2 + 1d Quantum Field Theories in Large N limit by 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 OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in 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 contri- butions to the fermions current-current correlation function h jm (x) jn (y)i by per- forming 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 con- tributing 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 f 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 dila- ton” is actually a tachyon. This indicates an instability of the phase of the theory with spontaneously broken approximate scale invariance. We rule out the exis- tence of Bardeen-Moshe-Bander phase. In this thesis, we show that Large N expansion is a powerful tool which 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 pub- lished. 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 pub- lished. In Chapter 4, we investigate f 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 jm (x) jn (y)i . 28 2.2.1 P1 .............................. 31 2.2.2 Evaluation of P1B ...................... 34 2.2.3 Evaluation of P1A + P1B . 35 2.3 Evaluation of P2 .......................... 36 v 2.4 Combining P1 and P2 ....................... 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 f 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 f2+1 ................... 91 4.5 Conclusion . 109 Bibliography . 110 A Trace over P1 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 f 6 in Large N . 131 E.1.1 ~j2 Term . 132 vi E.1.2 D(p) Term . 132 E.1.3 D(p)D(−p) Term . 133 E.1.4 D(p)~j2 Term . 134 E.1.5 D(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 f 4 theory. 11 Figure 2.1 Partial sum over fermionic loops to get the effective propaga- tor 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 deter- minant 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 measure- ments. (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 deter- minant 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 con- struction 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 num- ber 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(~f 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 cou- pling where in the infinite N limit scale symmetry breaking occurs is g2 = (4p)2 ≈ 158. 82 Figure 4.2 Spontaneous breaking of the internal rotation symmetry in f space. The field f chooses a ground state that violates the internal U(2) symmetry in the potential V(f) = f ∗f.