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Renormalization group analysis of phase transitions in the two dimensional Majorana-Hubbard model Wamer, Kyle Patrick 2018

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Renormalization group analysis ofphase transitions in the twodimensional Majorana-Hubbard modelbyKyle Patrick WamerB.Sc., The University of Toronto, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2018c© Kyle Patrick Wamer 2018Committee PageThe following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Renormalization group analysis of phase transitions in the twodimensional Majorana-Hubbard modelsubmitted by Kyle Patrick Wamer in partial fulfilment of the requirementsfor the degree of Master of Science in Physics.Examining Committee:Ian Affleck, Physics and Astronomy SupervisorFei Zhou, Physics and Astronomy Supervisory Committee MemberiiAbstractA lattice of interacting Majorana modes can occur in a superconductingfilm on a topological insulator in a magnetic field. The phase diagram asa function of interaction strength for the square lattice was analyzed re-cently using a combination of mean field theory and renormalization groupmethods, and was found to include second order phase transitions. One ofthese corresponds to spontaneous breaking of an emergent U(1) symmetry,for attractive interactions. Despite the fact that the U(1) symmetry is notexact, this transition was claimed to be in a supersymmetric universalityclass when time reversal symmetry is present and in the conventional XYuniversality class otherwise. Another second order transition was predictedfor repulsive interactions with time reversal symmetry to be in the sameuniversality class as the transition occurring in the Gross-Neveu model, de-spite the fact that the U(1) symmetry is not exact in the Majorana model.We analyze these phase transitions using a modified -expansion, confirmingthe previous conclusions.iiiLay SummaryAt zero degrees Celsius, water freezes. This is an example of a phase tran-sition between two phase of the molecule H20: liquid water and ice. Phasetransitions occur in all materials, and their classification has led to greatdiscoveries across all branches of physics. Recently, a class of materials hasbeen discovered with a unique feature: on their surfaces, these materialscan exhibit a new type of particle, called a Majorana particle, that has beenobserved nowhere else in nature. In this thesis, we use a model of Majoranaparticles to predict the phase transitions that may occur on the surface ofthese novel materials. This research may have applications in the field ofcomputer science, where scientists are attempting to use Majorana parti-cles to create the first quantum computer – a machine that uses quantummechanics to solve problems faster than a conventional computer.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Majorana-Hubbard Model . . . . . . . . . . . . . . . . . 32.1 The Majorana-Hubbard Model on the Square Lattice . . . . 42.2 Low Energy Field Theory . . . . . . . . . . . . . . . . . . . . 62.2.1 Hubbard-Stratonovich Transformation . . . . . . . . 82.3 Symmetry Constraints on U(1) Breaking Operators . . . . . 92.3.1 Quadratic Operators . . . . . . . . . . . . . . . . . . 102.3.2 Quartic Operators . . . . . . . . . . . . . . . . . . . . 102.3.3 Fermion-Boson Operators . . . . . . . . . . . . . . . . 113 Renormalization Group Methods . . . . . . . . . . . . . . . . 123.1 Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Wilson’s Approach to Renormalization . . . . . . . . . . . . 133.3 Dimensional Regularization . . . . . . . . . . . . . . . . . . . 153.3.1 Modified Minimal Subtraction Scheme . . . . . . . . 163.4 The Epsilon Expansion . . . . . . . . . . . . . . . . . . . . . 163.5 Modified Epsilon Expansion . . . . . . . . . . . . . . . . . . 183.5.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . 203.5.2 An Expansion in d = 3 + (1− ) Dimensions . . . . . 21vTable of Contents4 Renormalization of U(1) Breaking Operators . . . . . . . . 234.1 U(1) Breaking Operators with Attractive Interactions . . . . 244.1.1 Feynman Diagrams . . . . . . . . . . . . . . . . . . . 244.1.2 Beta Functions of U(1) Breaking Operators . . . . . . 304.2 U(1) Breaking Operators with Repulsive Interactions . . . . 314.2.1 Feynman Diagrams . . . . . . . . . . . . . . . . . . . 314.2.2 Beta Functions of U(1) Breaking Operator . . . . . . 325 Relevance of the Fermion Mass Operator . . . . . . . . . . 335.1 The Power of Supersymmetry . . . . . . . . . . . . . . . . . 335.1.1 Superspace Formalism . . . . . . . . . . . . . . . . . 345.1.2 Relating the fermion beta function and the stabilitycritical exponent . . . . . . . . . . . . . . . . . . . . . 355.2 Fermion Mass Beta Function . . . . . . . . . . . . . . . . . . 375.3 Consequence of a Relevant Fermion Mass Operator . . . . . 386 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41AppendicesA Derivation of the Low Energy Field Theory . . . . . . . . . 44A.1 Quadratic Hamiltonian . . . . . . . . . . . . . . . . . . . . . 44A.2 Quartic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 45A.3 Leading U(1) Breaking Operator . . . . . . . . . . . . . . . . 46B Promoting ψ to a Dirac Fermion in Four Dimensions . . . 48C Two Loop Calculation of the Fermion Mass Beta Function 50C.1 One Loop Diagrams . . . . . . . . . . . . . . . . . . . . . . . 51C.1.1 Fermion Propagator . . . . . . . . . . . . . . . . . . . 51C.1.2 Boson Propagator . . . . . . . . . . . . . . . . . . . . 53C.1.3 Interaction Vertex . . . . . . . . . . . . . . . . . . . . 54C.2 Two Loop Diagrams . . . . . . . . . . . . . . . . . . . . . . . 54C.2.1 Boson Counterterm Diagram . . . . . . . . . . . . . . 54C.2.2 Fermion Counterterm Diagram . . . . . . . . . . . . . 55C.2.3 Internal Boson Bubble Diagram . . . . . . . . . . . . 56C.2.4 Internal Fermion Bubble Diagram . . . . . . . . . . . 58C.2.5 Fermion Mass Beta Function . . . . . . . . . . . . . . 60viList of Figures2.1 Proposed phase diagram of the Majorana-Hubbard modelwith time reversal symmetry . . . . . . . . . . . . . . . . . . 54.1 Fermion self energy diagram in Wilson RG for g > 0 . . . . . 254.2 Boson self energy diagram in Wilson RG . . . . . . . . . . . . 264.3 First diagram renormalizing h3 and h4 in Wilson RG. . . . . 274.4 Second diagram renormalizing h3 and h4 in Wilson RG . . . 284.5 Fermion self energy in Wilson RG for g < 0 . . . . . . . . . . 315.1 Diagram generating φ4 + h.c. when the fermion mass is relevant 39C.1 Fermion self energy in renormalized perturbation theory . . . 51C.2 Boson self energy in renormalized perturbation theory . . . . 53C.3 Boson counterterm diagram in renormalized perturbation the-ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55C.4 Fermion counterterm diagram in renormalized perturbationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C.5 Two loop diagram with internal boson bubble in renormalizedperturbation theory . . . . . . . . . . . . . . . . . . . . . . . 57C.6 Two loop diagram with internal fermion bubble in renormal-ized perturbation theory . . . . . . . . . . . . . . . . . . . . . 58viiAcknowledgementsI would like to thank my supervisor, Ian Affleck, for his support and men-torship during this project. I would also like to thank Joseph Maciejko andIgor Klebanov for helpful comments.viiiDedicationTo my mother, for her encouragement, enthusiasm and unwavering optimismthat has made me never doubt what is possible.To my father, for his tireless hard work that has given me the opportunityto pursue an education, and has inspired me to produce my best work.To Gaby He´bert, for her loving support, and sense of adventure that hasleft me with so many great memories from these past two years.ixChapter 1IntroductionIn 1928, PAM Dirac proposed the existence of the positron: a particle withthe same mass and spin as the electron, but with opposite charge [1]. Thepositron (and electron) are examples of Dirac fermions: particles whichhave distinct antiparticles of the same mass, but with opposite physicalcharges. Dirac fermions can be contrasted with fermions that equal theirown antiparticle, known as Majorana fermions [2]. While the positron wasdetected shortly after Dirac’s prediction [3], a Majorana particle has neverbeen observed in particle physics.In condensed matter physics, a Majorana fermion can arise despite theabsence of fundamental Majorana particles, as an emergent phenomenon.Following the discovery of topological materials, it has been predicted thatMajorana excitations appear in various situations at topological defects andboundaries of topological insulators [4, 5]. In fact, this prediction has ledto an intense effort to develop a topological quantum computer that utilizesthe physics of Majorana fermions [2, 4].A setting in which a macroscopic number of interacting Majorana fermionsis predicted to occur is a layer of ordinary superconductor on a strong topo-logical insulator in a transverse magnetic field. The resulting vortex lattice ispredicted to have a Majorana mode localized at every vortex core [2]. Whilethere has been renewed interest in Majorana physics over the past decade,interaction effects have not been considered until recently. Much of thework on this subject has stemmed from the development of the Majorana-Hubbard model – a version of the Hubbard model involving Hermitian op-erators, and a four-site interaction term. [6–10]. This is the simplest in-teraction possible since a Hermitian Majorana operator obeying a canonicalanticommutation relation will square to unity.In this thesis, we study the critical behaviour of the Majorana-Hubbardmodel