(1 − s) 3 − s 4 − s

Again, we take the limit that s → 1. Around s = 1, π1B is given by

k 1 k 7 π = + + . (2.67) 1B 48π2 −1 + s 48π2 6

We found an exact form for π1 and the only missing piece is π2, it is given by

Tr/qγ µ (/q + /k)γ µ /q(/q + /p) π (k) = − 2 (q + p)2(q + k)2[q2]2 ps Tr/q(/q + /k)/q(/q + /p) = + . (2.68) (q + p)2(q + k)2[q2]2 ps

The above expression has the exact form of (2.57). We recopy the results here for readers’ convenience.

1 1 π (k) = (2.69) 2 8π2ks−2 (4 − s)(3 − s)(2 − s)(1 − s)

Around s = 1, it is given by

k π (k) = − (2.70) 2 48π2(s − 1)

45 Now, we have all of the pieces and we can add them to get π(k),

π(k) =π (k) + π (k) + π (k) 8 1A 1B 2 k = − 96π2 k 1 k 7 + + 48π2 −1 + s 48π2 6 k 1 − 48π2 −1 + s k = + . (2.71) 72π2

Recovering the tensor form of the self-energy, we ﬁnd that πµν is given by

1 k k π = + δ − µ ν . (2.72) µν 9π2 µν k2

Now we can add our next-to-leading order result to the leading-order result and ﬁnd the corrected polarization tensor. We add the new contribution coming from the condensate to the previous results in Eq.(2.48) and ﬁnd that " # N 1 92 1 k k Π¯ = + 1 − + + ... kδ − µ ν µν 16 4 9π2 9π2 µν k " # N k k = − 0.008 + 0.011 + ... kδ − µ ν . (2.73) 16 µν k

Or using the Kubo formalism, we can ﬁnd the conductivity

N 1 92 1 σ = + 1 − + + ... (2.74) 16 4 9π2 9π2

46 After rescaling the gauge ﬁeld so that our units match the standard units used, we ﬁnd that the conductivity of the phase with the condensate is given by

e2 σ = (1 − 0.03 + 0.04) + .... (2.75) 4

As discussed in the introduction, the dependence of the conductivity on the phase can be used experimentally to distinguish the phases that the Dirac semi-metal lives in.

2.6 Experimental Results Graphene’s optical conductivity has been determined experimentally by measuring the transmittance T and reﬂectance R of suspended graphene sheets [7–9]. The experimental results of these studies afﬁrm that graphene’s conductivity is very close to the conductivity of non-interacting 2 + 1d fermion gases.

In these experiments, the transmittance (T) and reflectance (R) of a sample are directly recorded by passing a beam of light through the sample. The conductivity is then derived using the measured T and R. The authors find the conductivity by using the transmittance through the formula T = (1 + 2πσ/c)−2. They find e2 that the optical conductivity of a single layer graphene is given by (1.0±0.15) 4h¯ , e2 e2 (1.0 ± 0.1) 4h¯ , and (1.01 ± 0.04) 4h¯ [respectively in [7], [8], and [9]]. These results are very close to the optical conductivity of non-interacting two-dimensional e2 Dirac fermions σ = 4h¯ , illustrated in Fig.2.6. The experimental results are in agreement with our theoretical result for graphene’s optical conductivity with- /without the presence of a condensate. In the phase that the condensate is absent, e2 we find that the conductivity of graphene is given by σ = 4h¯ (1 − 0.035) and in the e2 phase with the condensate, it is given by σ = 4h¯ (1 + 0.054). Both of the results are in the error interval of the experimental results and more accurate experimental results are needed for choosing the phase in which graphene lives in. As discussed in [9], surface contamination of graphene membranes by hydrocarbon

47 decreases the experiments’ accuracy. Measuring conductivity of cleaner samples of graphene and also using devices with wider ranges of frequencies can provide more 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 account a nonlinearity and triangular warping of graphene’s electronic spectrum. The gray area indicates the standard error for the measurements. (Inset) Transmittance of white light as a function of the number of graphene layers.

48 2.7 Conclusion We investigated a defect quantum field theory in 3 + 1d with large number of fermion flavors and studied the next-to-leading order contributions to the current- current correlation function h jµ (x) jν (y)i. We found that the next-to-leading order contributions from the fermion determinant to the effective kinetic term of photons are significantly suppressed. This suppression helps to validate the large N expansion of the interacting quantum field theory that models graphene. Our perturbation scheme seems to compete with the coupling expansion scheme for graphene. The large N expansion developed here however would work better for materials with larger number of fermion species. We found that this suppression is a result of a cancellation between two contributing Feynman diagrams. Understat- ing the reason behind this miraculous cancellation might be crucial in understating the physics of graphene, which is a strongly interacting system. We studied the conductivity of Dirac semi-metals in presence and absence of a condensate which might get turned on due to the short range interactions. The derived conductivity of graphene is very close in both phases and both of the results are inside the range of the error bars of the experimental results. To infer the phase that the material lives in by measuring the conductivity, one needs more accurate experimental results.

49 Chapter 3

The Beta Function of Charge Neutral 2 + 1d Dirac Semi-metal in the Large N

3.1 Introduction There are both experimental [18]-[21] and theoretical [22]-[45] indications that the Coulomb interaction renormalizes the Fermi velocity of the relativistic electrons in a Dirac semi-metal. Graphene is the best and most studied example. Experimentally, the dependence of the Fermi velocity on both density and energy have been investigated independently. The experiments find an increase in the Fermi velocity by varying the density of the carriers and approaching the charge neutral point. However, they find less than 1% change in the Fermi velocity in a decade change of the energy scale. As discussed in Section 3.8, the renormalization of the Fermi velocity can be thought of it acquiring a non-zero beta function, d βvF ≡ d lnΛ vF , and becoming an energy scale dependent parameter. The Fermi c velocity changes (runs) from its value of vF ∼ 300 to larger values in the lower energies. As well as providing a fascinating example of scale invariance breaking by renormalization in a quantum field theory, it has interesting implications for the

50 electronic properties of Dirac materials. Within an electron volt of the Fermi energy, neutral graphene is modeled by a 2+1-dimensional quantum field theory with four species of massless Dirac electrons [51], [52]. As the renormalization in the Fermi velocity due to the change in the energy scale is negligible, the free particle picture of graphene gives a remarkably good description of many of its physical properties and it is often said that graphene is a weakly interacting system. An enduring puzzle [54] is what happens to interactions such as the Coulomb interaction, which is strong before screening is taken into account, and which is not relativistic, but do not seem to ruin the relativistic nature of the electron spectrum. The parameter which characterizes the strength of the interaction is the graphene 2 2 fine structure constant, α = e = e c ≈ 300 ≈ 2.2 (we will set h¯ = 1). It is g 4πhv¯ F 4πhc¯ vF 137 large, therefore coupling constant perturbation theory is not reliable without some re-summation that takes screening into account. In this chapter, we focus on the renormalization in the Fermi velocity due to change in energy scale and will outline 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 to graphene [55], and which is very similar to the random phase approximation em- ployed and argued to have good convergence in reference [54]. Our essential new observation will be that, in the large N limit, the retarded nature of the Coulomb force, 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 when N is large, leaving the beta function small. Physically this implies that the material 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, we observe 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 in this case, the Lorentz symmetry of the leading contributions to the beta function makes it vanish. We ﬁnd that in the presence of the full Maxwell action, the lead-

51 ing 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 embedded in 3+1d space-time that was explained in the introduction chapter. Graphene electrons, with their emergent massless relativistic dispersion relations, are the degrees of freedom occupying the sheet. They interact by exchanging photons which are allowed to propagate in the surrounding bulk of space-time. We describe this theory by the defect quantum ﬁeld theory with the Euclidean space action

Z 3 h t ~ i S = d x ψ¯aγ (i∂t + At)ψa + vF ψ¯a~γ · (i∇ +~A)ψa εc Z 1 + d4x ~E2 + c~B2 . (3.1) 2e2 c

The integral in the ﬁrst line is over the 2 + 1d defect with the integrand being the Lagrangian density of N species of two-component spinor fermions. It has a global U(N) symmetry. For graphene, we should set N = 4 to compare our results with the experimental results for graphene’s Fermi velocity renormalization. There are topological insulators which are also described by defect ﬁeld theories of this kind with various values of N. The second line in Eq.(3.1) is the Euclidean Maxwell action integrated over the bulk of the 3 + 1d space-time surrounding the defect.

The fields (At,~A) are the usual Maxwell theory vector potential and the electric ˙ ~ ~ and magnetic fields are ~E = ~A − ∇At and ~B = ∇ ×~A, respectively. The vector potential also appears in the defect fermion action in the first line of Eq.(3.1) where it is minimally coupled to the fermion field. Note that the emergent semi-metal’s speed of light, the Fermi velocity vF , differs from the vacuum speed of the photon c which appears in the Maxwell action. Here c is the speed of light in the surrounding 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 is different.

52 In the following, we will be interested in an expansion of the quantum ﬁeld theory described by the action in Eq.(3.1) in the limit where N → ∞. Of course, as stated above, to describe graphene N should be set equal to 4. One might question the validity of expanding about the large N limit. The value of N = 4 is near the strong 1 coupling regime of the N expansion, although it is thought to be at least slightly above the value of N where the U(N) symmetry is broken dynamically by strong interactions. That critical value of N has several estimates, all of which are less than 4 [10], [56], [57], [58]. Some topological insulators can be described by this action, and they have N = 2, or even N = 1, which means their large N expansions are in the strongly coupled regime where dynamical symmetry breaking of chiral symmetry is expected.

The photon ﬁeld, whose action is in the second line, interacts with the electron by minimal coupling. The parameter vF is the Fermi velocity and ε and c are the dielectric constant and the speed of light in the substance which surrounds the defect. The Dirac matrices are 2×2 and obey {γ µ ,γν } = 2δ µν . To model graphene, as well as taking N = 4, we should set the Fermi velocity to vF ≈ c/300 at the scale of the ultraviolet cut-off, ≈ 1A˚−1.

The quantum field theory in Eq.(3.1) has no dimensional parameters and it is scale invariant at the classical level. However, renormalization will introduce a scale. It is also clear that, since the photon field is free field theory in the region asymptotically far away from the defect, the coefficients of the E2 and B2 terms do not renormalize. In the end, we shall only require renormalization constants t ~ for the operators z0ψγ¯ (i∂t + At)ψ and z1vF ψ¯~γ · (i∇ +~A)ψ. Because the model is not Lorentz invariant, these terms can have different renormalization constants. z1 Their ratio defines a renormalization of the Fermi velocity,v ˜F ≡ vF . This renor- z0 malization is logarithmic in the cut-off and the Fermi velocity becomes a scale- dependent running coupling constant. The perturbative beta function, in the limit

53 Figure 3.1: The Feynman diagram of the expansion of the fermion determinant is depicted. The series is even due to particle-hole and time reversal symmetry. The Feynman integrals for diagrams with more than two legs are ﬁnite.

where c/vF is very large has been computed to two-loop order αg N 103 3 β = v + − + ln2 α2v + ..., vF 4 F 12 96 2 g F

and for graphene can be approximated as βvF ' 2vF .

As discussed previously, in this chapter we use a different perturbation scheme and use the large N expansion that we developed in the previous chapter to study the running of the Fermi velocity in our Lagrangian. To implement the large N expansion, we integrate out the fermions to get the effective ﬁeld theory for the gauge ﬁelds. The fermions contribute terms in the effective action given by the Feynman diagrams in the series depicted in Fig. 3.1. The model describing our Dirac semi-metal after integrating out the fermions is given by the following

54 action   Z N 3 1/vF vF Seff = d x~E · r ~E +~B r ~B ~ 2 1 2 ~ 2 1 2 16 −∇ − 2 ∂t −∇ − 2 ∂t vF vF ∞ Z ...... + N ∑ dx1 dxnAµ1 (x1)...Aµn (xn)Γµ1...µn (x1, ,xn) n=4 Z " # εc 3 1/c c + d x ~E · q ~E +~B q ~B e2 −~∇2− 1 ∂ 2 −~∇2− 1 ∂ 2 c2 t c2 t Z 2 3 ~ + λ d x vF ∇ ·~A + A˙t Z 3 ¯ 1 − d xηa t ~ ηa. (3.2) z0γ (i∂t +At )+z1vF~γ·(i∇+~A)

The ﬁrst line in Eq.(3.2) is the expansion of the fermion determinant to quadratic order in the gauge ﬁelds. It has a factor of N in front of it. This term contains the leading order screening of the Coulomb interaction by the relativistic electrons, and we will refer to it as the “screening-generated term”. The second line in Eq(3.2) contains the higher order, multi-photon interaction terms coming from the fermion determinant. These terms are of order N. To emphasize this, we have

... explicitly extracted a factor of N. The Γµ1...µn (x1, ,xn) are connected irreducible multi-photon correlation functions arising from a single 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 (with photon-photon scattering). In the third line of equation Eq.(3.2) , we have presented the same Maxwell theory of the photon as in the second line of equation Eq.(3.1), but dimensionally reduced to 2 + 1d, with the component of the photon field that is perpendicular to the defect eliminated using its linear equation of motion. Note that this action has become non-local and resembles the screening-generated term in the first line, albeit with a different speed of light. The fourth line of Eq.(3.2) is a gauge-fixing term. In the last line of Eq.(3.2), we have introduced anti-commuting sources ηa,η¯a in order to generate correlation functions with fermion fields.

55 The first two lines are of order N and they dominate the large N limit. The vacuum kinetic term for the photon in the third line is ignored to the leading order in 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 only renormalization is that of the fermion wave-function and, because of the Lorentz invariance, the temporal and spatial parts get the same renormalization contributions, z0 = z1. This can be observed from the Feynman diagram in Fig.3.2 where the dotted line is the inverse of the suitably gauge fixed quadratic form in the first line of Eq.(3.2). This contribution could give the fermion operators non-trivial anomalous dimensions. The beta function for vF vanishes at order 1/N as a consequence of Lorentz invariance, with vF playing the role of light velocity.

Both Lorentz and conformal invariance are broken by turning on the third line in 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 Lorentz invariance of the remaining order 1/N2 contributions to the fermion self-energy, the other self-energy corrections do not contribute to βvF . Computing the beta function simply entails computing the ultraviolet divergent part of the contribution displayed in Fig.3.2. We will show that the infrared regime does not contribute to the beta function to the order of 1/N2.

3.2 Corrections to the Electron Propagator The symmetries of the system, including U(N), parity and time reversal, spatial rotation, and charge conjugation invariance limit the matrix form of the fermion propagator. If we deﬁne

Z 3 D † E d k −ik·(x−y) 0|Tψa(x)ψ (y)|0 = e S (k), (3.3) b (2π)3 ab

56 Figure 3.2: The leading contribution to the beta function in the large N limit comes from the Feynman diagram where the dotted line is the relativistic large N propagator and the insertion into the photon propagator is the tree-level classical Coulomb action which is non-relativistic. This diagram is of order 1/N2.

then

−1 h t 0 i S (k)ab = δab Σt(k0,|~k|)γ k + Σs(k0,|~k|)vF~γ ·~k . (3.4)

Generally, the electron propagator is highly gauge dependent. However, we expect the pole in the propagator to be gauge invariant. This sufﬁces to deﬁne the speed of the fermions. In fact, the dispersion relation is the solution of the following equation

Σt(iω(k),k) iω(k) = ± Σs(iω(k),k)vF k. (3.5)

We rewrite the above formula as

Σs(iω(k),k) vF (k) = vF Σt(iω(k),k) in which vF is the bare Fermi velocity and vF (k) = vF + δvF (k). In perturbation theory, ω(k) = vF k + δvF (k)k and we can write the self-energies as

Σt = 1 + δz0 + δΣt(k0,k) , Σs = 1 + δz1 + δΣs(k0,k). (3.6)

57 t δz0 and δz1 are the contributions coming from counter-terms δz0ψγ¯ (i∂t + At)ψ ~ and δz1vF ψ¯~γ ·(i∇+~A)ψ. Then we can plug the above into the dispersion relation and ﬁnd

δvF (k) = δz1 + δΣs(ivF k,k) − [δz0 + δΣt(ivF k,k)]. (3.7)

The counter-terms δz0 and δz1 must be chosen so as to cancel the logarithmically ultraviolet divergent terms in Σt and Σs, respectively. Then, the ﬁnite part of δz1 − δz0 can be chosen by the condition that vF is the Fermi momentum at some scale k = µ. Then

δvF (k) =δΣs(ivF k,k) − δΣt(ivF k,k)−

δΣs(ivF µ, µ) + δΣt(ivF µ, µ). (3.8)

In deriving the renormalization contributions we use a speciﬁc gauge, and one needs 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 absence of infrared divergences, it can be shown that the poles of the fermion self-energy are gauge invariant [60], [61]. Nielsen identities can be written as independence ∂Σ−1 of the on-shell self-energy from the gauge parameter λ, ∂λ = 0. Implementing this condition on our expression for the self-energy shows that the contribution to the Fermi velocity renormalization is gauge invariant. In the presence of infrared divergences, the gauge invariance of the Fermi velocity is not guaranteed by Nielsen identities. We investigate infrared divergences in the following sections and conclude that they are not present. As we discuss in the section devoted to an investigation of infrared divergences, turning on the Coulomb interaction regulates the infrared divergences and leaves the Fermi velocity gauge invariant.

58 3.3 Electron Self-Energy We now start our perturbative analysis of the corrections to the electron propagator. The fermion field gets a divergent (and gauge dependent) wave-function renormalization. This should be canceled by adding local counter terms δz0ψγ¯ 0(i∂t + ~ At)ψ and δz1ψ¯~γ · (∇ +~A)ψ to the original Lagrangian. As we discussed, renormalization of the speed of light appears because of the different contributions to the spacial and temporal parts of the electron wave-function. Logarithmic divergences in the wave-function coefficients result in infinite renormalization of graphene’s speed of light and makes it a scale-dependent parameter with a beta function. For readers’ convenience, we recopy our effective action Eq.(3.2) here   Z 2 ε 3 ~ 1 ~ ~ c ~ S = 2 d xE q E + Bq B e ~ 2 1 2 ~ 2 1 2 −∇ − c2 ∂t −∇ − c2 ∂t   Z 2 N 3 1 vF + d x~E · r ~E +~B r ~B. (3.9) ~ 2 1 2 ~ 2 1 2 16vF −∇ − 2 ∂t −∇ − 2 ∂t vF vF

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 the following sections where we will choose a gauge that makes the computations more 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- energy

3 Z d q Γ [γ (p − q )+vF γ · (~p −~q)]Γ Σ(p , p) = ∆ (q) µ 0 0 0 ν (3.10) 0 3 µν 2 2 2 (2π) (p0 − q0) +vF (~p −~q) where ∆µν (q) is the photon propagator. ∆µν (q) is clearly gauge dependent. We need to ﬁx a gauge to go further; we choose a gauge that makes the term induced by fermion 1-loop summation diagonal. This can be accomplished by imposing 2 ~ the gauge condition vF ∇ · A + ∂tA0 = 0. In Appendix D, we derive the propaga-

59 tor and check its instantaneous limit versus the previously studied instantaneous propagator. We find that our propagator indeed reduces to the instantaneous propagator in the instantaneous limit. After imposing our gauge, we can find that the new propagator for the gauge field has the following complicated form

16 ∆(p ,~p) = 0 N  2 2  c pP+ξ p0 ξ p0 p1 ξ p0 p2 2 2 2 η1 p η1 p η1 p  + 2Pp2− c2 p(p2+c2 p2)   ξ p0 p1 η4 ξ 1 ξ 0 2 ξη3 p1 p2)  2 2 2 , (3.11)  η1 p p η1η2 η1η2 p   2 2 2 2 2 2  ξ p0 p2 ξη3 p1 p2 η4−ξc p(p0+c p1)+ξ Pp2 2 2 2 η1 p η1η2 p p η1η2

2 η1 = (ξ p + c P), 2 η2 = (c p + ξP), 4 η3 = (c p + ξP), 4 2 2 2 2 η4 = c p P + ξc p(p + P ). (3.12)

2 2 2−→2 2 −→2 32εc3 We have set vF = 1 and deﬁned P = p0 + c p , p = p0 + p and ξ = e2N . To extract information about the running of the Fermi velocity, vF , due to the interactions, we should rewrite the calculated form of the fermion self-energy in the form 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 counter terms. However, the leading term of order 1/N in Eq.(3.9) is equivalent to the limit 1 in which e2 → 0. This contribution is Lorentz invariant resulting in δΣt = δΣs and, consequently, δvF = 0. To get a non-Lorentz invariant contribution, we must re- 1 store the electrodynamic terms (ﬁnite e2 ) to the action, where we can consider the added term as a perturbation. Their leading contribution appears in the diagrams depicted in Fig.3.2, and is of order 1/N2.

60 1 Orthogonality of the γ matrices can be used to check that p0Σt = 2tr(γ0Σ) and 1 piΣs = 2tr(γiΣ). Ultimately, to find the dispersion relation we need to impose the on-shell constraint. In the following computations to evaluate δvF , we can make derivations easier by imposing q2 = 0 and restricting the variables to live on-shell. As we are computing the first non-zero contributions to the free field theory dispersion relation, we can safely use the zeroth order dispersion relation 2 2 (q0 +q1 = 0) for imposing the on-shell constraint. With a small amount of manipulation, the following simplified expressions for the δΣs and δΣt coefficients can be found. Let’s start with computing δΣs, the spatial part of the self-energy

Z −→ −→ ( 16 | p |d| p |dp0dt −→ 4 −→ 2 δΣs = − 3 | p | c p(−pP + ξ| p | Nq1 R3 (2π) 2 2 2 2 −→ 2 ) + ξc p(−P + p0(p0 − 2iq1)) + ξ P(p0 + | p | − 2ip0q1) 4 2 2 2 2 2 2 4 cost + q1 c p P + ξ Pp0 + ξc p(P + p0) − ξ(c p + ξP) ) 1 1 1 |−→p |2 cos2t 2 −→2 −→ 2 2 2 p0 + p − 2ip0q1 − 2| p |q1 cost p (c p + ξP) (ξ p + c P) (3.13)

−→ −→ in which we have defined p2 = | p |sint and p1 = | p |cost. We first integrate over the angle variable t, and then scale the internal momentum by q1. The scaling in q1 cancels the 1/q1 factor in Eq.(3.13)(q1 appears in bounds of integral, the hard cut-off regulator). Due to our interest in the ultraviolet limit, we then expand the result in momenta and keep only the divergent terms. The original integral is linearly divergent, however, the linearly divergent term vanishes and leaves only a logarithmic divergence as the leading term. In other words, the first order terms of

61 the δΣs Taylor expansion around small q1 are the only divergent terms that survive

Z ∞ Z ∞ q 16 4 2 2 2 2 δΣs = − 2 |~p|d|~p| dp0 c p0 p0 + c |~p| + (2π) N 0 −∞ q q 2 2 2 2 2 2 2 2 2 2 2 2 ξ (p0 + |~p| ) p0 + c |~p| + ξc p0 + |~p| (2p0 + c |~p| ) 1 1 1 . (3.14) r 2 2 2 r 2 2 2 (p2 + |~p|2)3 2 p0+c |~p| 2 p0+c |~p| 0 ξ + c 2 2 c + ξ 2 2 p0+|~p| p0+|~p|

To proceed with the evaluation of the spatial part of the self-energy, we perform a change of variable by deﬁning p0 = x|~p|. The integral can be separated in terms of x and p integrations; the x integral is ﬁnite and the p integral is logarithmically divergent

Z ∞ Z ∞ √ 16 d|~p| 4 2p 2 2 2 2 p 2 2 δΣs = − 2 dx c x x + c + ξ (x + 1) x + c + ξc x + 1 (2π) N 0 |~p| −∞ 2 2 1 1 1 (2x + c ) q q 2 3 . (3.15) + c2 x2+c2 c2 + x2+c2 (x + 1) ξ x2+1 ξ x2+1

We regulate the integral by introducing a hard cut-off Λ. The result can be sum- marized as "( q 16 2 2 2 4 2 δΣs =− ξ ξ − (1 + ξ )c + c − 2ξ + ξ(−1 (2π)2N q + 2ξ)c2 + 2c4 − c2 −ξ 2 + c2(2c4 + ξ 2(−2 + c2))tan−1 √ 2 p 2 3 2 4 −1 ξ −1 + c −1 [ −1 + c ] + 2c (ξ − c )(tan [ p ] − tan [ c −ξ 2 + c2 √ ) # c −1 + c2 1 ]) ln[Λ/q1]. (3.16) p 2 2 3 p −ξ + c ξ 3(−1 + c2) 2 −ξ 2 + c2

The temporal part of the self-energy, δΣt, can be calculated in the same way as δΣs. As the calculation is fairly similar to the calculation for the spatial part, we

62 simply report the result of the calculations "( q 8 1 2 2 2 4 8 4 2 −1p 2 δΣt =− 4 −ξ + c − ξ c + 2c + ξ (−1+ c ) tan −1 + c N(2π)2 ξ 3c2 q + 4c4 (ξ 2 − 2ξc2) (−1 + c2)(−ξ 2 + c2) − c−2c4 + ξ 2(1 + c2)(tan−1[ √ √ ) ( ξ −1 + c2 −1 + c2 1 ] − tanh−1[c ]) + 4ξ 5 p 2 2 p 2 2 3 p c −ξ + c −ξ + c (−1 + c2) 2 −ξ 2 + c2) √ √ 2 2 2 ) −1 −1 + c −1 c −1 + c 1 tan [ξp ] − tanh [ p ] p ]ln[Λ/q1]. −ξ 2 + c6 −ξ 2 + c6 (−1 + c2)(−ξ 2 + c6) (3.17)

Recall from the previous sections, the measurable quantity which needs to be gauge invariant is the difference of these two coefﬁcients. The Fermi velocity beta function, which describes the rate of change of the Fermi velocity with a δvF change of scale can be found by applying its deﬁnition βvF = δ lnΛ . We compare our results with the results in [55] by taking the instantaneous Coulomb interaction 1 limit ( c2 → 0) [To compare our results, we restoring vF by the help of dimensional analysis.]. The beta function in the instantaneous limit is give by the following expression h i s 2 2 p 2 2 v v2ξ 2 8v ξ ArcCoth 1 − v ξ inst(v) = − 8 + 4 v [−1 + ] + . βvF 2 π ξ 2 2 p Nπ −1 + v ξ 1 − v2ξ 2 (3.18)

Taking into account that in [55] the author works in a four dimensional representation of the Clifford algebra instead of our two dimensional representation, the results are in agreement (Nf = 2 in [55]). In the last section, we will come back to the beta function in this limit, and whether there will be any contributions from the infrared regime.