on the square lattice in two spatial dimensions. In Chapter 2, weintroduce the model, discuss its features, and review the mean-field predic-tions made in [10]. We then review and extend the low energy field theorydescribing the predicted gapless phases of this model, for both repulsive andattractive interactions. A unique feature of this model is that it possesses an1Chapter 1. Introductionemergent U(1) symmetry, unlike the exact U(1) that is assumed in relatedmodels [11–16]. One of the major tasks of this thesis is to the determinethe role that U(1) breaking operators may have on the phase diagram. InChapter 3, we introduce various renormalization group methods that will beused to study these operators. Then, in Chapter 4, we show that all U(1)breaking operators are irrelevant to one loop order, using an -expansionand Wilson’s approach to the renormalization group . Finally, in Chap-ter 5, we consider the effects of a time reversal breaking perturbation, afermion mass term, on the Majorana-Hubbard model. Using a combinationof renormalization group and supersymmetry methods, we show that sucha perturbation is relevant, and results in a critical point in the conventionalXY universality class. If time reversal is an approximate symmetry, the crit-ical point may exhibit signs of N = 2 supersymmetry, which has previouslybeen realized in [11–15]. Just like the Majorana particle, supersymmetryis a prediction from high energy theory that has not yet been verified ex-perimentally. Based on our results, we propose that the Majorana-Hubbardmodel on a square lattice is a candidate system for realizing the signaturesof supersymmetry in an experimentally realizable set-up.2Chapter 2The Majorana-HubbardModelIn the theory of topological materials, a Majorana mode is predicted to occurat the core of a vortex on the surface of a superconducting layer placed ona strong topological insulator [2]. A lattice of vortices will occur when thislayered material is placed in a transverse magnetic field, and it is believedthat the corresponding Majorana lattice will exhibit interactions that falloff exponentially with the superconducting coherence length [9]. We denoteby γj the Majorana fermion operator at lattice site j. Since these operatorsare Hermitian, the canonical anticommutation relation{γi, γj} = 2δij (2.1)implies that the shortest possible range interaction term must occur on 4sites (an odd number of sites would lead to a Hamiltonian that is not abosonic operator). The simplest model describing a lattice of interactingMajorana fermions is then the Majorana-Hubbard model – a Hamiltonianwith nearest neighbour hopping and shortest possible range 4-site interactionterm:H = it∑〈ij〉eiφijγiγj + g∑[ijkl]γiγjγkγl (2.2)Here 〈ij〉 denotes nearest neighbour lattice sites, and [ijkl] denotes sets of 4closest lattice sites. The phase factor eiφij is determined by the requirementthat one superconducting flux quantum passes through each lattice point,giving rise to one Majorana fermion at each site [9, 17]. This model was firstintroduced by Stern and Grosfeld in the context of the fractional quantumHall effect [17].In one dimension, the Majorana-Hubbard model was shown to have aninteresting phase diagram [6–8]. At large enough interaction strength g,the Majorana modes on the chain dimerize to form (non-Hermitian) Diracfermions, breaking translational symmetry. In the case of g > 0, this phasetransition was shown to be described by the tricritical Ising model, which32.1. The Majorana-Hubbard Model on the Square Latticeexhibits supersymmetry. As we will see below, this manifestation of super-symmetry is not unique to one dimension.More recently, the Majorana-Hubbard model has been studied in twodimensions on the square lattice [10, 18] and the honeycomb lattice [19]. Inthis thesis, we restrict our attention to the square lattice in two dimensions.2.1 The Majorana-Hubbard Model on theSquare LatticeOn the square lattice, the interaction term occurs on plaquettes:H = it∑m,nγm,n[(−1)nγm+1,n + γm,n+1] + g∑m,nγm,nγm+1,nγm+1,n+1γm,n+1(2.3)The phase factor eiφij in (2.2) has been fixed so that there is pi magnetic fluxthrough each plaquette, corresponding to one vortex at each site, accordingto [9, 17]. The sign of t can be changed by a Z2 gauge transformationγm,n → sm,nγm,n sm,n = ±1 (2.4)but cannot be completely removed [10]. Without loss of generality, we as-sume t > 0. When g > 0, the underlying physical interactions are attractive,as can be shown using mean field theory, or by mapping the theory to itscontinuum limit (see below).This model was studied in detail in [10], where it was shown to havea rich phase diagram as a function of gt−1 (Figure 2.1). The large g limitwas also studied recently in [18]. As in the one dimensional Majorana-Hubbard model, at strong enough coupling, the Majorana modes like topair up on neighbouring sites to form Dirac fermions, breaking translationsymmetry in either the horizontal or vertical direction. For g > 0, the Diracfermions’ energy levels are empty, while they alternate being empty andoccupied for g < 0. We call these dimerized phases ‘ferromagnetic’ (FM)and ‘antiferromagnetic’ (AFM), respectively. Furthermore, two second orderphase transitions were predicted to occur at g = gc,1 ≈ −0.9t and g = gc,2 ≈+0.9t. The dotted line in Figure 2.1 is a first order phase transition thatdoes not have an interpretation in terms of Majorana pairings.By deriving the low energy continuum limit of (2.3), emergent Lorentzand U(1) symmetries were found to occur at these transitions. In terms ofa 2-component complex fermion ψ, the imaginary time Lagrangian densitywas found to beLU(1) = ψ¯γµ∂µψ + 64gΛ−20 (ψ¯ψ)2. (2.5)42.1. The Majorana-Hubbard Model on the Square Lattice0 gSUSYGross-Neveugaplessbroken T-reversal‘AFM’ MM pairing ‘FM’ MM pairingFigure 2.1: Proposed phase diagram of the Majorana-Hubbard model withtime reversal symmetryHere ψ¯ := ψ†γ0, and the Dirac gamma matrices, γµ, satisfyγµ := {σy, σx,−σz} {γµ, γν}ab = 2δab. (2.6)The coefficient Λ0 = a−1 is a bare cutoff defined by the inverse latticespacing, a, and the imaginary time coordinate has been rescaled so that thevelocity v = 4ta ≡ 1. Using (2.5), it was argued that the phase transition atgc,1 is in the universality class of the Gross-Neveu model, while the transitionat gc,2 corresponds to the N = 2 supersymmetric (SUSY) universality class.When a fermion mass term ψ¯ψ is present, it was further argued thatsupersymmetry is broken, and the transition at gc,2 falls into the XY uni-versality class (the gc,1 transition is not present in the massive case). Such aterm can be generated by adding a second-neighbour hopping to the Hamil-tonian:H → H + it2∑m,n∑s,s′=±1γm,2nγm+s,2n+s′ (2.7)This term breaks time reversal symmetry (see (2.28)) and should be includedin any model hoping to describe a vortex lattice in a transverse magneticfield.In this thesis, we use renormalization group methods to check the uni-versality class predictions for gc,1 and gc,2 made in [10]. In particular, wewill use an -expansion to determine the relevance of leading U(1) break-ing operators and the fermion mass term. In the next section, we beginby re-deriving the low energy field theory and calculating the leading U(1)breaking corrections. We then introduce a boson (real or complex, depend-ing on the sign of g), using a Hubbard-Stratonovich transformation. Thesefermion-boson models will be the starting point of our renormalization groupanalysis in later chapters.52.2. Low Energy Field Theory2.2 Low Energy Field TheoryDue to the alternating nature of the nearest neighbour hopping in (2.3), theunit cell spans two lattice sites, so we defineγm,2n = γem,2n γm,2m+1 = γom,2n+1. (2.8)These definitions of γe/o are slightly different than those of [10], and arechosen to simplify the form of the U(1) breaking operators. To derive alow energy field theory, we start with the dispersion relation of the non-interacting model (g = 0):E± = ±4t√sin2 kx + sin2 ky. (2.9)We then replace each Majorana operator γe/o with a combination of twoslowly varying Majorana fields χe/o,±, according toγe/o(~r) ≈ 2√2Λ−10 [χe/o+(~r) + (−1)xχe/o−(~r)]. (2.10)These fields χ± consist of the momenta modes of γ near the two Dirac pointsof the non-interacting theory, which occur at ~k = (0, 0) and ~k = (pi/a, 0).The coefficient Λ−10 = a is the lattice spacing, and its inverse defines abare energy cutoff of the theory. To derive the continuum limit, we Taylorexpand the quadratic and quartic pieces of (2.3) in Appendix A. We expandthe quartic operator to two derivatives, while keeping only leading orderquadratic terms, since the underlying symmetry of the lattice model forbidsany quadratic operator from breaking the U(1) symmetry (as proven below).The resulting Hamiltonian density isH = 4ita∑±[± χe±∂xχe± ∓ χo±∂xχo± + 2χe±∂yχo±]+Hint (2.11)where164gΛ−40Hint = (2.12)−4Λ20χe−χe+χo−χo+−∑s,s′=±ss′χes∂xχesχos′∂xχos′+2∂y(χe−χe+)∂y(χo−χo+)+2χe−χe+∂xχo−∂xχo+ + 2∂xχe−∂xχe+χo−χo+ + ∂x(χe−χe+)∂x(χo−χo+).We introduce two-component Majorana fermions χ+ := (χe+, χo+)T andχ− := (χo−, χe−)T, so that the first term of (2.11) becomesH0 := H−Hint = 4ita∑±χ±T [σz∂x + σx∂y]χ±. (2.13)62.2. Low Energy Field TheoryThese two-component Majorana fermions satisfy the canonical anti-commutationrelations {χi(~r), χj(~r′)} = δijδ(~r− ~r′). Since Hint is not a function of ∂tχ±,we have Lint = Hint in imaginary time, and the Lagrangian density corre-sponding to (2.11) isL =∑±χ¯±γµ∂µχ± +Hint. (2.14)We’ve set the velocity v = 4ta to unity, used the Euclidean gamma matricesdefined in (2.6), and defined χ¯± := χ±Tγ0. In order to identify any emergentU(1) invariance of (2.14), we define a complex fermion ψ according toψ = χ+ + iχ− =(χe+ + iχo−χo+ + iχe−). (2.15)In this language, the most relevant U(1) breaking operator in (2.14) is16gΛ−40(ψ1ψ2[∂xψ1∂xψ2 − ∂yψ1∂yψ2] + h.c.)