63 The beta function can be written in terms of v by taking c → 1 while keeping F vF εc e2 ﬁxed, and then setting c = 1. We ﬁnd that the Fermi velocity beta function is given by the following complicated function

3 q 4v 2 2 2 2 2 2 2 βv (v) = − − ξ (ξ − v )(−1 + v )(−6v + 3ξv + 2ξ (−1 + v ))+ F ξ 3Nπ2 q q n v −ξ 2 + v2(−6v2 + 2ξ 4(−1 + v2) + ξ 2(−1 + 4v2))tan−1[ −1 + 1/v2] − 2v2 √ √ 2 2 2 2 2 o −1 1 − v −1 1 − v − 3v + ξ (1 + 2v ) (tan [p ] − tan [ξ p ]) −ξ 2 + v2 −ξ 2 + v2 √ √ 2 2 2ξ 5 tanh−1[√ 1−v ] − tanh−1[√ξ 1−v ]! 1 1−ξ 2v2 1−ξ 2v2 + 3 p p 2 2 2 (−1 + 1/v2) 2 v5 −ξ 2 + v2 (−1 + v )(−1 + ξ v ) (3.19)

32ε where, as before, ξ = e2N . We have plotted our improved beta function versus the beta function found in [55] in Fig.3.3. As we expected, they start to deviate at larger Fermi velocities when the retarded nature of the interactions becomes more important.

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 have plotted the beta function in Fig.3.4. The dashed lines correspond to the instantaneous approximation results. We observe that the beta function gets smaller for large N, and the difference between the instantaneous approximation and the inclusion of the full Maxwell Lagrangian becomes more signiﬁcant.

Let us look at the large the N expansion of our beta function. We need to expand the complicated expression above in terms of ξ as it depends on N though 32ε ξ = e2N . We ﬁnd that, as expected from the Lorentz invariance of the lowest order

64 Figure 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 light velocity is our measure for velocity), we ﬁnd that the Lorentz symmetry prevents the Fermi velocity from running. The beta function in [55] violates this condition as its Lorentz symmetry is violated by construction but not the presence of the two velocities in the Lagrangian. term in 1/N, the ﬁrst non-zero contribution is of order 1/N2 √ " −1 2 # 1 32ε 2 2 4 tan [ 1 − v /v] βv (v) = − (5 − 2v ) + (5 − 16v + 8v ) √ . (3.20) F N2 e2π2 v 1 − v2

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 a result the expansion in [55] works better.

65 Β

v 0.2 0.4 0.6 0.8 1.0

-0.0005

-0.0010

-0.0015

Figure 3.4: We have plotted the beta function in [55](dashed lines) vs. Eq.(3.19) (solid lines). The orange, purple and green lines respectively correspond to N = 4,10,100 .

3.4 Infrared Contributions to the Fermi Velocity Beta Function In this section, we investigate the model with the instantaneous Coulomb inter- 1 action ( c2 → 0) with more caution. We will see that the fermion self-energy, to leading order, suffers from infrared (IR) divergences. These infrared divergences get regulated by the presence of the Coulomb interaction automatically, and the final result is infrared divergence free. Let’s start by writing down the action in this limit. By looking at the coefficient of the magnetic kinetic term, it can be 1 seen that in the instantaneous limit ( c → 0) , the fluctuations of the spatial component of the gauge field get suppressed and only the temporal part survives. The

66 instantaneous action can then be written as   Z 3 1 1 N 1 Sinst = d x ~E ~E + ~E r ~E (3.21) g2 p 2 32v ~ 2 1 2 −~∇ F −∇ − 2 ∂t vF

1 ε where we have deﬁned g2 = e2 . We are interested in studying the infrared behavior of the following integral, setting vF = 1 we ﬁnd that the fermion self-energy is given by

3 Z d q γ0(/p − /q)γ0 1 Σ = − 3 2 2 2 (2π) (p0 − q0) + (p − q) N √ q + 2q 16 2 2 g2 q0+q 3 Z d q (p0 − q0)γ0 − (~p −~q).~γ 1 = − 3 2 2 2 . (3.22) (2π) (p0 − q0) + (p − q) N √ q + 2q 16 2 2 g2 q0+q

µ We have used Feynman slash notation defined by /p = γ pµ , and for the rest of 2 2 2 2 this chapter we redefine p as p = p1 + p2 so that it only has the spatial components of the momenta in it. In the infrared regime, we expect q0 and q terms in the numerator 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 explicit integration of the angle variable using the complex coordinates and looking at the regime where q0 and q are both small. Before getting into the full calculation of the self-energy in the presence of the Coulomb interaction, let us do an approximate calculation. In the infrared regime, q p, the fermion self-energy can be approximated by a simplified integral given by

Z ∞ Z ∞ 2 2 IR dqdq0 1 p0 − 3p 2 p0γ0 −~p.~γ δΣ = −2π 3 2 + 2 q0 2 , (3.23) 0 −∞ (2π) p + p2 (p + p2)3 N √ q + 2q 0 0 16 2 2 g2 q0+q

1 suggesting that in the limit of g2 → 0, the fermion self-energy is infrared divergent. As we discussed before, vF renormalization depends on δΣt − δΣs. As a

67 result of the above observation, the running of vF is proportional to both δΣs and 1 δΣt (δΣt=-δΣs). By turning off the Coulomb interaction, g → 0, both δΣt and δΣs suffer from infrared divergences, therefore vF will also suffer from the same divergences. We will see that the Coulomb interaction smooths out these divergences.

Now, it is time for a rigorous treatment of the presence of possible infrared divergences. We start by integrating over the angle variable. We choose the external momenta to have no component in the y direction, (p0, p,0). Using this simpliﬁcation, we can simplify δΣt and δΣs a bit. δΣt is given by

Z ∞ Z ∞ Z 2π qdqdq dφ −1 (p − q ) δΣ .p = 0 0 0 t 0 3 q2 2q 2 2 2 0 −∞ 0 (2π) N √ + (p0 − q0) + p + q − 2pqcosφ 16 2 2 g2 q0+q ∞ ∞ Z Z qdqdq0 I dz 1 (q0 − p0) = 3 2 2 2 2 −1 0 −∞ (2π) iz N √ q + 2q (p0 − q0) + p + q − pq(z + z ) 16 2 2 g2 q0+q Z ∞ Z ∞ dq2dq 1 π(q − p ) = 0 0 0 . 3 N q2 2q p 2 2 2 2 0 −∞ (2π) √ + ((p − q) + (p0 − q0) )((p + q) + (p0 − q0) ) 16 2 2 g2 q0+q (3.24)

We can use the same trick to integrate the angle variable in the spatial contribution to the self-energy. δΣs is then given by

68 ∞ ∞ 2π Z Z Z qdqdq0dφ 1 (p − qcosφ) δΣs.p = 3 2 2 2 2 0 −∞ 0 (2π) N √ q + 2q (p0 − q0) + p + q − 2pqcosφ 16 2 2 g2 q0+q ∞ ∞ −1 Z Z qdqdq0 I dz 1 (p − 2q(z + z )) = 3 2 2 2 2 −1 0 −∞ (2π) iz N √ q + 2q (p0 − q0) + p + q − pq(z + z ) 16 2 2 g2 q0+q (3.25)

∞ ∞ " π 2 2 2 # Z Z dqdq0 1 p (p − q − (p0 − q0) ) = 3 N q 2 p + 1 . 0 −∞ (2π) √ + 2 2 2 2 16 2 2 g2 ((p − q) + (p0 − q0) )((p + q) + (p0 − q0) ) q0+q

By expansion of these equations in the region where p q, we are able to verify the results in [55]. As is mentioned in [55] , there is no infrared divergence in that regime; however, this does not prove that the theory is infrared divergence free. By zooming into this particular regime, we only check that a given theory with an infrared limit that corresponds to the ultraviolet regime of Eq.(3.21) will be 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 an infrared limit that corresponds to the ultraviolet regime of Eq.(3.21) will be infrared divergent free.

2 The integrals could be expanded in terms of g2 , but the expansion would not 2 be well deﬁned. As the integrals are infrared divergent at g2 → 0, the perturba-

69 tive expansion would be invalid and non-perturbative terms would show up. By exploring the most dominant terms in the strict infrared limit, we see that logarithmic terms in g2 show up. As a result, we can expand the self-energies in the following manner and look at their behavior in the infrared regime.

IR −2 −2 δΣt = αt ln[g ] + O(g ), (3.26) IR −2 −2 δΣs = αs ln[g ] + O(g ). (3.27)

αt and αs can be found by taking the logarithmic derivative of Σt and Σs respectively.

∂Σt αt = ∂ ln(g−2) −2 N √ q + 2 Z ∞ Z ∞ −2 16 2 2 g2 1 dqdq0 4π(p0 − q0)g q0+q = − lim 3 p p p0 1 →0 0 −∞ (2π) (p − q)2 + (p − q )2 (p + q)2 + (p − q )2 g2 0 0 0 0 δ( √Nq ) 1 Z ∞ Z ∞ dqdq 2π(p − q ) 16 q2+q2 = − 0 0 0 0 (3.28) 3 p 2 2 p 2 2 p0 0 −∞ (2π) (p − q) + (p0 − q0) (p + q) + (p0 − q0)

q 2 ∞ ∞ 16 q + q2 1 Z Z dqdq0 2π(p0 − q0) 0 1 αt = − δ(q) 3 p 2 2 p 2 2 p0 0 −∞ (2π) (p − q) + (p0 − q0) (p + q) + (p0 − q0) N ∞ 1 Z dq0 2π(p0 − q0) 16|q0| = − 3 2 2 . (3.29) p0 −∞ (2π) p + (p0 − q0) N

Above, we have used the following representation of the Dirac delta distribution to push the calculations forward

a δ(x) = lim . a→0 (x + a)2

As the computation for the spatial part is similar to the temporal part, we report

70 only the results here. The spatial part can be found in the same way, and we copy the result

∞ Z dq0 2π 16|q0| αs = 3 2 2 . (3.30) −∞ (2π) p + (p0 − q0) N

The remaining integrals can be evaluated explicitly as well,

Z dx (a − x)x a − x a = −x − b tan−1[ ] − ln[b2 + (a − x)2], b2 + (a − x)2 b 2 Z dx x a a − x 1 = − tan−1[ ] + ln[b2 + (a − x)2], b2 + (a − x)2 b b 2 and we ﬁnd that the difference between the α coefﬁcients is given by

2 2 ! 4 p0 + p −1 Λ − p0 −1 p0 −1 Λ + p0 αt − αs = − 2 tan [ ] + 2tan [ ] − tan [ ] , π N p0 p p p p (3.31) where Λ is the ultraviolet cutoff. Although each of αs and αt are ultraviolet divergent, the subtraction is ultraviolet finite. Then, at the limit that the ultraviolet 8 1 2 2 −1 p0 cut-off goes to infinity, we have αt − αs = − (p + p )tan ( ). If we π2N p0 p 0 p cast this result into our expansion, and restore the vF by dimensional analysis, we would find that the infrared contribution to the Fermi velocity renormalization is given by

2 2 2 IR 8 p0 + vF p −1 1 p0 −2 δ vF = − 2 tan [ ]ln[g vF ]. (3.32) π N p0 p vF p

The above equation should be evaluated on-shell, for which we need to use the 2 2 2 definition of the renormalized Fermi velocity p0 + (vF + δvF ) p = 0. We find that 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 the

71 infrared contribution to the renormalization equation of vF starts with a higher order in 1/N when comparing to the ultraviolet contributions. We observe that in the instantaneous model the contributions from the ultraviolet divergences are of the order 1/N, UVv = − 8 v δΛ , however, IRv is of order O(lnN/N2). δ F π2N F Λ δ F

As the infrared contribution to the self-energy turned out to be ﬁnite and independent of the infrared cut-off, it does not contribute to the beta function. This lack of contribution can be observed by the use of Eq.(3.8) and observing that the infrared terms cancel out due to their independence of the scale.

3.5 Experimental Results The Fermi velocity in graphene has been measured through measurements of transmittance T and reﬂectance R of suspended graphene sheets [7], scanning tunneling spectroscopy [19, 20], angle-resolved photo-emission spectroscopy measurement of the Dirac cones [21] and effective cyclotron mass [18] . Experimen- tally, the dependence of the Fermi velocity on both density and energy have been investigated independently. The experiments ﬁnd an increase in the Fermi velocity 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 dependence on density. In [18], the authors used the cyclotron mass to measure the Fermi velocity. A significant renormalization of the Fermi velocity is observed 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 introduction, in theory presented above, we are interested in not the change of the Fermi velocity by changing density but instead for fixed density as a function of energy, our calculations were performed assuming zero density. Although, the Fermi velocity shows substantial renormalization by changing the density, its behavior under change of energy scale at fixed density can be quite different. As such, we need

72 to compare our results with the experiments that investigate the renormalization of the Fermi velocity by changing the energy scale of the charge-neutral graphene in which the density is minimal (n 1015cm−2) and ﬁxed. In this measurement however, they do not investigate the renormalization of the Fermi velocity at constant densities and their results can not be used as an experimental check for our results.

Figure 3.6: Adapted from [18]. (a) Cyclotron mass as a function of Fermi wave-vector. 1 The dashed curves are the best linear ﬁts with assumption that mc ∼ n 2 . The dotted line is the behavior of cyclotron mass derived from the standard value of Fermi velocity. Graphene’s spectrum renormalized due to electron-electron interactions is expected to result in the dependence shown by the solid curve. (b) Cyclotron mass

plotted as a variable of vF .

In [20], the authors studied the renormalization of the Fermi velocity in graphene using scanning tunneling spectroscopy and measuring the dispersion relation of the excitations as a function of their energies. In graphene, the linear dispersion √ yields the Landau level (LL) spectra that is given by EN = E0 +vF sgn(N) 2ehNB¯ , where E0 is the energy for the zeroth level, N is the landau level number and B is the magnetic field. This dependence allows us to study the dispersion of Fermi velocity at fixed density and magnetic field by studying the energy bands for dif-

73 ferent LLs. Chae et al. investigated the LLs of graphene and found that at fixed √ density 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 from √ EN = E0 +vF sgn(N) 2ehNB¯ . As a result of the linearity, we can infer that changing the energy scale does not renormalize the Fermi velocity substantially and the excitations 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 Fermi velocity of the excitations gets renormalized. They confirmed the previous results that 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 a renormalization of 0.2% in a decade, which is consistent with the experimental results.

3.6 Conclusion We conclude this chapter by summarizing our main results. It was shown that 2 + 1d fermions in large N interacting via 3 + 1d electrodynamics acquire a beta function for their Fermi velocity. Due to Lorentz invariance, the beta function starts at second order in 1/N, in contrast to models with an instantaneous Coulomb interaction that acquire a beta function of order 1/N. We made a careful investigation of the infrared regime. It was argued that possible infrared divergences get regulated by the presence of the Coulomb interaction, and the infrared regime is divergence free. The gauge invariance of the Fermi velocity in presence of the interactions was discussed, and it was argued that, based on the absence of the infrared divergences, the Fermi velocity is gauge invariant.

74 Figure 3.7: Adapted from [20]. N = 1 to N = 6 LLs’ energy as a function of level number for different values of carrier density and B= 2T. For ﬁxed density the curves are highly linear, resulting in a possible negligible renormalization of the Fermi velocity. (Inset) Residuals from the linear ﬁt showing very good linearity in the LLs.

75 Chapter 4

φ 6 Theory in the Large N Limit

4.1 Introduction In this chapter, we will consider scalar field theories in 2 + 1d and study their behavior in large N limit. We first review the relevance of these model for studying 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 parametrically small beta function resulting in an approximate scale invariance has attracted attention, particularly when the approximate scale symmetry can be spontaneously broken, generating a pseudo-Goldstone boson in the form of a light dilaton. For example, the notion that the tree level scale invariance of the SU(2) ×U(1) electroweak theory is softly broken by the Higgs potential or dynamically broken by some physical mechanism beyond the Standard Model leaves the Higgs boson it- self as the dilaton, with some testable physical consequences [62]-[75]. Walking techni-color [65]-[71], confining and chiral symmetry breaking gauge field theories with approximate infrared conformal symmetry [72] are also scenarios where spontaneous breaking of approximate scale invariance could play an important role.

76 One of the prototypical examples of spontaneously broken scale symmetry occurs in the large N limit of the tri-critical O(N) symmetric g2(~φ 2)3-theory in three space-time dimensions which is an interesting quantum ﬁeld theory in its own right. The phase diagram of g2(~φ 2)3-theory in three space-time dimensions is of importance in studying graphene. The importance is a consequence of the renormalization of the coupling constant under renormalization group.

Let us consider the effect of short-range interactions by adding a Hubbard term to our Lagrangian of graphene. Suggested by the seminal work of Hertz [76], Hubbard interactions could be studied using the Hubbard-Stratonovich transformation. Let’s consider the half-ﬁlled Hubbard Hamiltonian given by the following on-site interaction term in the Hamiltonian,

1 1 HHubbard = U ∑(ni,↑ − )(ni,↓ − ). (4.1) i 2 2

In the above Hamiltonian, U is the Hubbard parameter which implies the importance 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 the Hubbard-Stratonovich transformation and decouple the spin-up/spin-down operators,

2 Z ∞ dx 2 ea /2 = √ e−x /2−ax. (4.2) −∞ π

The decoupled Hubbard Hamiltonian then would be given by the following formula in which S is the spin operator.

U 2U 2 HHubbard = ∑ − Si . (4.3) i 4 3

77 We then transform the Hubbard interaction Hamiltonian using the Hubbard-Stratonovich transformation and add a real scalar field to our Hamiltonian. In path integral language, the original Hubbard term gets substituted by a new scalar field and a coupling term that couples this added scalar field to the spin operator of the fermions U 2U 2 ∑i − S eHHubbard = e 4 3 i Z U 3U 2 ∑i 4 − 8 φi +φi.Si = [dφi]e . (4.4)

The coupling between the scalar ﬁeld and the fermions spin is given by

† ∑φi.Si = ∑φicisσss0 cis0 , (4.5) i i which in our Lagrangian would correspond to a φ · ψσψ term. Following the renormalization group scheme, the Lagrangian for the Hubbard-Stratonovich field, φ, would have all of the interaction terms that are relevant or marginal. This is a consequence of renormalization of the coupling constants under renormalization group. Here, φ 6 being the marginal interaction in 2 + 1d, makes the investigation of g2(~φ 2)3-theory phase structure in three space-time dimensions an important question for a graphene material that has Hubbard interactions in its Lagrangian. The introduced Hubbard-Stratonovich field φ, has three components and enjoys an O(3) symmetry. To study the phase diagram of the effective theory describing the φ field, we study the O(N) symmetric version of the theory and use N as an expansion parameter.

As N here is small, one might be skeptical about the results that are derived by studying the large N limit theories. In the real world, N is three so an expansion in powers of l/N may not seem like such a good idea. However, it is possible that the l/N expansion might actually be applied to infer valuable informations about quantum ﬁeld theories with a ﬁnite and even small N. The actual expansion

78 parameter would be proportional to 1/N and that coefﬁcient might play an important role in the validity of the 1/N expansion. This can be observed by reminding ourself about the well known coupling expansion for quantum electrodynamics in 3 + 1d. Although e ' 0.3, the actual expansion parameter α, ﬁne-structure constant, is given by the following formula

1 e2 α = 4πε0 hc¯ 1 ' . (4.6) 137

We observe that although the coupling constant in the Lagrangian, e, is not very small, the actual expansion parameter, the ﬁne-structure constant turns out to be small. At the same-time, as we observed in Chapter 2, contributions with the same order in 1/N might cancel the other contributions and leave a small contribution after these cancellations.

The Hubbard-Stratonovich transformation of a fermionic system transforms it into a fermionic system that is coupled to bosons. Recently bosonic fields coupled to the fermionic fields that enjoy a non-abelian symmetry with a large rank have received more attentions [77]. In Ref.[77], in search of non-Fermi liquid they start with a fermionic field with a four-point interaction and after introduction of a scalar field, who plays the role of Hubbard-Stratonovich field, they continue their investigation without fully integrating out the fermions. They employ the Wilsonian mechanism to study the coupled fermions with bosons and enhance the 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 of

79 a Hubbard-Stratonovich ﬁeld to the original Lagrangian density is given by

L = Lψ + Lφ + Lψ,φ , σ σ σ 0 Lψ = ψ (∂0 + µ − iv∇)ψσ + λψ ψ ψσ ψ ψσ 0 , λφ L = (∂ φ)2 + c2(∇φ)2 + m2 φ 2 + φ 4, φ 0 φ 4! σ Lψ,φ = λψ,φ ψ ψσ 0 φ, (4.7) where ψ is the fermionic field, φ is the Hubbard-Stratonovich field, v and c are the velocities of the fermionic and bosonic field respectively, µ is the chemical potential, λψ,φ , λψ and λφ are the coupling constants. In Ref.[77], the authors promote the model to have N fermion species and an N × N complex scalar field that belongs to the adjoint representation of the resulting fermions’ SU(N) symmetry group. The SU(N) invariant model then can be written as

i i j Lψ = ψ (∂0 + µ − iv∇)ψi + λψ ψ ψiψ ψ j, (1) (2) n o λφ λφ L = tr (∂ φ)2 + c2(∇φ)2 + m2 φ 2 + tr(φ 4) + (tr(φ 2))2, φ 0 φ 4! 4! i j Lψ,φ = λψ,φ ψ ψ jφi . (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 have (1) an enhanced symmetry for λφ = 0. The scalar ﬁeld would enjoy an enhanced 2 SO(N ) symmetry. The interaction term, λψ,φ , breaks this enhanced symmetry however, it is an approximate symmetry of the Lagrangian. In 2 + 1d, this model could generate more terms, including phi-six terms that are marginal in that dimension. To study this model in 2 + 1d then one needs to study the effect of other terms in the Lagrangian including the following term in the bosonic ﬁeld Lagrangian

2 2 3 Lφ 6 = g (tr(φ )) . (4.9)

80 In this chapter, we will study a critical model deﬁned by the following Lagrangian

1 g2 L = ∂ ~φ∂ ~φ + (tr(φ 2))3 −~j ·~φ, (4.10) 2 µ µ N2 where the φ field is a vector representation of SO(N) group. The g2(~φ 2)3 interaction is scale invariant at the classical level. Its beta function is of order 1/N and, it is therefore suppressed at large N. As a result, g2(~φ 2)3 remains exactly marginal at the leading order in the large N expansion and, at the next-to-leading order, it becomes a marginally irrelevant slowly running coupling. The theory is approximately scale invariant. The beta function, depicted in Fig. 4.1, has a trivial infrared fixed point at g2 = 0. In addition, as was argued long ago [12]-[15], it exhibits a nontrivial ultraviolet fixed point at g2 = 192. The ultraviolet fixed point renders the field theory asymptotically safe in that the ultraviolet cut-off can be removed without forcing triviality [14].

However, Bardeen, Moshe and Bander [16] showed that, with sufficiently strong coupling, the infinite N limit of g2(~φ 2)3 theory has a quantum phase tran- 1 ~ 2 sition to a phase where an O(N) singlet composite operator, N φ , gets an expectation value. This operator has a non-zero scaling dimension and its expectation value breaks the scale symmetry. The phase transition occurs at a critical value of the coupling, g∗2 = (4π)2 ≈ 158 that is somewhat smaller than the ultraviolet 2 fixed point, gUV = 192. They also showed that the ultra-violet cut-off can be removed only when g2 is tuned to their fixed point, g2 → g∗2. The condensate breaks the exact scale symmetry of that limit and they showed that the spectrum of the theory contains a massless dilaton. David, Kessler and Neuberger [78, 79] did a careful analysis of the phase diagram of the infinite N model. They also conjectured that, “for all finite N the BMB (Bardeen-Moshe-Bander) phenomenon does not survive.” The latter is something that we shall demonstrate to be so in the following. In a subsequent lattice study, Kessler and Neuberger demonstrated that the infinite N model with nearest neighbour lattice interactions does not exhibit

81 Β N 800

600

400

200

g 2 4 6 8 10 12 14

Figure 4.1: N× the beta function of large N regime of g2(~φ 2)3 theory in 2 three dimensions. The infrared fixed point is gIR = 0 and the ultra- 2 violet fixed point occurs at gUV = 192. The critical coupling where in the infinite N limit scale symmetry breaking occurs is g2 = (4π)2 ≈ 158. the Bardeen, Moshe and Bander phenomena [80]. Their conclusion suggests that the BMB phase is regulation scheme dependent and this dependency is a hint of instability 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 symmetry becomes approximate, in their strong coupling phase, the latter is represented as a spontaneously broken approximate symmetry. The dilaton becomes a quasi- Goldstone boson, acquiring a mass of order 1/N. In this chapter, we will examine this 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 interplay between the loss of tune-ability of the coupling constant when it has a renormalization group flow and dynamical breaking of scale invariance which is driven by strong coupling dynamics and occurs at a specific fixed point. Our central con-

82 clusion will be that the “light dilaton” of this theory is actually a tachyon. This indicates an instability of the phase of the theory with spontaneously broken approximate 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 that this 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 originally computed by Townsend [12] and our results are in agreement with his where they overlap. The main difference is that he studied O(N) symmetry breaking whereas we study the the massive phase which occurs near the tri-critical point.

4.2 Spontaneous Symmetry Breaking Spontaneous symmetry breaking occurs when the ground state is not invariant under the symmetry transformations of a Lagrangian like the cartoon in Fig.4.2. Spontaneous symmetry breaking surprisingly provides a large amount of information about the spectrum of a theory. The phenomena of spontaneous symmetry breaking plays an important role in our understanding of various phenomenas in nature. Most simple phases of matter and phase transitions, like crystals, magnets, and conventional superconductors can be simply understood from the viewpoint of spontaneous symmetry breaking. The mechanism of spontaneous broken symmetry plays an important role in the context of the strong interactions, speciﬁcally chiral symmetry breaking. In the Standard Model of particle physics, spontaneous symmetry breaking of the SU(2) × U(1) gauge symmetry associated with the electroweak force generates masses for several particles, and separates the electromagnetic and weak forces.