(2.16)as shown in Appendix A. Including this term, the low energy field theorydescribing (2.3) isL = ψ¯γµ∂µψ+Mψ¯ψ+64gΛ−20 (ψ¯ψ)2+16gΛ−40(ψ1ψ2∂rψ1∂rψ2+h.c.)(2.17)where we’ve introduced the notation∂rψa∂rψb := ∂xψa∂xψb − ∂yψa∂yψb. (2.18)and we’ve also introduced a fermion mass term: As shown in [10], when thesecond-neighbour hopping term is included (see (2.7)),L → L+Mψ¯ψ M := 8t2 (2.19)Since(ψ¯ψ)2 = −ψ∗1ψ∗2ψ2ψ1 (2.20)we see from (2.17) that g > 0 corresponds to underlying physical interactionsthat are attractive. As a last comment, we note that the Nielson Ninomiyatheorem [20] is not violated here, even though we have achieved a singleDirac fermion on the lattice, since the U(1) symmetry is only emergent, andnot exact.72.2. Low Energy Field Theory2.2.1 Hubbard-Stratonovich TransformationIn the absence of the U(1) breaking operator, the interaction term in (2.17)is proportional to (ψ¯ψ)2. In this case, we expect a massless boson to appearat the phase transitions gc,1, gc,2, whose expectation value provides the orderparameter of the transition [10, 16]. Such a boson can be introduced usinga Hubbard-Stratonovich transformation. This procedure depends on thesign of the (ψ¯ψ)2 interaction: in the case of attractive interactions (g >0), a complex charge-2 boson is introduced, while in the case of repulsiveinteractions (g < 0), a real boson is introduced. To promote these bosonicvariables to dynamical fields, we reduce the energy scale of the continuumtheory from Λ0 down to some reduced scale Λ  Λ0. Using the samesymbols to denote these renormalized fields, we arrive at the following twoimaginary time Lagrangian densities, depending on the sign of g:• Repulsive Interactions (g < 0):L1 = ψ¯γµ∂µψ+(∂µσ)2+r2σ2+η1σψ¯ψ+η22σ4+h1 [ψ1ψ2∂rψ1∂rψ2 + h.c.](2.21)• Attractive Interactions (g > 0):L2 = ψ¯γµ∂µψ+Mψ¯ψ|∂µφ|2+m2|φ|2+λ1[φ∗ψTCψ + h.c.]+λ22|φ|4+L′2(2.22)where C = iγ0 andL′2 := h2ψ1ψ2∂rψ1∂rψ2+h3φ∂rψ1∂rψ2+h4φ[∂2rψ1ψ2+ψ1∂2rψ2]+h.c. (2.23)We have only included a fermion mass in the case of attractive interactions;the phase transition for g < 0 vanishes as soon as time reversal symmetryis broken, according to mean field theory [10]. Note that in the case of at-tractive interactions, two additional U(1) breaking operators are generatedduring this renormalization procedure. Such terms do not occur for a realboson σ, since they violate an underlying pi2 -rotation symmetry of the lattice,as explained in Section 2.3. The Greek coupling constants {λi, ηi} precedeU(1) preserving operators, while the Latin coupling constants {hi} precedeU(1) breaking operators. Equations (2.21) and (2.22) will be the startingpoint for all of our calculations that follow. We will assume that the symme-try breaking parameters {hi} and M are small, so that the theories are closeto their quantum critical points. This is not an unreasonable assumption forthe lattice model: the U(1) breaking operators are superficially irrelevant,82.3. Symmetry Constraints on U(1) Breaking Operatorsand are preceded by a factor of Λ−40 . At a reduced cutoff Λ Λ0, the cou-pling constants will be suppressed by four factors of Λ/Λ0. Of course, thisargument is incomplete, as it ignores higher order renormalization effects.If the {hi} and M are not small, their flow will depend on the the presenceof additional fixed points in parameter space.We have assumed that under this renormalization, the velocities of theboson and fermion flow to a common value. This has been shown to be thecase in the U(1) invariant version of these models, and to linear order inM and {hi}, we expect the same result to hold [12, 13]. The irrelevanceof Lorentz breaking operators has also been established for fermion-bosonmodels on the honeycomb lattice.[21–23] The fermion and boson velocitieswould be identical if Lorentz invariance was exact.In [10], the nature of the transitions at gc,1 and gc,2 was predicted usingthe U(1) symmetric versions of (2.21) and (2.22), and invoking universality.In the fermion-boson models, the transitions are driven by reducing thesquared boson mass, and letting it change sign. The U(1) symmetric versionof (2.21) was considered in [16], and the transition was shown to correspondto that of the Gross-Neveu model, with spontaneous breaking of the Z2symmetryσ → −σ ψ¯ψ → −ψ¯ψ (2.24)It is not the Ising transition, because an additional massless fermion fieldψ is present. The U(1) version of (2.22) involving the charge-2 boson φ,(2.22), has been studied as well [11–13, 15, 16], and the transition is knownto exhibit N = 2 supersymmetry when M = 0. This should not be confusedwith the N = 1 supersymmetry that is present in [24].2.3 Symmetry Constraints on U(1) BreakingOperatorsTo complete this chapter, we comment on the symmetries of (2.17). Theauthors of [10] identified various exact symmetries of the lattice Hamiltonian(2.3), which must be obeyed at the continuum level. We label them C forcharge conjugation, P for parity, and R for pi2 -spatial rotation. Explicitly,they are:C : ψ(x, y) 7→ ψ∗(x, y) (2.25)P : ψ(x, y) 7→ −iγ1ψ∗(−x, y) (2.26)R : ψ(x, y) 7→ e− ipi4 e ipi4 γ0ψ(−y, x) (2.27)92.3. Symmetry Constraints on U(1) Breaking OperatorsAdditionally, in the special case of t2 = M = 0, the model is also invariantunder time reversal, T :T : ψ(x, y) 7→ −γ0ψ∗(x, y), i 7→ −i (2.28)In the following, we demonstrate how C and R are sufficient to en-sure that all quadratic operators in the continuum theory preserve the U(1)symmetry, as is in the case in (2.11). We then demonstrate how the U(1)breaking term in (2.16) is the only possible quartic operator, with two or lessderivatives, that satisfies the symmetries C,P and R. These results applyeven when the fermion mass is nonzero, since neither argument requires theuse of T . Finally, we explain how R limits the U(1) breaking fermion-bosoninteractions to the ones present in (2.21) and (2.22).2.3.1 Quadratic OperatorsThe most general U(1) breaking quadratic operator (with or without deriva-tives) is of the formψTAψ + ψ†A†ψ∗ (2.29)for some differential operator A(x, y). Under C,C : ψTAψ + ψ†A†ψ∗ 7→ ψTA†ψ + ψ†Aψ∗ (2.30)which forces A to be Hermitian. Under R,R : ψTA(x, y)ψ 7→ − i2ψT (1− iσy)A(−y, x)(1 + iσy)ψ. (2.31)The right hand side of (2.31) cannot appear for nonzero A, since it is anti-Hermitian, and violates (2.30). Therefore, no charge 2 operator is allowedby symmetry.2.3.2 Quartic OperatorsOne-Derivative Quartic OperatorsA four-Fermi operator involving a single derivative can only have charge0 or ±2: terms with charge ±4 include at least three fermi fields with-out derivatives, and vanish by Fermi statistics. Since R is a combinationof spatial rotation by pi2 and U(1) rotation by −pi4 , these two possibilitiesrequire, respectively, a derivative operator that transforms trivially or onethat transforms with a prefactor of i. Of these, only the latter exists:∂x + i∂y (2.32)but such an operator breaks CP .102.3. Symmetry Constraints on U(1) Breaking OperatorsTwo-Derivative Quartic OperatorsRepeating the previous argument, the derivative operator of a charge 2 four-fermi term must transform with a factor of i to satisfy R symmetry. This isnot possible for a generic two-derivative operator Aab∂a∂b, ruling out charge2 operators. Charge 4 terms require a derivative operator that transformswith a prefactor of −1 to be invariant under R. By Fermi statistics, the twoderivatives must act on separate Fermi fields, so the most general operatorsareψ1ψ2[∂xψ1∂xψ2 − ∂yψ1∂yψ2] (2.33)andψ1ψ2[∂xψ1∂yψ2 − ∂yψ1∂xψ2] (2.34)Of these, only the former is allowed, since the latter breaks CP . Therefore,the U(1) breaking operator appearing in (2.14) is the only possible termwith two or less derivatives.2.3.3 Fermion-Boson OperatorsIn the case of attractive interactions, a complex boson φ ∼ ψ1ψ2 is intro-duced. Using (2.27), we see thatR : φ(x, y)→ iφ(−y, x) (2.35)Since ψTCψ also picks up a factor of i under R, the following two derivative,U(1) breaking operators are invariant under R-symmetry:φ[∂xψ1∂xψ2 − ∂yψ1∂yψ2] + h.c. (2.36)andφ[(∂2x − ∂2y)ψ1ψ2 + ψ1(∂2x − ∂2y)ψ2]+ h.c. (2.37)It is easy to check that the remaining symmetries (2.25 - 2.27) also leavethese operators invariant.In the case of repulsive interactions, a real boson σ ∼ ψ¯ψ is introduced,which is invariant under R:R : σ(x, y)→ σ(−y, x) (2.38)Using the above constraints on pure fermion operators, the most relevantU(1)breaking femrion-boson operator is thenσ2ψ1ψ2[∂xψ1∂xψ2 − ∂yψ1∂yψ2] (2.39)which is too irrelevant for our considerations.11Chapter 3Renormalization GroupMethodsIn the previous chapter, we derived the low energy field theories that char-acterize the Majorana-Hubbard model near its quantum critical points gc,1and gc,2 (equations (2.21) and (2.22)). Both of these theories contain oper-ators that a priori do not let us easily determine their universality classes.The main result of this thesis will be to use the renormalization group tocharacterize these problematic operators, and determine the role they playnear these two critical points. In this chapter, we review various methodsfrom renormalization group theory that we will apply throughout the fol-lowing chapters. This material is explained very clearly in the texts [25, 26],and in the paper [27].3.1 Beta FunctionsThe fundamental idea behind the renormalization group is to quantify howthe coupling constants of a theory depend on the choice of length scale. Ifa certain coupling constant increases as we increase the