It is a well known fact that spontaneous breaking of a symmetry results in emergent massless particles, called Goldstone modes. The existence of Goldstone

83 Figure 4.2: Spontaneous breaking of the internal rotation symmetry in φ space. The ﬁeld φ chooses a ground state that violates the internal U(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 possible to extend the following arguments to full quantum level. Consider a theory consisting of generic ﬁelds φi(x) given by a Lagrangian of the form

L (φi,∂φi) = (kinetic terms) −V(φi), (4.11) where we have assumed our Lagrangian is local and can be separated in kinetic and potential terms. We assume that there exist φ 0 such that it minimizes V(φ) and as a result is a stationary point as well

∂V = 0. (4.12) ∂φi φ(x)=φ 0

84 Let us expand V around its minimum, we ﬁnd that

2 ! 0 1 0 ∂ V 0 V(φi) = V(φi ) + (φ − φ )i (φ − φ ) j + .... (4.13) 2 ∂φi∂φ j φ(x)=φ 0

The term m2 = ∂V is symmetric and its eigenvalues give the mass of ﬁelds. The i j ∂φi∂φ j eigenvalues are by deﬁnition positive around the minimum. Goldstone’s theorem states that every continuous symmetry of the Lagrangian that is not a symmetry of minimum solution, φ 0, gives rise to a massless particle corresponding to a zero 2 eigenvalue of mi j.

Let us write the general form of a continuous symmetry transformation for the ﬁeld φ(x), assuming that α is an inﬁnitesimal parameter

φ → φ + α∆(φ). (4.14)

We consider constant ﬁelds. We drop the derivative terms in the Lagrangian and use the invariance of the Lagrangian under the symmetry transformation to ﬁnd

∂V ∆(φ) = 0. (4.15) ∂φi

Differentiating with respect to φ and imposing φ = φ 0 we ﬁnd that

∂ 2V ∆(φ) = 0. (4.16) ∂φi∂φ j φ=φ 0

We finally find that for any transformation that respects the ground state symmetries and consequently leaves φ 0 unchanged, the above equation is satisfied trivially, ∆(φ) = 0. For any spontaneously broken symmetry ∆(φ) 6= 0, ∆(φ) is the eigenstate with zero eigenvalue corresponding to a massless particle.

85 Here, we started with the assumption that the original Lagrangian enjoys an exact 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 symmetry present at Veff(φ) after including the loop corrections to classical potential, we might end up with spontaneous symmetry breaking scenario. For example, we can observe spontaneous breaking of O(N) symmetry in O(N) model and compute the mass of the Goldstone bosons explicitly. As O(N) is an internal symmetry of the model, it survives the quantum corrections. Indeed the effective potential turns out to be O(N) symmetric, resulting in an exact O(N) symmetry and a massless Goldstone boson. Later in this chapter, we study the dilaton, the Goldstone boson responsible for breaking an approximate conformal symmetry. Although we start with a classical conformal symmetry and find that to the leading order in our perturbation theory it stays exact, resulting in a massless dilaton, we find that higher corrections make the dilaton massive. The emergent mass of such dilaton stems form the fact that renormalization of fields introduces a scale in the effective potential and reduces our exact symmetry to an approximate symmetry. As it can be expected, the dilaton mass turn out to be proportional to the term that breaks the scale invariance.

4.3 Effective Action Technique The effective potential method is useful in studying spontaneous symmetry breaking in field theories. Effective potentials are generating functionals for single- particle irreducible Green functions of a field theory, and were introduced first by Euler, Heisenberg and Schwinger. Their usefulness was first pointed out by Jona-Lasinio [82] and was used extensively by several authors [83]-[87]. In this section, we review the effective potential and the technique suggested by DeWitt and Jackiw [86] for calculating the effective potential Γ, using functional methods in the multi-loop approximation.

Consider a ﬁeld theory given by a Lagrangian L (φi,∂φi). Here φi represents

86 all of the fields and for simplicity we assume that only the first derivative of φi is present in the Lagrangian. By adding a source term J(x) that couples to the fields in the Lagrangian, we can make a generating functional such that its functional derivatives result in single-particle reducible Green functions. In the presence of a source, the action in d + 1 dimensions is given by,

Z d+1 n o S [ϕ] = d x L (φi,∂φi) + Ji(x)φi(x) . (4.17)

Next, one can deﬁne a functional W[J] in terms of the probability amplitude for the vacuum state in the far past to go into the vacuum state in the far future in the presence of the external source

−W[J] + − e = 0 0 . (4.18)

We chose to work in Euclidean space and as a result we have a negative sign instead of i. The term W[J] is the generating functional for the connected Green functions; that is, we can write

1 Z d+1 d+1 d+1 n1...np W[J] = ∑ d x1 ...d xn1 ...d wnk G (x1 ...xp)J1(x1) {n} n1!...nk!

...J1(xn1 )...Jk(wnk ), (4.19)

n ...n in which G 1 p (x1 ...xp) is the sum of all connected Feynman diagrams with n1 external lines of type 1, n2 external lines of type 2, etc. A classical field can be defined as a 1-point correlator of the field (expectation value of field) and can be found using W[J],

h0+ |φ(x) |0−i ϕ (x) = i i h0+ |0−i δW[J] = . (4.20) δJi(x)

The term W[J] and the classical ﬁeld ϕi(x) which are functionals of source J can

87 be used to perform a Legendre transformation and obtain the effective action, Γ[ϕ], a functional of the classical ﬁeld Z d+1 Γ[ϕ] = W[J] − d x ϕiJi(x). (4.21)

Here, Γ[ϕ] is a generating functional for connected 1-particle irreducible Green functions. It can be expanded in classical ﬁelds

1 Z d+1 d+1 d+1 n1...np Γ[ϕ] = ∑ d x1 ...d xn1 ...d wnk Γ (x1 ...xp)ϕ1(x1) {n} n1!...nk!

...ϕ1(xn1 )...ϕk(wnk ). (4.22)

n ...n The Γ 1 p (x1 ...xp) are the 1-particle irreducible Green functions, deﬁned as the sum of all connected Feynman diagrams which cannot be disconnected by cut- ting a single internal line; these are evaluated without propagators on the external lines. It can be veriﬁed by the following observations.

For simplicity, we assume that we only have one species of field φ. The defi- 0 δϕ(x) nition of the classical field Eq.(4.20), can be used to deduce that G(x,x ) = δJ(x0) . The expansion for the effective action, W[J], Eq.(4.19,4.22), and the effective ac- p p tion definition Eq.(4.21) can be used to relate Γ (x1 ...xp) and G (x1 ...xp). For example, for 3-point functions, we find that,

Z 3 d+1 d+1 3 G (x1,x2,x3) = d w1 ...d w3 G(x1,w1)G(x2,w2)G(x3,w3)Γ (x1,x2,x3), (4.23) or diagrammatically for 3-point and 4-point functions, we can show the results as Fig.4.3 and Fig.4.4. In other words, the connected 3-point and 4-point functions can be constructed as a sum of irreducible 3-point and 4-point vertices with exact propagators in the external line. This procedure can be extended to an arbitrary n-point function and one may convince oneself that the one-particle structure was

88 Figure 4.3: Connected reducible three-point function in terms of irreducible vertices.

Figure 4.4: Connected reducible four-point function in terms of irreducible vertices.

fully analyzed such that Γ vertices are indeed irreducible.

As we are interested in spontaneous symmetry breaking of our symmetrical Lagrangian, we are interested in solutions of Eq.(4.20) with non-zero ϕ in the absence of source, J(x) = 0. We will focus our investigations on spatial and temporal

89 constant solution, ϕ(x) = ϕ0. In this case, one can deﬁne an effective potential as Z d+1 Γ(ϕ0) = −Veff(ϕ0) d x. (4.24)

Expansion around a constant classical ﬁeld conﬁguration is equivalent to expanding Γ in powers of the external momenta about the point where all external momenta are zero; effective potential is the zeroth-order term of such an expansion,

Z 1 Γ = dd+1x −V(ϕ ) + Z(ϕ )∂ϕ.∂ϕ + ... . (4.25) 0 2 0

By comparing the expansion in Eq.(4.22) and Eq.(4.25), it is easy to notice that nth derivative of the effective potential with respect to the classical ﬁeld is the sum of 1-particle irreducible graphs with n vanishing external momenta.

To approximate the effective potential, we use the “background ﬁeld” method developed by Jackiw and DeWitt [85], [86]. Their result can be written as

1 Γ[ϕ] = S[ϕ] + lndetD−1 + ..., (4.26) 2

−1 δ 2S[φ] where D (ϕ) = 2 . In deriving the above, authors assume that there δφ φ=ϕ exists a solution φ0, that satisﬁes the classical equations of motion.

S[ ] δ φ = 0. (4.27) δφ φ=φ0

In this method, essentially we expand the action around the stationary point of the action but later on we treat the field corresponding with the stationary point as the classical field. In other words we expand the action around the classical field and drop the linear-terms (tadpole). The power of this methods comes from the fact that in a compact way, it sums up all of 1-loop graphs which contribute to the effective potential. A task which is quite computationally hard and vulnerable against errors if one chooses to use the usual Feynman graph summation and take

90 into account the combinatorial factors. It is suggested that some mechanisms of spontaneous symmetry breaking only can be obtained by including an infinite subset of loop diagrams. As a result, it is crucial to employ a method that at least benefits from summing up an infinite subset of loop diagrams[83], [84].

6 4.4 Tachyonic Excitations in φ2+1 In this section, we find the effective potential for φ 6 model in 2 + 1 d. The O(N) vector field will be denoted ~φ(x). We will find it convenient to describe the theory 1 ~ 2 by two variables, the composite field χ(x) = N φ and an auxiliary field M(x), whose expectation value is proportional to the ~φ-field mass. Both χ(x) and M(x) M(x) have classical dimension one and χ(x) is dimensionless. Whenever hM(x)i is not zero, the ~φ-field is massive and it does not obtain an expectation value. We will find that, at the leading and next-to leading order in the 1/N expansion, the renormalized background field effective action [86], is

Z χ3(x) S = N d3x g2(M(x)) − g∗2 M(x) + 6 χ(x) ∂M(x) · ∂M(x) + + ... , (4.28) 96π|M(x)|

2 ∗2 M where g (M) is the running coupling at scale M and g (x), where x = χ , is the scale invariant part of the non-derivative terms in the effective action, containing 1 contributions of order one and of order N . The ellipses denote contributions of or- 1 der N2 or higher of any type and terms with more than two derivatives. Although χ is nominally a positive operator, an inﬁnite normal ordering constant has been subtracted from it so that it can now be either positive or negative. The couplings have been tuned so that terms proportional to (~φ 2)2 or ~φ 2 are absent.

To use the background ﬁeld effective action (4.28), we should ﬁrst solve the

91 equations which determine its extrema,

δS δS = 0 , = 0. (4.29) δ χ(x) δM(x)

Solutions of these equations are the classical fields which we shall denote by M0 and χ0. If there are more than one solution (there will not be in our example), we should 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 M0 and χ0 are non-zero and that they are slowly varying functions, sufficiently so that the expansion in their derivatives is accurate. (M0 and χ0 are usually constants.) Then, in order to compute a one-particle irreducible correlation function of the fields χ(x) and M(x), we take functional derivatives of the background field action S by the variables χ(x) and M(x), and we subsequently evaluate the resulting functions “on-shell” by setting χ(x) and M(x) to χ0 and M0, respectively. This yields the renormalized, connected, one-particle-irreducible multi-point correlation functions of the quantum fields χ(x) and M(x). For example, the connected two-point correlation functions are found by inverting the one-particle irreducible two-point functions which are obtained as functional second derivatives of the effective action. They are thus given by " # hχχi − hχihχi hχMi − hχihMi = hMχi − hMihχi hMMi − hMihMi −1 " δ 2S δ 2S # δ χ2 δ χ∂M = 2 2 . δ S δ S δ χ∂M δM2 M,χ=M0,χ0

For example, we obtain the composite operator correlation function

1 2 1 2 1 2 1 2 N φ (x) N φ (y) − N φ (x) N φ (y) 1 Z d3 p 48πχ2/M = eip(x−y) 0 0 . (4.30) 3 24πχ3 N (2π) p2 + 0 β(g2(M )) M0 0

92 Let us review a few interesting features of our results:

1. We are putatively working in the leading and next-to-leading orders of the large N expansion. The quantities in brackets in (4.28) are of order one and 1 2 of order N . The running coupling constant, g (M), on the other hand, is the solution of the renormalization group equation using the beta function 1 1 1 which is of order N . If expanded in N , it contains all orders of N , multi- plied by powers of logarithms of the mass scale ratio. This “sum of leading logarithms” is needed in order to accommodate possible very small or very large values of the condensate, M ∼ µ exp(N · ...).

2. At this order in the large N expansion, the only renormalization group function entering the affective action (4.28) is the running coupling constant g2(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 on the details of the function g∗2(x) in equation (4.28). It relies only on the fact that its leading contribution at large N is independent of N and the fact that it is scale invariant, that is, it is a function of only the dimensionless M 2 ratio χ and g . Validity of the derivative expansion also requires non-zero χ0 and M0 as the classical solutions. When evaluated on the solutions of the ∗2 2 1 equations of motion (4.29), g = (4π) + O( N ). At leading order in large N, g∗ = 4π, the value of the coupling at the Bardeen-Moshe-Bander ﬁxed point.

4. When N goes to infinity, the beta function vanishes and g2(M) becomes M-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 does not determine the scale. The fluctuations of the scale, M or χ, is in a flat direction and forms a massless dilaton. This is the Bardeen-Moshe-Bander solution.

5. The signature of the dilaton is the presence of the pole in the correlation

93 function of φ 2(x)φ 2(y) in equation (4.30). Note that the mass of this pole is proportional to the beta function, β(g2(M)). The latter vanishes at inﬁnite N, leaving the dilaton massless in that limit.

6. For large but ﬁnite N, the pole in the two-point function (4.30) occurs at

3 2 24πχ0 2 −p = β(g (M0)). M0

2 The beta function is positive, β(g (M0)) > 0. However, the values χ0 and M0 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 propagator in (4.30) having a negative sign and the dilaton has become a tachyon. The tachyonic mass indicates that the phase that we are describing is unstable to ﬂuctuations.

In the following, we will present our derivation of equation (4.28) and the simple computation leading from equation (4.28) to (4.30). We consider the Euclidean quantum ﬁeld theory which has N real scalar ﬁelds ~φ = (φ 1,φ 2,...,φ N) and O(N) symmetry in three space-time dimensions. The classical Landau-Ginzburg potential is given by

r u g2 V(~φ 2) = ~φ 2 + (~φ 2)2 + (~φ 2)3, (4.31) 2 4N 6N2 where N ~φ 2 ≡ ∑ φ aφ a. a=1 When u > 0, there is a line of second order phase transitions at r = 0 as depicted in Fig. 4.5. When u < 0 there is a line of ﬁrst order phase transitions. These lines of transitions terminate at the tri-critical point O where u = r = 0. At the classical level this phase structure persists for all positive values of g2 and the g2(~φ 2)3 coupling is exactly marginal.

94 Figure 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 of second order phase transitions and a line of ﬁrst order phase transitions g2 ~ 2 3 where the potential is equal to 6 (φ ) .

To examine ﬂuctuations, we consider the Euclidean functional integral

Z R 3 ~ ~ Z[ j] = [d~φ] e− d xL[φ, j], (4.32) with the Lagrangian density

1 L = ∂ ~φ∂ ~φ + NV(~φ 2/N) −~j ·~φ, (4.33) 2 µ µ where µ = 1,2,3 and repeated indices are summed and we have introduced a source ~j(x) in order to use the functional integral as a generating functional for correlators of ~φ(x). In order to study the large N limit, we introduce two auxiliary

95 ﬁelds by inserting

Z ∞ 1 = [dχ(x)]δ(χ(x) −~φ 2/N) (4.34) −∞ Z ∞ Z i∞ 2 R 1 m2(N −~ 2) = [dχ(x)] [dm (x)]e 2 χ φ (4.35) −∞ −i∞ 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 to note that the integration for m2 is on the imaginary axis. We will find out and explicit form for the scale M2 that was present in (4.28) as a function of m2. With these additional fields, the Lagrangian becomes

1 m2 m2 L = ∂ ~φ∂ ~φ + ~φ 2 − N χ + NV(χ) −~j ·~φ. (4.36) 2 µ µ 2 2

The ~φ-ﬁelds now appear in a quadratic form and we integrate them exactly to get an effective action

N S[m2, χ,~j] = Trln(−∂ 2 + m2) 2 Z m2 1 1 + NV(χ) − N χ − ~j ~j . (4.37) 2 2 −∂ 2 + m2

To ﬁnd the partition function, it remains to integrate χ and m2,

Z 2 ~ Z[ j] = [dm2dχ] e−S[m ,χ, j] (4.38)

. This would yield a generating functional where functional derivatives with respect 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 not spontaneously broken. Instead, there will be a condensates hχ(x)i and m2(x) which will result in a mass gap for the ~φ-ﬁeld. To begin, it is instructive to put the

96 source j(x) to zero and to write the effective action in (4.37) in an expansion in derivatives of the variable χ(x) and m2(x) [88, 89],

S Z Λm2 |m|3 m2χ ∂m.∂m = − +V(χ) − + + ... , (4.39) N 4π2 12π 2 96π|m| where Λ is the ultra-violet cut-off and the ellipses represent terms with more than two derivatives of m. We have dropped a constant term that is m2 and χ independent. The effective action in (4.39) has an ultra-violet divergent Λ-dependent term which must be removed by renormalization. We can renormalize the expression by introducing counter-terms. This is accomplished by replacing V(χ) by V − Λ . Then, after a field translation, (x) = ˜ (x)+ Λ , the cut-off depen- χ 2π2 χ χ 2π2 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, third and fourth terms in (4.39) are the effective potential for m and χ at the leading order in the large N expansion. In the remainder of this chapter, we will choose the potential V(χ) to be the specific dimension-three operator

g2 V(χ) = χ3(x) + counterterms, (4.40) 6 where the counterterms will be needed to cancel divergences at higher orders in 1 N . 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 quantum level in the leading order in the large N expansion.

In the large N limit, we can use the saddle-point technique to evaluate the remaining functional integral (4.38). The saddle points are field configurations which solve the equations of motion derived from the effective action (4.37). We will use the notation χ0 and m0 to denote fields which satisfy the equations of motion. When the fields are constant, the saddle points are extrema of the renormalized effective potential obtained from (4.39), The potential in (4.39) has a line

97 of extrema, located at |m0| = −4πχ0 and these extrema exist only when the coupling constant g is set to the Bardeen-Moshe-Bander ﬁxed point at g → g∗ = 4π. To see this, consider the equation of motion for m. This equation does not involve the coupling constant. It has the solution ( −4πχ χ < 0 |m0| = 0 χ > 0 .

A massive solution exists only when χ is negative. Let us assume this is so. We can plug this solution into the effective action to get (We use Sˆ to distinguish this partially on-shell action from S deﬁned elsewhere.)

Z |χ|3 Sˆ = N − g2 − (4π)2 + ... , (4.41) 6 and ask whether there is now a solution for χ. When g > g∗ this expression has no 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 the background ﬁeld technique. To implement this technique, we do the substitution

χ → χ + δ χ , m2 → m2 + iδm2 (4.42) and, following the recipe in [86], we drop the linear terms in δ χ and δm2. Then, the action expanded to quadratic order is

N Z Λm2 S = Trln(−∂ 2 + m2) − N 2 4π2 Z m2 1 1 + NV(χ) − N χ − ~j ~j 2 2 −∂ 2 + m2 " #" # N Z V˜ 00(χ) −i/2 δ χ + [δ χ,δm2] + ..., (4.43) 2 −i/2 ∆/2 + J [~j] δm2

98 where

When m2 is a constant,

3 (x,y) = R d p eip(x−y) (p) ∆ (2π)3 ∆ 1 p ∆(p) = 4π p arctan 2|m| . (4.46)

Before we proceed, we can use the action (4.43) to study the spectrum of fluctuations in the infinite N limit. For this purpose, we invert the quadratic form in (4.43) and find the propagator

D E D ED E 1 ~ 2 1 ~ 2 1 ~ 2 1 ~ 2 N φ N φ − N φ N φ = hδ χ δ χi 2 2 1 p ∆(p) N 4π p arctan 2|m| 3m 1 = N = ≈ , (4.47) 1 + 2V 00( )∆(p) 2m p N p2 χ 1 − p arctan 2m π where, in the last equality, we have put the condensate on shell and the coupling constant equal to the ﬁxed point value, 4π. The last expression reproduces the sum of bubble diagrams which would be expected from studying the Feynman diagrams for this correlation function. The massless pole is due to the dilaton which is a Goldstone boson for spontaneous breaking of the scale symmetry which is exact at this order in the large N expansion. We can see that this massless pole is the only pole by studying the denominator of (4.47).

2m p Z 1 x2 1 − arctan = dx 2 (4.48) p 2m 0 4m 2 p2 + x

99 in the complex −p2-plane. It is easiest to see from the integral representation of the function that the only zero is at −p2 = 0. There is also a cut singularity on the positive −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 integral over the ﬂuctuations in (4.43) to get the effective action

N S = Trln(−∂ 2 + m2) 2 Z m2 1 1 + NV(χ) − N χ − j j 2 2 −∂ 2 + m2 " # 1 V˜ 00(χ) −i/2 + lndet + ..., (4.49) 2 −i/2 ∆/2 + J where the ellipses stand for corrections of order 1/N and higher. When we assume that the source j and the classical ﬁelds m2 and χ are constants, we obtain the effective action evaluated on constant ﬁelds, ( Z 1 3 m2 ~j2/N S = N − m2 2 +V(χ) − χ − 12π 2 2m2 " !# 1 Z d3 p 2~j2/N + ln 1 + 2V 00(χ) ∆(p) + 2N (2π)3 m4(p2 + m2) +...}. (4.50)

Corrections represented by the ellipses in the last line of (4.50) are functions of 1/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 is the next-to-leading order. The integral in the next-to-leading order is ultra-violet divergent and renormalization is required. The linear term in the effective action

100 in a Taylor expansion in ~j2/N is

~j2 1 4 g2χ Z d3 p 1 1 − − 2N m2 N m4 (2π)3 p2 + m2 1 + 2g2χ∆(p) ~j2 1 4 g2χ Z d3 p 1 − 2g2χ∆(p) = − − 2N m2 N m4 (2π)3 (p2 + m2) 4 g2χ Z 0 d3 p 1 1 − N m4 (2π)3 p2 + m2 1 + 2g2χ∆(p) ~j2 1 ≡ − , (4.51) 2N M2 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 renormalization is finite. The prime on the integration in the third line means that the first two divergent terms which are written before it have been subtracted, resulting in a finite integral. (These divergent terms are the first two terms in a Taylor expansion of the integral in g2.) Keeping M finite as the ultra-violet cut-off is scaled to infinity requires that we take m2 to be a divergent function of M2,

4 2 2 2 4 2 Λ M g χ Λ m = M − g χ 2 − + 2 ln N 2π 4π 2π N ξ1M 4 Z 0 d3 p 1 1 − g2χ , (4.52) N (2π)3 p2 + M2 1 + 2g2χ∆(p) where ξ1 parameterizes the ﬁnite part of the logarithmically divergent integral.

101 The effective action is ( Z 1 M2 ~j2/N S = N − M3 +V(χ) − χ − 12π 2 2M2 4 2 χ M 4 2 Λ M g χ Λ + + g χ 2 − − 2 ln 2 8π N 2π 4π 2π N ξ1M 4 Z 0 d3 p 1 1 + g2χ N (2π)3 p2 + M2 1 + 2g2χ∆(p) " !# 1 Z 0 d3 p 2~j2/N + ln 1 + 2g2χ ∆(p) + 2N (2π)3 M4(p2 + M2) 3 1 Z d p 2 4 3 + V 00∆ − V 00∆ + V 00∆ + ... . (4.53) N (2π)3 3

As before, the prime on the integral in the fourth line indicates that the term of order ~j2/N and the divergent terms which are written in the fifth line (these are the first, second and third order terms in a Taylor expansion in g2) have been subtracted to render the integral finite. The terms that have been introduced by χ M the mass renormalization are proportional to 2 + 8π which vanishes on-shell. Here, we will first renormalize the effective action off-shell and then later on we will put the variables on-shell. We will find that the action is both on-shell and off-shell renormalziable. The divergent terms in the fifth line are

2 2 1 Z 2 4 3 g χ Λ M V 00∆ − V 00∆ + V 00∆ = − N 3 N 2π2 4π

4 2 π Λ − 4M ln Λ 6 3 g χ 2 Mξ2 g χ Λ − 6 3 + 8 2 ln , N 2 π 3 · 2 π Mξ3 where ξ2 and ξ3 are constants which parameterize the ﬁnite parts of divergent integrals.

Putting these in the effective action, we ﬁnd a miraculous cancellation. All divergent terms with a power of M in the numerator cancel. The remaining divergent

102 terms can be canceled by counter-terms added to V(χ) alone. The counter-terms introduce the scale µ in the action. What remains is ( Z 1 g2 M2 ~j2/N S = N − M3 + χ3 − χ − 12π 6 2 2M2 2 2 2 4 2 M g χM g χM g χ M ξ2 − χ + + 2 − 3 ln 4π 2πN 16π N 16π N ξ1 M g2χM Z 0 p2 1 + χ + dp 4π π2N p2 + 1 g2χ arctan p/2 1 + M 2π p " !# M3 Z 0 g2χ arctan p/2 4~j2/NM6 + dpp2 ln 1 + + 4π2N M 2π p (p2 + 1) g4χ3 µ g6χ3 µ − 2 ln + 8 2 ln + ... . (4.54) 4π N ξ1M 3 · 2 π N Mξ3

We shall set j2 = 0 and seek solutions of the equations of motion

δS = 0 (4.55) δM δS = 0. (4.56) δ χ

Here we used hard cut-off in order to regulate our integrals. In Appendix.E, we use dimensional regularization to regulate all of the diverging integrals and verify our results’ independence from the choice of the regulator.

There are three important lessons to be learned from the form of the effective action (4.54).

1. First of all, to this order in 1/N the theory is off-shell renormalizable. The effective action that we have computed can be used to ﬁnd the renormalized correlation functions of φ, χ,im2-ﬁelds where all external lines have vanishing momenta.

103 2. 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). From 2 those terms we can ﬁnd the beta function for the g (~φ 2)3 interaction. The 6N2 effective action is a physical quantity, the volume times the energy of the theory 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 g depends on µ in such a way that the action does not depend on µ explicitly. This yields

∂ 1 3g4(µ) g6(µ) µ g2(µ) − − ln = 0, ∂ µ N 2π2 27π2 M

d 1 3g4 g6 β(g) = µ g2(µ) = − + ..., (4.57) dµ N 2π2 27π2

where the ellipses denote contributions of order 1/N and higher. This result matches 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 argument 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 depend on 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, δ δM S is a homogeneous function of χ and M and it is therefore solved by M = αχ whence it gives an equation for α. That equation is solved by 1 α = −4π + δα where δα ∼ N .