scale, we say thecorresponding operator is relevant. If a coupling constant decreases at largerlength scales, we say the corresponding operator is irrelevant. Finally, anoperator whose coupling constant does not evolve is called marginal. Incondensed matter theory, where there is often a great difference of scalesbetween the ‘bare’ scale of the lattice constant and the observable scale ofmacroscopic phenomena, an irrelevant operator can safely be excluded fromthe effective Lagrangian. Thus, our task will be to show whether or not theU(1) breaking and T -breaking operators of (2.21) and (2.22) are irrelevantor not. If so, the classification of the related U(1) and T symmetric modelsof [10] may be applied here.The equations describing the evolution of coupling constants as a func-tion of length scale are known as beta functions. Given a family of lengthscales b−1Λ, parametrized by b = eδl, the beta function of coupling constant123.2. Wilson’s Approach to RenormalizationX is defined to beβX :=dXd log b=dXdδl(3.1)Relevant operators have positive beta functions, while irrelevant operatorshave negative ones. In some applications, such as those introduced in Section3.3, it is more natural to quantify how coupling constants depend on a energyscale parameter µ, instead of a length scale. In this case,βX = − dXd logµ(3.2)Beta functions can also be used to identify critical points. At a criticalpoint, the system exhibits scale invariance, and the coupling constants donot evolve. In other words, critical point correspond to ‘fixed-points’ inparameter space, which are precisely the zeros of the beta functions.We will calculate these beta functions using two different approaches tothe renormalization group: 1) the Wilson, or ‘momentum shell’, approach,and 2) dimensional regularization.3.2 Wilson’s Approach to RenormalizationMuch of our modern understanding of the renormalization group is thanksto Ken Wilson [28]. In his approach, one calculates beta functions by slightlyreducing the energy scale of the theory by integrating out fields whose mo-mentum modes lie in a thin shell in momentum space. This shell consistsof all momenta with magnitude lying within {b−1Λ,Λ}, where Λ is the UVcutoff of the original theory.We will explain how this integration is carried out for the case of asingle scalar field ϕ. This will introduce the notation that we will use in thefollowing chapters for the more complicated field theories (2.21) and (2.22).The Lagrangian density is taken to beLϕ = 12(∂ϕ)2 +∑iXiOi[ϕ] (3.3)where Oi[ϕ] is a generic operator involving the field ϕ and its derivatives.We begin by separating the field into a slow and fast componentϕ = ϕs + ϕf , (3.4)where ϕs contains the momentum modes of ϕ with magnitude less thanb−1Λ, and ϕf contains the modes that lie within the shell. In terms of these133.2. Wilson’s Approach to Renormalizationnew variables, the Lagrangian density can be reorganized as follows:Lϕ = Ls + L0f + Lsf . (3.5)The first term, Ls, equals the original Lagrangian density, but with ϕ re-placed with ϕs. The second term, L0f , equals the free fast Lagrangian density,12(∂ϕf )2. The remaining term contains all operators that mix slow and fastcomponents. The partition function can then be rewritten asZ =∫Dϕsϕfe−∫ddx(Ls+L0f+Lsf ) (3.6)= Z0,f∫Dϕse−∫ddx(Ls+δL)) (3.7)whereZ0,f :=∫Dϕfe−∫ddxLf 〈· · · 〉f := Z−10,f∫Dϕf · · · e−∫ddxLf (3.8)ande−∫ddxδL := 〈e−∫ddxLsf 〉f (3.9)In other words, integrating out the fast modes has generated new terms inthe Lagrangian. Since the operators appearing in (3.3) were generic, we canwriteδL = 12δZϕ(∂ϕs)2 +∑iδZiOi[ϕs] (3.10)in terms of renormalization constants δZϕ and δZi. To compare Ls + δL tothe original theory, we rescale coordinatesx→ b−1x (3.11)so that the new UV cutoff is once again Λ, and rescale the fieldϕs → ϕs√(1 + δZϕ)bd−2 (3.12)so that the new kinetic term is once again 12(∂ϕ)2. Defining di and ni to bethe mass dimension and number of factors of ϕ, respectively, of Oi, we findthat the coupling constants of the reduced theory, {Xi(b)}, satisfyXi(b) = Xi(1 +δZiXi)(1 + δZϕ)−nibdi (3.13)143.3. Dimensional RegularizationThis expression can now be differentiated with respect to log b, yielding thebeta functions of the theory, {βXi}.While very physical, the Wilsonian approach to the renormalizationgroup leads to complications beyond first order in perturbation theory, whennested momentum shell integrations are required. We now introduce a sec-ond approach to the renormalization group that is more suited for higherorder calculations [25, 27].3.3 Dimensional RegularizationInstead of explicitly changing the length scale of the theory, beta functionscan also be calculated using a properly regularized theory at a fixed energyscale. Consider the following example Lagrangian density of a real scalarfield:Lϕ4 =12(∂ϕ)2 +m2ϕ2 +Xϕ4 (3.14)This theory is not regularized: a perturbative expansion of its correlationfunctions will lead to divergences, order by order. To resolve this, we usea procedure known as dimensional regularization, in which we continue thespacetime dimension away from an integer, rendering momentum loop inte-grals finite. We then subtract off these contributions by introducing coun-terterms into the Lagrangian density, before continuing back to an integerdimension. This is performed at a given energy scale µ, which we fix.For example, at one loop, the ϕ self energy receives a contribution pro-portional to ∫ddp(2pi)d1(p2 +m2)∝ Γ(1− d2)(3.15)where Γ(x) is the gamma function with poles at non-positive integers. Fornon-integer d, this expression is finite, and can be cancelled by introducinga countertermδZϕ12(∂ϕ)2 (3.16)into the Lagrangian, withδZϕ ∝ −Γ(1− d2)(3.17)Repeating these steps for all divergences at a given order, we arrive at arenormalized Lagrangian density at scale µ,Lϕ4,r =12Zϕ(∂ϕr)2 + Zmµ2m2rϕ2r + ZXµ4−dXrϕ4r , (3.18)153.4. The Epsilon Expansionin terms of renormalized field ϕr and renormalized coupling constants mrand Xr. The renormalization constants Zi contain the introduced countert-erms δZi according toZi = 1 + δZi (3.19)The explicit energy scale µ enters to make the renormalized coupling con-stants dimensionless. Now, to extract the beta functions, we must relate thetwo Lagrangian densities (3.14) and (3.18). Matching kinetic terms, we findϕ =√Zϕϕr (3.20)and then rescaling, we findmr = mµ−1ZϕZ−1m (3.21)Xr = Xµ4−dZ2ϕZ−1X (3.22)These equations are the dimensional regularized analogues of (3.13). Differ-entiating with respect to − logµ generates the desired beta functions.3.3.1 Modified Minimal Subtraction SchemeExactly how the counterterms in (3.19) are defined leads to further choice inrenormalization scheme. If only the divergent parts of the loop diagram areincluded in the counterterm, the scheme is known as ‘minimal subtraction’.In our calculations, we use the more common ‘modified minimal subtrac-tion’ scheme, or MS, which adds to the counterterm the universal constantlog (eγE/4pi) that always occurs in Feynman diagrams. This is implementedby rescaling the energy scale µ→ µ eγE4pi in (3.18) [25, 27].3.4 The Epsilon ExpansionIn both the Wilsonian and dimensional regularization pictures of the renor-malization group, we are tasked with calculating beta functions and de-termining their fixed points. In practice, these fixed points are often notaccessible in two space dimensions and one time dimension; instead, onemust consider the theory close to its upper critical dimension (UCD), andexpand about this point. This procedure is known as the -expansion, andwas first introduced to study the theory Lϕ4 (defined in (3.14)) [25]. In dspacetime dimensions, the scaling dimension of the interaction term ϕ4 is4× d− 22= 2d− 4163.4. The Epsilon Expansionand its upper critical dimension is 4. In other words, the methods of meanfield theory are expected to break down for the physically interesting casesof d = 1, 2, 3. To resolve this problem, theorists promoted the parameter dto a continuous variable, and considered the model in d = 4−  dimensionsfor  1, and expanded in powers of . This was said to correspond to thetheory ‘close to 4 dimensions’. Formally, this procedure of promoting d to acontinuous variable is done at the level of Feynman diagrams, carrying outmomentum integrals using d-dimensional spherical coordinates. The idea isthat for  small, the interaction is only ‘slightly’ relevant, and mean fieldtheory might not be so bad.In this expansion, to O(), the beta function of X in (3.14) can be shownto be [26]βX = X − 92pi2X2 (3.23)revealing a nontrivial fixed point X∗ = 2pi29 . This showcases the true powerof the -expansion: the ability to search for new phase transitions that areinaccessible using mean field theory.While we might expect the -expansion to be valid in the limit of in-finitesimal , it has shown to be surprisingly predictive in the limit  → 1.For example, the theory Lϕ4 , which corresponds to the classical Ising model,predicts the specific heat to scale with critical exponent α = 1/6 at O() andα = .109 at O(5) [26]. Even at O(), the agreement with the experimentalrange of 0–0.14 is impressive. It is results like these that have resulted inphysicists regarding the -expansion as a very important tool in the studyof critical phenomena.For the U(1) symmetric versions of (2.21) and (2.22), both the Gross-Neveu and N = 2 SUSY critical points were identified using the -expansion(the upper critical dimension of both σψ¯ψ and φ∗ψTCψ+ h.c. is also four).These fixed points occur at the following values:[16, 27]λ21,∗(4pi)2=12+O(2) (3.24)η1,∗(4pi)2=8+O(2) (3.25)To linear order in the fermion mass and the U(1) breaking couplings {hi},the value of these fixed points will not change, and so it makes sense to askthe following question:What are the beta functions of M and {hi}, evaluated at the criticalpoints λ1,∗ and η1,∗?173.5. Modified Epsilon ExpansionWe will provide an answer to this question in Chapters 4 and 5. However,in addressing this question, an issue arises involving Lorentz invariance. Inthe original -expansion of ϕ4 theory, ϕ is a Lorentz scalar, transforming asa singlet under the SO(d) Lorentz group in every dimension d. For a theoryinvolving fermions, more care is needed, since not all operators invariantunder SO(3) are invariant under the larger SO(4) Lorentz group in fourdimensions. In the following section, we introduce the necessary formalismto treat these issues.3.5 Modified Epsilon ExpansionSince the upper critical dimension of the fermion-boson interactions σψ¯ψand φ∗ψTCψ + h.c. is four, we will carry out an -expansion about fourdimensions. In four dimensions, a two-component complex fermion is a Weylfermion. To derive the Weyl Lagrangian, we start from the four dimensionalDirac theory in real time,LD = iΨ¯Γa∂aΨ. (3.26)The gamma matrices are in the Weyl basis, and can be written in terms oftwo sets of Pauli matrices {σi} and {τi}:Γ0 = τx ⊗ σ0 Γk = iτy ⊗ σk (3.27)where σ0 := 1. These matrices satisfy{Γa,Γb} = 2diag(1,−1,−1,−1). (3.28)Writing Ψ asΨ = (ψR ψL)T (3.29)and expanding (3.26), the fields ψL and ψR decouple. Defining ψ¯ = ψ†σy,the ψL sector can be written asLW = iψ¯σy∂0ψ+ iψ¯[σyσx∂1 +∂2 +σyσz∂3]ψ ψ¯ := ψ†γ0 = ψ†σy (3.30)where we’ve suppressed the ‘L’ subscript, and inserted σ2y = 1 in each term.By relabelling coordinates ∂2 ↔ ∂3, and performing a Wick rotation, wefind that the imaginary time Lagrangian density for the Weyl fermion isLW = ψ¯[∂µγµ + i∂3]ψ µ = 0, 1, 2 (3.31)183.5. Modified Epsilon ExpansionSince (3.31) is a Lorentz scalar, and (∂µ, ∂3) is a 4-vector, we see that theobject ψ¯ψ is no longer invariant under the Lorentz group. Instead, it is acomponent of the 4-vector,A =(ψ¯γµψψ¯ψ), (3.32)that is contracted with (∂µ, ∂3) in (3.31). This can also be seen explicitly,using the general form of a Lorentz transformation in the Weyl basis:[25]Λ(α) = e~α·~σ ~α ∈ C (3.33)Under this transformation,ψ¯ψ 7→ ψ†e~α∗·~σγ0e~α·~σψ = ψ¯e−~α∗·~σT e~α·~σψ (3.34)which does not equal ψ¯ψ for general ~α. It is only invariant under asubset of operators,{eλσx , eλσz , eiλσy}, λ ∈ Rwhich generate the three dimensional Lorentz group.The breaking of Lorentz invariance creates difficulties when studying thefermion mass operator Mψ¯ψ, as well as the Gross-Neveu interaction ψ¯ψσin (2.21). While these operators are invariant under the three dimensionalEuclidean Lorentz group SO(3), they transform nontrivially under the fullSO(4) Euclidean Lorentz group. As a consequence, additional operatorsthat are invariant only under the SO(3) ⊂ SO(4) subgroup can be generated,including (for k ∈ Z+)ψ¯(i∂3)kψ |∂k3φ|2 (∂k3σ)2 (φ∂k3φ∗ + h.c.) σ∂k3σ (3.35)We will only discuss the role of the most relevant operators, with k = 1.Then in four dimensions, we should replace the Lagrangian densities (2.21)and (2.22) with the following:L′1 = ψ¯[/∂+ i∂3 + if1∂3]ψ+Mψ¯ψ+(∂aσ)2 +f2(∂3σ)2 +η1σψ¯ψ+η2σ4 (3.36)+f3σ∂3σ + · · ·L′2 = ψ¯[/∂+i∂3+if1∂3]ψ+Mψ¯ψ+|∂aφ|2+f2|∂3φ|2+λ1[φψTCψ+h.c.] (3.37)+λ22|φ|2 + f3(φ∂3φ∗ + h.c.) + · · ·193.5. Modified Epsilon ExpansionThe ‘· · · ’ represent the U(1) breaking operators present in (2.21) and(2.22), which are unchanged. Since the parameters {fi} are not presentin the three dimensional model, they only appear in the four dimensionalmodel after at least one renormalization step, and are suppressed by atleast one factor of M or η1 (the Lorentz breaking operators in (3.36) and(3.37)). In either case, terms O(f2i ) and O(fiM) are beyond our order ofapproximation, and should be dropped from the calculations that follow.3.5.1 PropagatorsInverting the quadratic forms in (3.36 3.37), we find the following propaga-tors, to linear order in M and fi:G(p) = 〈ψ(p)ψ¯(p)〉 = ip + f1p3 +Mp2− 2p3(M + f1p3) ip + p3p4(3.38)where we’ve introduced a four dimensional ‘slash notation’A := Aµγµ − iA3 (3.39)We write this propagator as a sum of a Lorentz invariant (G1) and a non-Lorentz invariant (G2) part:G(p) = G1(p) +G2(p) (3.40)G1(p) =ip +Mp2G2(p) =p3p2[f1 − 2(M + f1p3) ip + p3p2](3.41)Only the first term is a Lorentz invariant. Likewise, the boson propagatorsareD(p) = 〈σ(p)σ(−p)〉 = 〈φ(p)φ∗(p)〉 = D1 +D2 (3.42)whereD1(p) =1p2D2(p) = −f2p23p4− f3ip3p4(3.43)The presence of these non-Lorentz invariant terms in the fermion andboson propagators may seem problematic. This is because when we thinkof an -expansion, we usually continuously change the dimension withoutaltering the underlying symmetries of the theory. However, this is not feasi-ble when working with fields that transform nontrivially under the Lorentzgroup, such as spinors: the terms are still invariant under SO(3), but this isno longer the Lorentz group in four dimensions.203.5. Modified Epsilon ExpansionOf course, this is not the first time an -expansion has been attemptedon these models. In the case of attractive interactions, the conventional ap-proach is to relate (2.22) to the Nambu-Jona-Lasinio model in four dimen-sions, involving a 4-component Majorana fermion χ, and two real bosons φ1and φ2 [15, 16, 29]. The interaction term in this model isχ¯(φ1 + iγ5φ2)χ (3.44)where γ5 is the fifth gamma matrix in four dimensions. In the massless case,this theory possesses a continuous U(1) chiral symmetry:χ→ eiαγ5χ φ→ e−2iαφ (3.45)In the Majorana representation, γ5 is pure imaginary, so that this trans-formation leaves the Majorana real. In three dimensions, this model corre-sponds to the U(1) version of (2.22), with the chiral U(1) mapping to thecharge U(1) symmetry in the three dimensional theory. However, since a Ma-jorana mass breaks the chiral U(1), we are unable to adopt this approach toour model when a fermion mass term is present.Another popular approach in the literature, for the U(1) versions ofboth (2.21) and (2.22), is to extend the theory to one of N Dirac fermionsin four dimensions, and then continue N → 12 in the -expansion [16]. Thisapproach is difficult to justify, since a four dimensional Dirac mass doesnot correspond to a three dimensional Dirac mass in this limit. See forinstance, [30]. Instead, the four dimensional mass couples different chiralsectors together. Using a change of basis, we can decouple the sectors, butin this case the three dimensional masses occur with opposite signs, and thelimit N → 12 is ill-defined. This is explained in more detail in Appendix B.Therefore, we are forced to develop a new approach in order to calculaterenormalization group functions in these theories. In the end, this approachwill agree with the naive N → 12 limit in a conventional -expansion, butis arguably more reliable, since it keeps the form of all operators fixed as dis continued back to three dimensions. Perhaps there is a simple argumentjustifying the N → 12 limit, but we haven’t been able to produce one.3.5.2 An Expansion in d = 3 + (1− ) DimensionsIn this thesis, we use a different approach to extract only the Lorentz in-variant contributions to our Feynman diagram calculations. It is a modified-expansion that isolates the Lorentz breaking direction (‘p3’ in momentumspace), and shrinks it to zero extent in the → 1 limit. To understand this,213.5. Modified Epsilon Expansionrecall that the conventional -expansion is carried out at the level of internalmomentum integrals. In a Lorentz invariant theory, all momentum integralswill have the structure ∫d4p(2pi)4F (p) (3.46)for some function F depending only on the magnitude of momentum. Now,we continue from four to d dimensions, by writing∫d4p(2pi)4F (p)→∫ddp(2pi)dF (p) = Ωd∫dppd−1F (p) (3.47)where Ωd is the surface area of the sphere Sd−1. Both Ωd and the radialintegral are well-defined as functions of a continuous parameter d.Now, let us turn to our non-Lorentz invariant theory, which has propaga-tors modified by terms proportional to p3p2. To lowest order in p23, any Lorentzbreaking contribution to a momentum integral will have the structure∫d4p(2pi)4F (p)p23 (3.48)since odd powers of p3 vanish by the symmetric integration. Higher powersof p23 will be at least quadratic in the small parameters M and fi. In the con-ventional -expansion, we would now promote p to a d-dimensional vector,write p23 = p2F ′(θi) in terms of spherical coordinates, and find some nonzerocontribution. But this is unphysical, since all Lorentz breaking contributionsshould vanish when we return to the three dimensional theory. Instead, wepromote p to a 3 + d′ dimensional vector, and p3 to a d′ dimensional vector,with d′ = 1− :∫d4p(2pi)4f(p)p23 →∫d3+d′p(2pi)3+d′F (p)|p3|2 =∫d3+d′p(2pi)3+d′F (p)2+d′∑i=3p2i (3.49)= d′∫d3+d′p(2pi)3+d′F (p)p21 =d′3 + d′Ω3+d′∫dpp3+d′+1F (p) (3.50)In the limit  → 1, d′ → 0, this integral vanishes. Since this applies toall Lorentz breaking contributions to the Feynman diagrams, the modified-expansion amounts to replacing the propagators in (3.40, 3.42) with theirLorentz invariant pieces, and carrying out the conventional -expansion:G(p)→ G1(p) D(p)→ D1(p) (3.51)Throughout the following chapters, we implement this modified scheme,and drop the subscript ‘1’ in the fermion and boson propagators.22Chapter 4Renormalization of U(1)Breaking OperatorsIn this chapter, we determine the relevance of the U(1) breaking operatorspresent in (2.21) and (2.22) to one loop order in the modified -expansion.This is done by calculating the beta functions of these operators in theWilsonian renormalization scheme. To begin, we decompose fields into slowand fast components, following the conventions of Chapter 3. We separatethe Lagrangian density into a slow, fast, and ‘mixed’ piece, according to(3.5), and keep mixed terms that have exactly two fast fields; operatorswith more or less fast fields do not contribute at one loop order. For thecase of repulsive interactions in the Majorana model (2.21),Lsf,1 = η1σsψ¯fψf + η1σf (ψ¯sψf + ψ¯fψs) + 6η22σ2sσ2f + 2σ2sσ2f (4.1)+h14[CabCcdψa,sψb,s∂rψc,f∂rψd,f + ψa,fψb,f∂rψc,s∂rψd,s + h.c.]