We can write the effective action in the form

S Z χ3 h µ i = g2(µ) − g∗2( M ,g) − β(g(µ))ln + ... , (4.58) N 6 χ M

104 where

1 1 g∗2 (x,g) = x3 + 3x2 + O , (4.59) 2π N and the g-dependence is only in the higher orders in 1/N and can be substituted for its leading order g = 4π. Also, on-shell,

1 g∗2 (x ,g) = (4π)2 + O . (4.60) 0 N

χ3 ∗2 In − 6 g , we have gathered all of the terms in the effective action (4.54) except µ g3 3 those proportional to ln M in the last line and the 6 χ term in the ﬁrst line. In the last equality, we have used the leading order solution of the equation δS/δM = 0, which is , M = −4π + O(1/N) in g∗2. χ0

Then minimum of (4.58) occurs at

g∗2 − g2 1 M = µ exp − , (4.61) β(g) 3 where we use equation (4.60) for g∗. This solution is non-perturbative, both in 1 the sense that, since β ∼ N , it does not have a Taylor expansion in 1/N, and in the sense that, when it is substituted into the effective action, the logarithm produces a 1 factor of β ∼ N which invalidates the large N expansion. In higher orders, powers M of ln µ will produce factors of N which can cancel their large N suppression. This is similar to the phenomenon in the scalar ﬁeld theory example in Coleman and Weinberg’s work [87] on dynamical symmetry breaking. There, they used the renormalization group to re-sum higher order logarithmic terms to obtain a more accurate result. When they did, the minimum went away - there was no longer a symmetry breaking solution. In the present case, we will be more fortunate. This allows us to ﬁnd a solution is the presence of g∗ in the action. To begin, we will use the renormalization group to sum the leading logarithms of perturbation theory to

105 all orders. In this particular case, it is very simple. We replace the combination which occurs in the effective action,

µ g2(µ) − β(g(µ))ln , (4.62) M by the running coupling at scale M, g2(M), which is deﬁned by integrating the beta function

Z g2(M) dg2 M = ln . (4.63) g2(µ) β(g) µ

The result of the integral, g2(M), has a 1/N expansion and the leading terms repro- 1 µ duce (4.62). The corrections have higher orders in N ln M . The renormalization group improved potential energy of the effective action is then the one given in equation (4.28), which we recopy here for the reader’s convenience,

Z χ3(x) S = N d3x g2(M(x)) − g∗2 M(x) + 6 χ(x) ∂M(x) · ∂M(x) + + ... . 96π|M(x)|

We will now study the states of the theory using this effective action. The equations of motion are,

δS χ2 χ M 0 = = 0 g2(M ) − g∗2 + 0 0 g∗20 (4.64) δ χ(x) 2 0 6 δS χ3 1 1 0 = = 0 β − g∗20 , (4.65) δM(x) 6 M0 χ0

∗20 ∗2 where g is a derivative of g by its argument M/χ. M0 and χ0 are the solutions of these equations. We have dropped the derivative terms since we assume that the solutions will be constant ﬁelds.

106 Equations (4.64) and (4.65) imply

∗20 χ0 2 g = β(g (M0)), (4.66) M0 1 g2(M ) − g∗2 = − β(g2(M )). (4.67) 0 3 0

Equation (4.66) is an algebraic equation containing terms of order one and of order 1 and the variables M0 and g2. g2 appears only in the terms of order 1 and N χ0 N it can therefore be regarded as a constant, and set to 4π. In the leading order, equation (4.66) has the solution M0 = −4π + O(1/N) and the order 1 terms are χ0 N easily computable.

We recall that g∗2 has a similar structure to equation (4.66), it contains terms of order one and of order 1 and the variables M0 and g2, and g2 appears only in N χ0 the terms of order 1 . We can then plug the solution for M0 which we discussed in N χ0 ∗ 1 the above paragraph into g to obtain a N 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 coupling solves the equation. This yields the value of the condensate M0 and the 1 1 1 above considerations then determine χ0 = − 4π M0 + O( N ). Due to the order N violation of scale invariance, and unlike the scale invariant inﬁnite N limit, the values of these condensates are no longer arbitrary, but they are ﬁxed by the value of the running coupling constant at some reference scale. We substitute the solution into the effective action and then obtain

Z M3 S = N 0 β(g(M )) + ... , (4.68) on−shell 18(4π)3 0 where the ellipses are terms of order 1 and higher. N2

To examine the fate of the dilaton, we return to the action (4.28) and we con-

107 sider the ﬂuctuation matrix about the solution that we have found,

2 2 1 δ S χ0 M0 ∗200 2 = β − g , (4.69) N δ χ 3 6χ0 1 δ 2S χ2 M = 0 β + 0 g∗200, (4.70) N δ χ∂M 6M 6 2 3 2 1 δ S χ0 χ0 ∗200 p 2 = − 2 β − g + , (4.71) N δM 6M0 6 48π|M0| 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,

2 2 " δ S δ S # 2 2 1 2 M p det δ χ δ χ∂M = 0 β − , (4.72) N2 δ 2S δ 2S 32π2 12 δ χ∂M δM2 where we have used the fact that

1 M3 3M2 g∗2 = + + ... = (4π)2 + ... (4.73) 2π χ3 χ2 and,

3 M g∗200 = + 6 + ... = −6 + .... (4.74) π χ

The determinant of the the ﬂuctuation matrix is proportional to the inverse propagator of the χ- and M- ﬁelds. The beta function is positive over the interesting range of g2 (see Fig. 4.1). Clearly, from equation (4.72), we see that these excitations are tachyonic with mass given by,

3M2 m2 = − 0 β. (4.75) dilaton 8π2

108 4.5 Conclusion We conclude this chapter by summarizing our results. We found that, although at leading order in 1/N, phi-six theory in three dimensions exhibits spontaneous scale 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 spontaneously broken phase is therefore unstable. We found that BMB phase is not stable. Our background field technique found the perturbative beta function of the O(N) symmetric g2(φ 2)3 theory. The result agreed with the beta function originally 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 the phase in which the interactions induce a condensate is unstable in finite N.

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116 Appendix A

Trace over Π1 Dirac Matrices

In this appendix, we compute the trace over the Dirac matrices which appear in the numerator of Π1 integrand. The numerator of the integral is given by the following

µ ρ µ ρ Num1 =Tr(/q + /p)γ (/q + /p + /k)γ (/q + /k)γ /qγ = + 2Tr(/q + /p)γρ (/q + /k)(/q + /p + /k)/qγρ − Tr(/q + /p)(/q + /p + /k)γ µ γρ (/q + /k)γ µ /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)

117 " Num1 = 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 identities

2(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)

118 we can simplify the numerator, Num1, and get more simple form for it. The numerator then can be simpliﬁed to the following expression " 1 Num = (−k2 + (q + p)2 + (q + p + k)2){−k2+ 1 2 (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)

" 1 = (q + p + k)2 − k2 + (q + k)2 + q2 + −p2 2 + (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)

" 1 = (q + p + k)2 4k2 − 4(q + k)2 + 2q2 + 4p2 2 − 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 − 10k2 p2 . (A.6)

Now, this expression would be integrated using a measure which has the following denominator, (q + p)2(q + p + k)2(q + k)2q2 ps. We can use some symmetries of this denominator to simplify the above expression. For example, q → −q−k then

119 p → −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 as

2 2 2 2 2 2 4 4 Num1 = + (q + p + k) 8k − 4(q + k) + 2q + 8p − 4(q + p) − 2k − 2p − 5k2 p2. (A.7)

120 Appendix B

Recurrence Relations for Is

Consider the integral

1 I = . (B.1) s (q + p)2(q + p + k)2(q + k)2q2 ps

By dimensional analysis, it is of order k−s−2 and it therefore obeys the identity

d k · I + (s + 2)I = 0, (B.2) dk s s or

2k · (q + p + k) 2k · (q + k) (s + 2)I = + . (B.3) s (q + p + k)2 (q + k)2

We do the transformation q → q − p then p → −p in the ﬁrst term to get

2k · (q + k) (s + 2)I = 2 . (B.4) s (q + k)2

Then we use 2k · (q + k) = −q2 + k2 + (q + k)2 to get

s k2 − q2 I = . (B.5) 2 s (q + k)2

121 Now, we can also use

2p · (q + p) 2p · (q + p + k) 0 = 3 − − − s (q + p)2 (q + p + k)2 ∂ p = . (B.6) ∂ p (q + p)2(q + p + k)2(q + k)2q2 ps

Again, in the second-last term, we transform q → −q − p − k to get

2p · (q + k) (3 − s)I + 2 = 0. (B.7) s (q + k)2

Now, using 2p · (q + k) = (q + p + k)2 − p2 − (q + k)2, we have

(1 − s) (q + p + k)2 − p2 I = − . (B.8) 2 s (q + k)2

Finally,

2q · (q + p) 2q · (q + p + k) 2q · (q + k) 0 = 1 − − − (q + p)2 (q + p + k)2 (q + k)2 ∂ q = , (B.9) ∂q (q + p)2(q + p + k)2(q + k)2q2 ps

I q · (q + k)+(q + p) · (q + k)+(q + k + p) · (q + k) s = . (B.10) 2 (q + k)2

After a bit of simplifying Is we ﬁnd that

1 (q + p + k)2 + q2 − p2 − k2 I = − . (B.11) 2 s (q + k)2

122 Now, we assemble the results:

s k2 − q2 I = , (B.12) 2 s (q + k)2 1 − s (q + p + k)2 − p2 I = − , (B.13) 2 s (q + k)2 1 (q + p + k)2 + q2 − p2 − k2 I = − . (B.14) 2 s (q + k)2

Not all of these equations are independent, indeed (B.12)+(B.13)=(B.14). Using (B.12),

s − 2 k2 p2 − q2 p2 I = 2 s−2 (q + k)2 −(q + p + k)2 + p2 k2(q + p + k)2 − q2 p2 =k2 + (q + k)2 (q + k)2 1 − s k2(q + p + k)2 − q2 p2 =k2 I + , (B.15) 2 s (q + k)2 or

1 − s 2 k2(q + p + k)2 − q2 p2 I = −k2 I − . (B.16) s−2 2 − s s 2 − s (q + k)2

123 Now, using our integral formula

k2(q + p + k)2 1 = k2 (q + k)2 [(q + k)2]2(q + p)2q2 ps 2 1 1 = k 2 2 2 s [(q + k) ] q (q + p)2[p2] 2 2 s 1 1 3 s * + k Γ[ 2 − 2 ]Γ[ 2 ]Γ[ 2 − 2 ] 1 = 3 s s s 1 2 2 2 + (4π) 2 Γ[ 2 ]Γ[2 − 2 ] [q + k) ] [q ] 2 2 2 s 1 1 3 s s 1 s k Γ[ 2 − 2 ]Γ[ 2 ]Γ[ 2 − 2 ] Γ[ 2 + 1]Γ[− 2 ]Γ[1 − 2 ] = s s s s 1 1 s 3 2 2 +1 Γ[ ]Γ[2 − ] (4π) [k ] 2 2 Γ[2]Γ[ 2 + 2 ]Γ[ 2 − 2 ] 1 s = (B.17) 32π2ks 2 − s and

q2 p2 1 = (q + k)2 [(q + k)2]2(q + p + k)2(q + p)2 ps−2 1 1 = 2 2 s −1 (q + k) q [(q + p + k)2]2[p2] 2 s 1 1 5 s * + 1 Γ[ 2 − 2 ]Γ[− 2 ]Γ[ 2 − 2 ] 1 = 3 s s s 1 2 + 2 (4π) 2 Γ[2]Γ[ 2 − 1]Γ[2 − 2 ] [(q + k) ] 2 2 q 1 Γ[ s − 1 ]Γ[− 1 ]Γ[ 5 − s ] Γ[ s ]Γ[ 1 ]Γ[1 − s ] = 2 2 2 2 2 2 2 2 (4 )3ks Γ[2]Γ[ s − 1]Γ[2 − s ] s 1 3 s π 2 2 Γ[ 2 + 2 ]Γ[ 2 − 2 ] 1 3 − s = − . (B.18) 32π2ks 1 − s

Now we can plug in the above result in (B.16 to ﬁnd an explicit recurrence relation for Is−2. 2 1 − s 2 1 s 3 − s I = −k Is − + . (B.19) s−2 2 − s 2 − s 32π2ks 2 − s 1 − s

124 In evaluation of two-loop integrals, we will need to have a recurrence relation for

Is−4. We can work it out using our relation for Is−2 and the result is

" 2 4 (3 − s)(1 − s) 1 1 s(3 − s) (3 − s) I =k Is + + s−4 (4 − s)(2 − s) 16π2ks−2 4 − s (2 − s)2 (2 − s)(1 − s) # 5 − s 2 − s − + . (B.20) 3 − s 4 − s

125 Recurrence Relations

2 1 − s 2 1 s 3 − s I = −k Is − + s−2 2 − s 2 − s 32π2ks 2 − s 1 − s

" 2 4 (3 − s)(1 − s) 1 1 s(3 − s) (3 − s) I = k Is + + s−4 (4 − s)(2 − s) 16π2ks−2 4 − s (2 − s)2 (2 − s)(1 − s)

# 5 − s 2 − s − + 3 − s 4 − s

Table B.1: Table of Recurrence Relations for Is

126 Appendix C

Calculation of I1

In this appendix we evaluate I1 using Schwinger parameters. Lets start by writing a more general form of the integral I1. Using our bracket notation, a more general form of I1 is given by

1 I(a1,...,a5;d) = . ((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 in the following form [53].

−d Z a−3d/2 al−1 (4π) Γ(a − d) U ∏l αl I(a1,...,a5;d) = dα1 ...dα5 δ 1 − αl , 2 a−d 5 ∑ a−d (k ) ∏l Γ(al) R>0 V (C.2) in which U and V are given by

U = (α1 + α2 + α3 + α4)α5 + (α1 + α2)(α3 + α4),

V = (α1 + α2)α3α4 + α1α2(α3 + α4) + α5(α1 + α3)(α4 + α2). (C.3)

127 We use the Cheng-Wu theorem [90] to change the delta function to δ(α5 = 1) and focus on I1 by imposing the following constraints a1 = a2 = a3 = a4 = 1, a5 = 1/2 and d = 3. We are then left with the following integral for I1

1 Z 1 I1(k) = dα1 ...dα5 3 3 5 128π k R>0 V (α1,α2,α3,α4,1) 1 Z = dα1 ...dα4 3 3 4 128π k R>0 1 . (C.4) (α1 + α2)α3α4 + α1α2(α3 + α4) + (α1 + α3)(α4 + α2)

Fortunately, the above integral can be evaluated explicitly. The integration can be done by integrating α1 and α3 ﬁrst and then the remaining integrals. We could evaluate I1 and it has a simple form

1 I (k) = . (C.5) 1 64k3

128 Appendix D

Instantaneous Limit of Graphene in Large N

In this section, we derive the propagator that we used in calculating the graphene’s beta function in the large N. We then take the limit in which the light velocity goes to inﬁnity and check that our propagator reduces to the more studied instantaneous limit propagator.