+h1 [CabCcdψa,fψb,s∂rψc,f∂rψd,s + h.c.]For the case of attractive interactions in the Majorana model (2.22),Lsf,2 = λ2[φ2sφ∗f2 + 2|φs|2|φf |2]+ λ1Cab[2φ∗fψa,fψb,s + φ∗sψa,fψb,f ] (4.2)+h24CabCcd [ψa,sψb,s∂rψc,f∂rψd,f + ψa,fψb,f∂rψc,s∂rψd,s + 4ψa,fψb,s∂rψc,f∂rψd,s]+h32Cab [φs∂rψa,f∂rψb,f + 2φf∂rψa,f∂rψb,s]+h4Cab[φs∂2rψa,fψb,f + 2φf∂2rψa,fψb,s]plus Hermitian conjugate terms. Integrating out the fast fields in the abovetheories will generate a series of Feynman diagrams, which we calculatebelow. These diagrams will contribute to renormalization constants Zi, interms of which the coupling constants are:h1 = h1,0Zh1Z−2ψ b−d h2 = h2,0Zh2Z−2ψ b−d (4.3)234.1. U(1) Breaking Operators with Attractive Interactionsh3 = h3,0Zh3Z−1/2φ b−d/2 h4 = h4,0Zh4Z−1/2φ b−d/2 (4.4)Above, the {hi,0} are the ‘bare’ couplings defined at scale Λ, while the{hi} are the couplings defined at the scale b−1Λ. The explicit factors of b aregenerated from the rescaling (3.11). Differentiating these expressions withrespect to log b, we obtain the desired beta functions.Throughout our calculations, we consider all one loop diagrams thatare O(hi),O(λ2i ),O(η2i ) and O(M). We define the operator ∗ on momentavectors a, b asa ∗ b := axbx − ayby (4.5)and we use faint/bold propagator lines to denote slow/fast fields in ourFeynman diagrams. We also use the notation p := /p − ip3, introduced inChapter 3. From the outset, we set the boson masses to zero, since thismarks the phase transitions of interest. All Feynman diagrams have beendrawn using the package [31].4.1 U(1) Breaking Operators with AttractiveInteractionsIn this section, we evaluate the one loop diagrams corresponding to (2.22).Using the modified -expansion, the fermion and boson propagators areG(p) =ip +Mp2D(p) =1p2(4.6)We use solid lines to represent the fermion propagators, and dashed lines torepresent the boson propagators. An arrow is used to indicate the directionof charge; this charge is +1 for the fermion, and +2 for the boson. Finally, weinclude the operators of the external legs in the definitions of our Feynmandiagrams.4.1.1 Feynman DiagramsFermion PropagatorThe single one loop diagram that renormalizes the fermion propagator toO(hi) is shown in Figure 4.1. Including the external legs, it equals=∫ddk(2pi)dψ¯s(k)Σψ(k)ψs(k) (4.7)244.1. U(1) Breaking Operators with Attractive InteractionskpFigure 4.1: Fermion self energy diagram in Wilson RG for g > 0whereΣψ(k) = −4λ21∫fddp(2pi)dD(p)CTGT (p− k)C (4.8)and the p integration is over the Wilson shell. Using the modified -expansionpropagators, and expanding to linear order in the slow momentum k, this isΣψ(k) = −4λ21∫fddp(2pi)d1p2[−ik† −Mp2+ 2p · k ipp4](4.9)Since the region of integration is symmetric, we can replacep · kip† → p2dik† (4.10)to findΣψ(k) = −ik†4λ21[1− 2d] ∫fddp(2pi)d1p4− 4λ21M∫fddp(2pi)d1p4(4.11)Using∫fddp(2pi)d1p4= Ωd∫ Λb−1Λdppd−5 =2(4pi)d/2Γ(d/2)Λd−4δl +O(δl2) (4.12)for b = eδl, we find the following renormalization constants for the fermionkinetic term and fermion mass term:Zψ = 1 +8λ21(4pi)d/2Γ(d/2)[1− 2d]Λ−δl ZM = 1− 8λ21(4pi)d/2Γ(d/2)Λ−δl(4.13)254.1. U(1) Breaking Operators with Attractive InteractionskpFigure 4.2: Boson self energy diagram in Wilson RGBoson PropagatorThe unique one loop diagram that renormalizes the boson propagator tolinear order in O(hi) is shown in Figure 4.2. It equals=∫ddk(2pi)dφ∗s(k)Σφ(k)φs(k) (4.14)whereΣφ(k) = 2λ21∫fddp(2pi)dtr[CG(p)CGT (k − p)] (4.15)Using= CGT (p)C =ip−Mp2(4.16)we haveΣφ(k) = 2λ21∫fddp(2pi)d1p2(k − p)2 tr[(ip +M)(ik† − p† −M)] (4.17)Since the phase transition occurs when the boson mass is tuned to zero,we isolate the terms proportional to k2, to extract Zφ. We need not beconcerned with the generation of terms proportional to k4 only, since thesedrop out of the modified -expansion. We findZφ = 1 +8(4pi)d/2Γ(d/2)[1− 2d]λ21Λ−δl. (4.18)Renormalization of h2At one loop, there is no diagram renormalizing h2. Therefore,Zh2 = 1 (4.19)264.1. U(1) Breaking Operators with Attractive InteractionsRenormalization of h3 and h4:There are two diagrams that contribution to the renormalization of h3 andh4 at one loop. The first is shown in Figure 4.3, and equalskpk2k1Figure 4.3: First diagram renormalizing h3 and h4 in Wilson RG.=∫ddk1(2pi)dd2k2(2pi)dφs(−k1 − k2)ψa,s(k1)Fab(k1, k2)ψb,s(k2) (4.20)where k := −k1 − k2, the solid vertex denotes an insertion of the U(1)breaking operator h2, andF (k1, k2) = −λ1h22∫fddp[C[p∗(k−p)+k1∗k2]tr[CG(p)CGT (k−p)] (4.21)−4k2 ∗ (k − p)CG(p)CGT (k − p)C]The integrand of this expression is, to O(M),−2C [p ∗ (k − p) + k1 ∗ k2]p2(k − p)2 p·(k−p)+4k2 ∗ (k − p)Cp2(k − p)2 [p·(k−p)+iM(p−k†+p†)](4.22)where we’ve dropped terms according to the modified -expansion procedure.In spherical coordinates,p ∗ p = p2x − p2y = p2 sin θ cos(2φ) (4.23)integrates to zero over angular coordinates when multiplied by any power of|p|, so we can drop such terms. Likewise, (p ∗ k)p integrates to zero since it274.1. U(1) Breaking Operators with Attractive Interactionsis odd in pz. Keeping at most two powers of slow momenta k, and droppingterms that vanish upon integration, we can replace the previous expressionwith2C3k1 ∗ k2 + 2k2 ∗ k2p2(4.24)so thatF (k1, k2) = −λ1h2C[3k1 ∗ k2 + 2k2 ∗ k2]∫fddp1p2(4.25)= −λ1h2C[3k1 ∗ k2 + 2k2 ∗ k2] 2(4pi)d/2Γ(d/2)Λd−2δl (4.26)The second diagram renormalizing h3 and h4 is shown in Figure 4.4, andequalsk2k1pkFigure 4.4: Second diagram renormalizing h3 and h4 in Wilson RG∫ddk1(2pi)dd2k2(2pi)dφs(−k1 − k2)ψa,s(k1)Gab(k1, k2)ψb,s(k2) (4.27)where k := −k1 − k2, and the solid vertex denotes the insertion of the U(1)breaking operators proportional to h3 and h4, andG(k1, k2) = −4λ21∫fddp(2pi)dD(p)CG(p− k1)CGT (−p− k2) (4.28)284.1. U(1) Breaking Operators with Attractive Interactions× [h3k2 ∗ (−p− k2) + 2h4(p+ k2) ∗ (p+ k2)]Again, we drop terms proportional to p∗p and k∗p, since they will integrateto zero. The result is, to quadratic order in the slow momenta k,G(k1, k2)→ −4(2h4 − h3)k2 ∗ k2λ21∫fddp(2pi)dD(p)CG(p− k1)CGT (−p− k2)(4.29)= −4(2h4 − h3)k2 ∗ k2λ21C∫fddp(2pi)d1p4(4.30)= −4(2h4 − h3)k2 ∗ k2λ21C2(4pi)d/2Γ(d/2)Λ−δl (4.31)Adding this result to (4.25), we find the following renormalization con-stants:Zh3 = 1−6λ1h2h32Λ−δl(4pi)d/2Γ(d/2)(4.32)andZh4 = 1 +[2λ1h2 + 4(2h4 − h3)λ21] 2Λ−δl(4pi)d/2Γ(d/2)h4(4.33)The factors of Λ2 were removed by redefining the couplings constants tobe dimensionless from the start of the calculation.Remaining DiagramsFor all remaining diagrams, we cite the calculations of [27], since these donot receive corrections from the U(1) breaking terms or the fermion massto this order. As a result, the beta functions for λ1 and λ2 are unchanged,and we can use the critical value λ21 from [27]:λ21,∗(4pi)2=12+O(2) (4.34)Renormalization Constants at O()To determine the value of these renormalization constants to O(), we re-place λ21 in these expressions with λ1,∗ in (4.34). Any corrections from U(1)294.1. U(1) Breaking Operators with Attractive Interactionsbreaking operators or the fermion mass will be higher order in the parame-ters {hi,M}. We find, to O(), the following renormalization coefficients:Zψ =1 +3δl (4.35)ZM =1− 23δl (4.36)Zh2 =1 (4.37)Zh3 =1−6h2δlh3√3(4pi)√ (4.38)Zh4 =1 + 2δl[h2√h44pi√3+(2h4 − h3)3h4](4.39)(4.40)4.1.2 Beta Functions of U(1) Breaking OperatorsUsing (4.3 - 4.4), we findβM =M[1 +dZMdδl− dZψdδl]= M [1− ] (4.41)βh2 =− h2[d+ 2dZψdδl]= −h2[4− 3](4.42)βh3 =− h3[d2+32dZψdδl− dδZh3dδl]= −h3[2 +6h2√√3(4pi)h3](4.43)βh4 =− h4[d2+32dZψdδl− dδZh4δl]= −h4[2− 2h2√h44pi√3− 2(2h4 − h3)3h4](4.44)(4.45)Since βh2 is only a function of h2, and is negative for  = 1, we concludethat h2 flows to zero at large length scales, independent of h3 and h4. Thisimplies that βh3 is also negative at large length scales, so that h3 → 0.Finally, we are left withβh4 → −h4[2− 43]→ −23h4 < 0 (4.46)so that h4 also flows to zero. Therefore, at the critical point gc,2, all U(1)breaking operators are irrelevant. Meanwhile, the fermion mass operator ismarginal at one loop, and requires a higher order calculation. In the nextchapter, we set the U(1) breaking operators to zero, and carry out a fourloop study to address the relevance of a fermion mass.304.2. U(1) Breaking Operators with Repulsive Interactions4.2 U(1) Breaking Operators with RepulsiveInteractionsWe now calculate the renormalization constants for the theory (2.21). In thiscase, the only U(1) breaking operator is a four-fermi term. According (4.3),to determine the h1 beta function, we only have to calculate Zψ and Zh1 .Since there is no one loop diagram renormalizing h1, calculating the fermionpropagator will be sufficient. Note that we are using the same symbol Zψfor the renormalization constant in both (2.21) and (2.22), even though theyare different quantities.Using the modified -expansion, the fermion and boson propagators areG(p) =ipp2D(p) =1p2(4.47)The fermion mass is set to zero since time reversal symmetry is present at thetransition gc,1. We use solid lines (with an arrow indicating the direction ofcharge) to represent the fermion propagators, and dashed lines to representthe boson propagators. As before, we include the operators of the externallegs in the definitions of our Feynman diagrams.4.2.1 Feynman