The kinetic terms of the Lagrangian can be decomposed into fermion and photon parts, we ﬁx our gauge such that the fermion part becomes diagonal. The kinetic terms are given by the following matrices

 N p  16 0 0 =  N p , (D.1) Lfermion  0 16 0  N p 0 0 16

129  2 2  2c(p1 +p2 )ε 2cp0 p1ε 2cp0 p2ε e2P − e2P − e2P  2 2 2  2cp p ε 2c(p0 +c p2 )ε 2c3 p p ε Lphoton =  0 1 1 2 . (D.2)  − e2P e2P − e2P   3 2 2 2  2cp0 p2ε 2c p1 p2ε 2c(p0 +c p1 )ε − e2P − e2P e2P

We now ﬁnd the inverse of the sum of the above terms and derive the propagator that we used in previous sections to calculate the graphene’s beta function. The propagator has the following form

 2 2  c pP+ξ p0 ξ p0 p1 ξ p0 p2 2 2 2 η1 p η1 p η1 p 16  + 2Pp2− c2 p(p2+c2 p2)   ξ p0 p1 η4 ξ 1 ξ 0 2 ξη3 p1 p2  ∆(p0,~p) = 2 2 2 . N  η1 p p η1η2 η1η2 p   2 2 2 2 2 2  ξ p0 p2 ξη3 p1 p2 η4−ξc p(p0+c p1)+ξ Pp2 2 2 2 η1 p η1η2 p p η1η2 (D.3)

To investigate the instantaneous limit, we need to take the c → ∞ limit. It is a non-trivial task to take that limit in the propagator however, it can be easily done for the Lagrangian. Lets expand our Lagrangian around inﬁnite light speed, the following Lagrangian needs to get contracted with Aµ

2  2~p εc N p 2p0 p1εc 2p0 p2εc  e2P + 16 − e2P − e2P  2(p 2c2+p2)εc 3  = 2p0 p1εc 2 0 N p 2p1 p2εc . (D.4) Lkinetic  − e2P e2P + 16 − e2P   3 2 2 2  2p0 p2εc 2p1 p2εc 2(p1 c +p0)εc N p − e2P − e2P e2P + 16

3 As the spacial part of Aµ contributes a term of order c to the Lagrangian and the temporal has a contribution of order c, the spacial part (magnetic) gets 00 suppressed. We can then only keep Lkinetic in our Lagrangian. We then get back the instantaneous Lagrangian.

130 Appendix E

Dimensional Regularization of φ 6 in Large N

In this section, we analyze φ 6 theory in the large N limit using dimensional regularization. We need to introduce a scale µ in our Lagrangian so that the coupling constant has the correct dimensions in any dimension of space-time. The corrected Lagrangian is given by

1 m2 m2 g2 L = ∂ ~φ∂ ~φ + ~φ 2 − N χ + N µ−2(2ω−3)χ3 −~j ·~φ. (E.1) 2 µ µ 2 2 6 0 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 obtain the following effective potential in arbitrary dimensions

V 1 Z d2ω p g2 m2 ~j2/N eff = ln(p2 + m2) + µ−2(2ω−3)χ3 − χ − NV 2 (2π)2ω 6 0 2 0 2m2 ! 1 Z d2ω p 2~j2/N + ln[1 + 2g2χ µ−2(2ω−3) ∆(p) + ]. (E.2) 2N (2π)2ω 0 m4(p2 + m2)

Lets expand the log term resulting from the determinant. To simplify the re-

131 sult, we use the identity ∆(p) = ∆(−p)

" !# Z d2ω p 2~j2/N ln 1 + 2g2χµ−2(2ω−3) ∆(p) + ' (2π)2ω m4(p2 + m2) ( ! Z d2ω p ~j2/Nm4 2∆(p) + 4 g2χµ−2(2ω−3) (2π)2ω p2 + m2 ! ~j2/Nm4 − 2∆(p)∆(−p) + 8∆(−p) (g2χµ−2(2ω−3))2 p2 + m2 ) 8 + ∆(p)3(g2χµ−2(2ω−3))3 . (E.3) 3

We only kept the terms that are not ﬁnite in dimensional regularization. We evaluate the above term by term in the following sections.

E.1.1 ~j2 Term It is well known that this term is ﬁnite however we reconﬁrm the result by explicit calculations. This term is given by the following

Z d2ω p 1 1 1 1−ω = Ω[2ω] πCsc[ωπ] (2π)2ω p2 + m2 2 m2 2ω→3 −−−→−2|m|π2. (E.4)

E.1.2 ∆(p) Term We do not expect this term to be divergent either however, lets conﬁrm our guess. We observe that ∆(p) term is given by the following

2 Z d2ω p d2ω q 1 1 Z d2ω p 1 = , (E.5) (2π)2ω (2π)2ω p2 + m2 (p + q)2 + m2 (2π)2ω p2 + m2

132 which is ﬁnite due to our previous calculation.

E.1.3 ∆(p)∆(−p) Term This term is indeed divergent and contributes to the beta function. Lets start evaluating it, it is given by the following

Z d2ω p Z 4 d2ω p 1 ∆(p)∆(−p) = i (2 )2ω (Σp ) 2ω ∏ 2ω 2 2 π δ i (2π) i=1 (2π) pi + m Z ∞ Z 4 d2ω p 1 = dx i exp(ip .x). (E.6) ∏ 2ω 2 2 i −∞ i=1 (2π) pi + m

To evaluate the above equation, we need to ﬁnd the solution to the Poisson equation in general dimensions. Fortunately, it can be done in a close form given by

Z 2ω Z ∞ Z 2ω d p 1 ip.x d p ip.x−λ(p2+m2) 2 2 2 e = dλ 2 e (2π) ω p + m 0 (2π) ω Z ∞ 2m −1 dλ − |mx| ( + −1) = | |ω πω e 2 λ λ x 0 λ ω x 1−ω = (2π)−ω K (|mx|). (E.7) m ω−1

133 Using 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,

Z d2ω p ∆(p)∆(−p) = (2π)2ω Z ∞ 4 −ω x 1−ω 2ω−1 dx (2π) | | Kω−1 (|mx|) x Ω[2ω] = 0 m −4ω 1 −5−4 1 −8+2 1−3 √ 2 ω |m| ω π ω πCsc[πω] − 4ω (E.8) Γ[ω] m 3 7 Γ[4 − 3ω]Γ[3 − 2ω]Γ[−1 + ω]2F1 4 − 3ω, 2 − ω, 2 − 2ω,1 7 Γ[ 2 − 2ω] 16Γ[2 − 2ω]Γ[2 − ω]Hyp 1 ,1,3 − 2ω , 5 − ω,ω ,1 − 2 2 5 Γ 2 − ω ! Γ[1 − ω]Γ[2 − ω]Hyp[{1,2 − ω,− 1 + ω},{ 3 ,−1 + 2ω},1] + 2 2 . (E.9) 2−5+2ω

We expand the result around 2ω = 3 and keeping only the poles. We ﬁnd that ∆(p)∆(−p) term is simply given by

Z d2ω p |m| ∆(p)∆(−p) ∼ . (E.10) (2π)2ω 32π3(2ω − 3)

E.1.4 ∆(p)~j2 Term We follow the same strategy as our strategy for evaluation of ∆(p)∆(−p) term. We again ﬁnd that ∆(p)~j2 term can be written in terms of hyper-geometric functions,

134 Z d2ω p 1 ∆(−p) = (2π)2ω p2 + m2 Z 3 d2ω p 1 i (2 )2ω (Σp ) = ∏ 2ω 2 2 π δ i i=1 (2π) pi + m Z ∞ Z 3 d2ω p 1 dx i exp(ip .x), (E.11) ∏ 2ω 2 2 i −∞ i=1 (2π) pi + m

Z 2ω Z ∞ 3 d p 1 −ω x 1−ω 2ω−1 2 ∆(−p) 2 2 = dx (2π) | | Kω−1 (|mx|) x Ω[2ω] (2π) ω p + m 0 m −3−2ω −3ω 2 1 −6+ 3 −2 = − m ω π 2 ω Csc[πω]Γ[3 5 m Γ 2 − ω Γ[ω] 5 1 1 1 − 2ω] F [1,3 − 2ω, − ω, ] + 3 F [1,2 − ω,− , ] 2 1 2 4 2 1 2 4 1 1 − 2(−1 + ω) F [1,2 − ω, , ] . (E.12) 2 1 2 4

Expanding the result around 2 + 1d, we ﬁnd the following pole for ∆(p)~j2 term

Z d2ω p 1 1 ∆(−p) = − . (E.13) (2π)2ω p2 + m2 32π2(2ω − 3)

E.1.5 ∆(p)3 Term We approximate ∆(p) by its exact value at zero mass

Z d2ω q 1 ∆(p) = (2π)2ω q2(p − q)2 Γ[2 − ω]Γ2[ω − 1] = [p2]ω−2. (E.14) (4π)ω Γ[2ω − 2]

135 Due to dimensional analysis non-zero mass can not contribute to the divergent terms of ∆(p)3. The integral is originally logarithmic divergent at m = 0 and the correction for the non-zero mass can not be divergent. As a result, we regulate the infrared divergences by a mass term, M2. As the UV part is not sensitive to this regularization, this regularization is valid. The IR regularization consequently will not alter the UV divergent part.

3 Z d2ω p Z d2ω p Γ[2 − ω]Γ2[ω − 1] ∆3(p) ' [p2 + M2]3ω−6 (2π)2ω (2π)2ω (4π)ω Γ[2ω − 2] → 1 −−−→−2ω 3 . (E.15) (64π)2(2ω − 3)

E.1.6 All Terms Together Now, we have all of the parts of the puzzle and we can put them together to ﬁnd the effective potential. The ﬂuctuations’ determinant is given by

" !# Z d2ω p 2~j2/N 1 |m| ln 1 + 2g2χµ−2(2ω−3) ∆(p) + ' − (2π)2ω m4(p2 + m2) (2ω − 3) 16π3 ! 1 ~j2 g6χ3 µ−2(2ω−3) − g4χ2 µ−2(2ω−3) µ−2(2ω−3) − µ−4(2ω−3) 4π2 Nm4 6(16π)2(2ω − 3) ! |m| 1 ~j2 2 = − g4χ2 + g6χ3 ln µ. (E.16) 8π3 2π2 Nm4 3(16π)2

Here, we isolated g6χ3 µ−2(2ω−3) and g4χ2 µ−2(2ω−3) as the original interactions have a similar dependence on µ, g2χ3 µ−2(2ω−3). To add the corrections to the original interaction terms, we need to put them in a similar form.

Now, we incorporate the newly calculated contributions to our effective po-

136 tential and ﬁnd the renormalized mass, M

2 3 2 2 2 4 2 2 2 2 3 ! V (m ) 2 g m ~j /N g |m|χ χ ~j g χ eff = − + χ3 − χ − − − + − ln µ NV 12π 6 2 2m2 N 16π3 4π2 Nm4 3(16π)2 2 3 2 2 ~2 4 2 2 3 (m ) 2 g 3 m j /2N g |m|χ g χ = − + χ − χ − 4 2 + 3 + 2 ln µ. 12π 6 2 m2 − 1 g χ ln N 16π 3(16π) N 2π2 µ (E.17)

4 2 4 2 We ﬁnd that M2 = m2 − 1 g χ ln µ or equivalently m2 = M2 + 1 g χ ln µ . We N 2π2 m N 2π2 M rewrite our potential in terms of the renormalized mass, M

V M3 1 g4χ2 µ g2 M2 1 g4 µ ~j2/2N eff = − − M ln + χ3 − χ − ln χ3 − NV 12π N 16π3 M 6 2 N 4π2 M M2 g4χ2 M g2χ µ + + ln N 16π3 3(16π)2 M M3 M2 ~j2/2N g2 g4 µ g2 = − − χ − + χ3 + ln −1 + χ3. (E.18) 12π 2 M2 6 4Nπ2 M 3 .26

Our ﬁnal result for the effective potential is in agreement with our previous result in which we had used the hard cut-off regulator.

137