DiagramsFermion PropagatorThe single one loop diagram that renormalizes the fermion propagator toO(h1) is shown in Figure 4.5. It equalskpFigure 4.5: Fermion self energy in Wilson RG for g < 0=∫ddk(2pi)dψ¯s(k)Σψ(k)ψs(k) (4.48)314.2. U(1) Breaking Operators with Repulsive InteractionswhereΣψ(k) = η21∫ddp(2pi)dD(p)G(k + p) (4.49)We expand Σψ(k) in powers of k, and extract the linear piece to determineZψ:Σψ(k) = η21∫ddp(2pi)dD(p)G(k + p)→∫ddp(2pi)d1p2[ ikp2− 2p · k ipp4](4.50)= η21[1− 2d](ik)∫ddp(2pi)d1p4= η21[1− 2d](ik)2Λ−δl(4pi)d/2Γ(d/2)(4.51)so thatZψ = 1 +[1− 2d]2η21Λ−(4pi)d/2Γ(d/2)(4.52)where we’ve replaced k with k†, since the difference renormalizes the operatorψ¯k3ψ, which doesn’t enter into the modified -expansion.Since the beta functions for η1, η2 receive no O(h1) corrections, we cancite the results of [16] that at the critical point gc,1, η1 has a value ofη21,∗(4pi)2=8+O(2) (4.53)so that to O(),Zψ = 1 +η21(4pi)2= 1 +8(4.54)4.2.2 Beta Functions of U(1) Breaking OperatorUsing (4.3), we findβh1 = −h1[4− 34](4.55)at the phase transition gc,1. Therefore, to one loop order, the U(1) breakingoperator is irrelevant, and the phase transition falls into the Gross-Neveuuniversality class, as predicted in [10].32Chapter 5Relevance of the FermionMass OperatorIn this chapter, we determine the relevance of the fermion mass operator in(2.22) beyond one loop order in the modified -expansion. We treat M asa small parameter, so that terms O(M2) will be dropped. We also neglectall U(1) breaking operators, since these were shown to be irrelevant in theprevious section.A straightforward, but tedious approach to the problem is to calculateall two loop diagrams in the modified -expansion. This is done in AppendixC. A more efficient approach is to relate the fermion mass beta function tothe stability critical exponent in the massless theory, which allows us to goto O(4), using the following identity:βM = M [1− ω] ω := ddλ21dλ21d logµ∣∣∣λ1=λ∗1,(massless)(5.1)In words, ω is the derivative of the beta function for λ21, in the massless the-ory, evaluated at the critical point. To prove equation (5.1), we first developthe superspace formalism. This argument closely follows the derivation ofthe identityβm2 = m[2− ω] (5.2)for the boson mass operator m2|φ|2 in [27]. The identity (5.2) was firstclaimed in [11]).5.1 The Power of SupersymmetryIn this section, we derive (5.1) for the theory (2.22) at the critical pointgc,2, where the two U(1) invariant couplings λ1 and λ2 flow to a commonvalue, λ∗ [27]. We use the results of Chapter 4 to ignore all U(1) breakingoperators, so that the theory is supersymmetric in the massless limit.335.1. The Power of Supersymmetry5.1.1 Superspace FormalismOur first step will be to rewrite the massless theory (2.22) in the superspaceformalism. This is most easily done in real time. We introduce a chiralsuperfieldΦ(y) := φ(y) +√2θψ(y) + θ2F (y) (5.3)where θ, θ¯ are two-component Grassmann spinors, and y is the (real time)superspace coordinateyµ := xµ − iθγµRθ¯ (5.4)By real time, we mean that xµ is a real time coordinate, and the matricesγµR = {−γ0, iγ1, iγ2} satisfy the 2+1 dimensional Minkowski metric:{γµR, γνR} = 2diag(1,−1,−1) (5.5)Throughout, we use the following spinor summation convention:θα = αβθβ θα = αβθβ θ2 = θαθα = 2θ2θ1 (5.6)whereαβ :=(0 −11 0)αβ :=(0 1−1 0)(5.7)Within this convention, we have the following identitiesθαθβ = θαθβ =12θ2αβ θγµRθ¯θγνRθ¯ =14θ2θ¯2tr[γµRγνR] =14θ2θ¯2ηµν(5.8)Finally, the Grassmann integration measure is defined as follows:d2θ = −14dθαdθβαβ =⇒∫d2θθ2 = 1 (5.9)Using this formalism, the superfield can be expanded asΦ(y) = φ(x)−iθγµRθ¯∂µφ(x)−14θ2θ¯2∂µ∂νηµνφ+√2θψ(x)+iθ2√2∂µψ(x)γµRθ¯+θ2F (x).(5.10)where ∂2R := ∂µ∂νηµν .Using this, the free SUSY Lagrangian density isL0SUSY =∫d2θd2θ¯Φ†Φ = −14[∂2Rφφ∗+φ∂2Rφ∗]+|F |2+ 12∂µφ∂νφ∗ηµν (5.11)+i∫d2θ¯θψ¯∂µψγµRθ¯ − i∫d2θθγµR∂µψ¯θψ345.1. The Power of SupersymmetrySinceθ¯∂µγµRθ¯ =12θ¯2ψ¯γµ,TR ∂µψ θγµ∂µψ¯θψ = −12θ2ψ¯γµ,T∂µ (5.12)up to total derivatives, equation (5.11) equals|F |2 + ∂µφ∗∂νηµνφ+ iψ¯γµ,TR ∂µψ (5.13)To produce a boson-fermion interaction term, we add to (5.11) a superpo-tential termδLSUSY =∫d2θW (Φ) +∫d2θ¯W (Φ†) W (Φ) :=λ3Φ3 (5.14)and apply the equations of motion for the auxiliary field F :F = −λφ∗2 F ∗ = −λφ2 (5.15)We find, using θψθψ = −12ψT [iσ2]ψ, thatLSUSY := L0SUSY+δLSUSY = ∂µφ∗∂νηµνφ+iψ¯γµR∂µψ−λ2|φ|4−λ(φψTCψ + h.c.)(5.16)which is exactly the real time version of (2.22), at the critical pointλ1 = λ2 = λ ≡ λ∗. In other words:∫d2θd2θ¯Φ†Φ +∫d2θλ3Φ3 +∫d2θ¯λ3Φ†3 = L2,real (5.17)5.1.2 Relating the fermion beta function and the stabilitycritical exponentNow that we’ve rewritten (the real time version of) L2 in the superspace for-malism, we would like to introduce a fermion mass operator in this language.This is achieved by adding the following expression to (5.17):−∫d2θd2θ¯2MΦ†θθ¯Φ = −4M∫d2θd2θ¯θ¯ψ¯θθ¯θψ = −Mψ¯ψ (5.18)To linear order in M , this addition can be compensated by rescaling thesuperfield,Φ→ (1 +Mθθ¯)Φ (5.19)which shifts the coupling λ accordingly:λ→ λ˜(M) := λ+ 3Mθθ¯ (5.20)355.1. The Power of SupersymmetryIn other words, the massive theory with coupling λ is equivalent to themassless theory with coupling λ˜. Now, to access the scaling dimension ofψ¯ψ, we require the notion of bare and renormalized fields and masses. Wewrite the bare theory in terms of bare Φ0 and bare M0, λ0:Lbare =∫d2θd2θ¯Φ†0(1− 2M0θθ¯)Φ0 +∫d2θλ03Φ30 +∫d2θ¯λ03[Φ†0]3 (5.21)and the renormalized theory in terms of Φ and Mµ,λµ/2:L =∫d2θd2θ¯Z˜Φ†(1−2Mµθθ¯)Φ+∫d2θλµ/23Φ30+∫d2θ¯λµ/23[Φ†0]3 (5.22)Here the renormalization scale µ has been introduced so that M and λ aredimensionless. Notice that there is no renormalization constant Zλ – thisfollows from SUSY nonrenormalization theorems [32, 33]. In the masslesstheory, we can write down an equation similar to (5.22), replacing Z˜ withsome other renormalization constant Z. In general, these two functions willbe different; however, using (5.20), we haveZ˜(λ) = Z(λ˜) = Z(λ)[1 + 3Mµθθ¯λ∂ logZ∂λ]+O(M2) (5.23)Using this and comparing (5.21) to (5.22), and we find the relationM = M0µ−1[1− 32λ∂ logZ∂λ]−1(5.24)Writingλ∂ logZ∂λ= 2λ2∂ logZ∂λ2(5.25)to expand [1− 32λ∂ logZ∂λ]−1= 1 + 3λ2∂2 logZ∂λ2+O(λ4), (5.26)we can differentiate (5.24) with respect to logµ to find− βM := ∂M∂ logµ= M[−1− 3λ2 ∂γ∂λ2](5.27)where γ = − ∂Z∂ logµ is the anomalous dimension of the fermion in the masslesstheory. The unconventional negative sign is introduced so that these func-tions agree with their Wilson counterparts. Now, in the supersymmetrictheory, γ can be rewritten in terms of the beta function of λ2, sinceλ20 = λ2µ−Z(λ)3 (5.28)365.2. Fermion Mass Beta Functionbecause the superpotential is not renormalized. The beta function is− βλ2 = −dλ2d logµ= λ2[−− 3γ] (5.29)Differentiating with respect to λ2, and using the fact that to O(4), the valueof γ at the SUSY point (=: λ∗) is ([29])γ(λ∗) = − 3(5.30)we have− dβλ2dλ2= −− 3γ(λ∗)− 3λ2∗∂γ∂λ2= −3λ2∗∂γ∂λ2(5.31)Comparing this to (5.27), we findβM = M[1− dβλ2dλ2](5.32)proving (5.1).5.2 Fermion Mass Beta FunctionIn [29], ω has been evaluated in the massless theory to four loop order:ω = − 23+(118+2ζ33)3 +1540(420ζ3 + 1200ζ5 − 3pi4 + 35)4 +O(5)(5.33)Using Pade´ extrapolation (see [16]), the authors of [29] found the valuesω = 0.872 and ω = 0.870, depending on which Pade´ approximant is used.In [34], the value ω = .910 was obtained using the conformal bootstrap. Inall three approaches,βM = M [1− ω] (5.34)is positive, and the fermion mass operator is relevant. Therefore, at thephase transition gc,2, a time reversal breaking perturbation will destroy theemergent supersymmetry. The resulting universality class is determined inthe following subsection. However, if time reversal is an approximate sym-metry of the underlying lattice model, some signatures of supersymmetry,including equal scaling dimensions for the boson and fermion fields, may stillbe present. In passing, we note that our explicit two loop results, calculatedusing dimensional regularization, agree with (5.1) and (5.33) to O(2) (seeAppendix C).375.3. Consequence of a Relevant Fermion Mass Operator5.3 Consequence of a Relevant Fermion MassOperatorSince the fermion mass is relevant, a large mass will be generated near thecritical point. At energy scales  M , the fermion degrees of freedom canbe integrated out completely. To perform this integration explicitly, weuse a Hubbard-Stratonovich transformation to replace all of the four-Fermiinteractions in (2.17) withLint = −m2|φ|2 + (φ[ρ1ψ¯Cψ¯T + ρ2∂rψTC∂rψ] + h.c.) (5.35)whereρ1 = 4m√gΛ−10 ρ2 =m2√gΛ30(5.36)This expression (5.35) reproduces (2.17) to O(g) when φ is integrated out.The boson φ no longer corresponds to the Cooper pair φ ∼ ψ1ψ2 of (2.22);instead, it corresponds toφ ∼ ψ1ψ2 + 12∂rψ∗1∂rψ∗2. (5.37)We can use (5.37) to determine how φ transforms under the exact latticesymmetries (2.25 - 2.27). Explicitly, these transformations areC : φ(x, y) 7→ φ∗(x, y) (5.38)T : φ(x, y) 7→ −φ∗(x, y), i 7→ −i (5.39)P : φ(x, y) 7→ φ∗(−x, y) (5.40)R : φ(x, y) 7→ iφ(−y, x) (5.41)The most noteworthy equation is (5.41), since it implies that the mostrelevant U(1) breaking operator allowed by symmetry is φ4 +φ∗4. To deter-mine the coefficient of this operator, we integrate out the fermions explicitly,using the notation introduced in Chapter 4. The unique one loop diagramgenerating a φ4 interaction is shown in Figure 5.1. We are not interestedin derivative operators, so we can set all external momenta to zero. Thecontribution to the operator φ4 is then equal to= −8ρ21ρ22∫d3p(2pi)3(p ∗ p)2tr[G(p)CGT (−p)CG(p)CGT (−p)C] (5.42)where the integral is over all momentum modes up to a cutoff Λ ∼M . UsingC(/pT +M)C = /p−M , the trace equalstr[G(p)CGT (−p)CG(p)CGT (−p)C] = 2(p2 +M2)2(5.43)385.3. Consequence of a Relevant Fermion Mass OperatorFigure 5.1: Diagram generating φ4 + h.c. when the fermion mass is relevantWriting p ∗ p = p2 sin2 θ cos(2φ) in spherical coordinates, the expression(5.42) equals− 16ρ21ρ22∫d3p(2pi)3p4 sin4 θ cos2(2φ)(p2 +M2)2∝ ρ21ρ22M3 (5.44)Therefore, a φ4 + h.c. operator is generated, with coupling constant pro-portional toρ21ρ22M3 ∝ Λ−10(mΛ0)4g2(MΛ0)3(5.45)Since our original assumption was that the fermion mass is small com-pared to the bare cutoff, we see that the coefficient of φ4 is highly suppressed.Therefore, the low energy theory near the critical point gc,2 has the followingstructureL = |∂µφ|2 +m2|φ|2 + ρ|φ|4 + ρ˜(φ4 + φ∗4) ρ˜ ρ (5.46)This model was studied in [35, 36] using -expansion techniques andin [37] using Monte Carlo, where it was shown that ρ˜, which lowers thesymmetry from U(1) to Z4, is irrelevant in 3 spacetime dimensions and thecritical point is the XY one. Therefore, once the fermion mass becomesrelevant, the universality class of gc,2 will change from N = 2 SUSY to theconventional XY transition.39Chapter 6ConclusionIn this work, we have shown that the emergent U(1) symmetry present atthe critical points of the Majorana-Hubbard model is preserved when U(1)breaking corrections are taken into account. Moreover, we have shown thata fermion mass term, generated by a time reversal breaking perturbation,is a relevant operator at four loops in the -expansion. These results sug-gest that in the case of repulsive interactions, the Majorana-Hubbard modelhas a critical point in the Gross-Neveu universality class, and in the case ofattractive interactions, the model has a critical point in the N = 2 super-symmetric universality class when time reversal symmetry is unbroken, andin the XY universality class otherwise. These results agree with the clas-sification of Affleck et. al.[10]. 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Vicari. N-componentGinzburg-Landau Hamiltonian with cubic anisotropy: A six-loop study.Phys. Rev. B, 61:15136–15151, June 2000.[37] Jie Lou, Anders W. Sandvik, and Leon Balents. Emergence of u(1)symmetry in the 3d xy model with Zq anisotropy. Phys. Rev. Lett.,99:207203, Nov 2007.43Appendix ADerivation of the LowEnergy Field TheoryIn this appendix, we derive (2.11), which is the low energy continuum de-scription of (2.3).A.1 Quadratic HamiltonianRelabelling the Majorana operators γ according to (2.8), the first term of(2.3) becomesH0 = it∑m,nγem,2n[γem+1,2n+γom,2n+1]+γom,2n+1[−γom+1,2n+1+γem,2n+2]. (A.1)Now using the expansion (2.10), the first piece of (A.1) isγem,2n[γem+1,2n + γom,2n+1] ≈ (A.2)8Λ−20 [χe+(m, 2n)+(−1)mχe−(m, 2n)][χe+(m+1, 2n)+(−1)m+1χe−(m+1, 2n)+χo+(m, 2n+ 1) + (−1)mχo−(m, 2n+ 1)]where we’ve suppressed the lattice constant a in the arguments of χ.To derive a continuum field theory, we will Taylor expand the fields χabout the point a(m+ 12 , 2n+12). Let χ := χ(m+12 , 2n+12), and define∂± :=12(∂x ± ∂y). (A.3)Each derivative will contribute an additional factor of lattice spacing a =Λ−10 . Then (A.2) becomes, after an integration by parts,8Λ−20∑±±χe±ea∂xχe± + χe±ea∂yχo± (A.4)44A.2. Quartic Hamiltonianwhere we’ve dropped alternating terms that do not contribute to the lowenergy theory. Performing a similar expansion for the second piece of (A.1),and adding both contributions together yields8Λ−20∑±±χe±ea∂xχe± ∓ χo±ea∂xχo± + χe±[ea∂y − e−a∂y ]χo± (A.5)Note that even-derivative functions vanish when sandwiched between thesame Majorana operator, since integration by parts givesχ∂(2k)χ = (−1)2k∂(2k)χχ, (A.6)and {χ, ∂(2k)χ} = 0. Therefore we can replace ea∂x by its odd part. Finally,using ∑m,n7→ 12Λ20∫dxdy (A.7)we find that the quadratic Hamiltonian density isH0+H′2 = 4it∑±±χe± sinh(a∂x)χe±∓χo± sinh(a∂x)χo±+2χe± sinh[a∂y]χo±(A.8)In Section 2.3, it is shown that the underlying symmetry of the latticemodel forces all quadratic operators to preserve the emergent U(1) sym-metry. Therefore, in our leading order study of U(1) breaking operators, weneglect the effects of H′2.A.2 Quartic HamiltonianWe now repeat the steps of (A.1) for the interacting piece of (2.3), whichsplits into two pieces:Hint = g∑m,nγem,2nγem+1,2nγom+1,2n+1γom,2n+1+γom,2n+1γom+1,2n+1γem+1,2n+2γem,2n+2(A.9)We are only required to Taylor expand the following objectAe/o(x, y) := γe/o(x, y)γe/o(x+ a, y) (A.10)where (x, y) = a(m+ 12 , 2n+12). In terms of this function, Hint can bewritten asHint = −g2Λ20[Ae(x− a/2, y − a/2)Ao(x− a/2, y + a/2) (A.11)45A.3. Leading U(1) Breaking Operator+Ae(x− a/2, y + a/2)Ao(x− a/2, y − a/2)].Using (A.11), and expanding up to two derivatives,A(x−a/2, y−a/2) ≈ 8Λ−20∑±±[aχ±(∂++∂−)χ±−a2∂+χ±∂−χ±+a22(∂2+−∂2−)χ±χ±](A.12)+8Λ−20 (−1)m∑±±[χ∓χ±+aχ∓(∂−−∂+)χ±−a2∂+χ∓∂−χ±+a22χ∓(∂2−+∂2+)χ±]where we’ve used the notation introduced in the previous subsection. Usingthis result, (A.7), and integration by parts, the Hamiltonian density can bewritten as164gΛ−40Hint = (A.13)−∑s,s′=±1ss′χes∂xχesχos′∂xχos′−4Λ20χe−χe+χo−χo++2∂y(χe−χe+)∂y(χo−χo+)+2χe−χe+∂xχo−∂xχo+ + 2∂xχe−∂xχe+χo−χo+ + ∂x(χe−χe+)∂x(χo−χo+)which is (2.12).A.3 Leading U(1) Breaking OperatorIn this appendix, we derive (2.16), which is the U(1) breaking piece of (A.13)in the ψ notation. For each type of term of (A.13), we insert the (inversesof) (2.15), and extract the U(1) breaking part:• Type 1:χe−χe+∂xχo−∂xχo+ + ∂xχe−∂xχe+χo−χo+= −18(ψ2ψ∗1−ψ∗2ψ1)(∂xψ1∂xψ∗2−∂xψ∗1∂xψ2)−18(ψ2ψ1−ψ∗2ψ∗1)(∂xψ1∂xψ2−∂xψ∗1∂xψ∗2)→ −18[ψ2ψ1∂xψ1∂xψ2 + h.c.] (A.14)• Type 2:∂i(χe−χe+)∂i(χo−χo+) (no sum over i, and for i = x, y)= −14(∂iψ2∂iψ∗2ψ∗1ψ1+∂iψ∗1∂iψ1ψ2ψ∗2)−18[∂iψ1∂iψ2+∂iψ∗1∂iψ∗2][ψ1ψ2+ψ∗1ψ∗2]46A.3. Leading U(1) Breaking Operator−18[−∂iψ1ψ∗2 − ∂iψ∗1ψ2][ψ∗1ψ2 + ψ1ψ∗2]→ −18[∂iψ1∂iψ2ψ1ψ2 + h.c.] (A.15)• Type 3:(χe+∂xχe+ − χe−∂xχe−)(χo+∂xχo+ − χo−∂xχo−)= −14(∂xψ∗1∂xψ1ψ∗1ψ1+∂xψ2∂xψ∗2ψ2ψ∗2)−18(∂xψ2∂xψ1+∂xψ∗2∂xψ∗1)(ψ2ψ1+ψ∗2ψ∗1)−18(∂xψ∗2∂xψ1 + ∂xψ2∂xψ∗1)(ψ2ψ∗1 + ψ∗2ψ1)→ −18[∂xψ2∂xψ1ψ2ψ1 + h.c.] (A.16)Using equations (A.14-A.16), we find that the U(1) breaking piece of(A.13) is16g0Λ40ψ1ψ2[∂xψ1∂xψ2 − ∂yψ1∂yψ2] + h.c. (A.17)Since Hint = Lint for an imaginary time Lagrangian density, we’ve repro-duced (2.16).47Appendix BPromoting ψ to a DiracFermion in Four DimensionsOne idea to resolve the issue of breaking Lorentz invariance in the -expansionis to promote ψ to a Dirac fermion in four dimensions. If this Dirac theorycan be decoupled into two Weyl sectors, then we may obtain the Weyl renor-malization group functions by continuing N , the number of Dirac fermions,from 1 to 12 in this theory. We now show that this limit is ill-defined.To generate the interaction term φ∗ψTCψ in each Weyl sector, we con-sider following operatoriφ∗ΨT(C 00 −C)Ψ + h.c. C = iγ0 (B.1)To show that it is Lorentz invariant, it is sufficient to consider ψTCψ, sinceLorentz transformations do not couple Weyl sectors in the Weyl basis. Using(3.33),ψTCψ 7→ ψT e~α·~σTCe~α·~σ = ψTCe−~α·~σe~α·~σψ = ψTCψ (B.2)under a general Lorentz transformation. Adding this interaction to the freeDirac Lagrangian density, we haveL = Ψ¯[∂aΓa +M ]Ψ + [iφ∗ΨT(C 00 −C)Ψ + h.c.] (B.3)By rotating ψR → γ0ψR, so that both Weyl fermions propagate in the samedirection, (B.3) becomes2∑i=1[ψ¯i[∂µγµ + i∂3]ψi + [iφ∗ψTi Cψi + h.c.]]+Mψ¯LψR + ψ¯RψL (B.4)where we used (3.31). The two Weyl sectors can be decoupled by introducingψ± :=1√2(ψL ± ψR). (B.5)48Appendix B. Promoting ψ to a Dirac Fermion in Four DimensionsThis doesn’t affect the interaction term, but it modifies the mass terms toM [ψ¯LψL − ψ¯RψR] (B.6)This relative sign in the mass terms cannot be removed, implying thatthe two Weyl sectors are distinct. Any continuation of the Dirac numberN → 12 would have to choose between one of these two distinct sectors,rendering the limit ill-defined.49Appendix CTwo Loop Calculation of theFermion Mass Beta FunctionIn this appendix, we calculate βM at the critical point gc,2 in the massivetheory, to two loop order in the modified -expansion. These calculationsverify (5.33) to this order. While our one loop calculations were carried outin the Wilson picture of the renormalization group, it is easier to use dimen-sional regularization for higher order calculations. Following the conventionsoutlined in Section 3.3, we introduce counterterms, order-by-order, to can-cel the divergences appearing in loop diagrams. Our task is to determinethe renormalization constants that appear in the renormalized LagrangiandensityL = Zψψ¯[/∂+i∂3]ψ+MµZM ψ¯ψ+Zφ|∂aφ|2+λ1Zλ1µ/2[φψTCψ+h.c.]+λ2Zλ2µ|φ4|(C.1)Here µ is an energy scale characterizing the RG flow; it plays a role analogousto b−1 in the Wilsonian picture. The factors of µ appearing in L ensure therenormalized couplings and renormalized mass M are dimensionless. Thebeta function of M can be determined using the formulaeβM = − [−1− γψ + γM ]M (C.2)βX := − dXd logµγX := − dZxd logµ(C.3)The negative signs present in the definitions of βX and γX ensure that thesefunctions have the same signs as their Wilsonian counterparts, dXd log b anddZxd log b , where b is the length scale characterizing the size of a Wilsonianmomentum shell. The anomalous dimensions satisfyγx = βλ21d logZxdλ21+ βλ22d logZxdλ22+ βMd logZxdM(C.4)and provide a system of equations to find the beta functions at a givenorder. To determine the 2 loop anomalous dimensions, we require the 250C.1. One Loop Diagramsloop renormalization functions ZM , Zψ, as well as the 1-loop beta functionsβλ21 , βM , and βλ22 .In fact, it will turn out that ZM and Zψ only depend on λ21 to two loops,and so we only need to know βλ21 . Because ZM and Zψ are independent ofM , we can use the massless one loop β function from [15]:βλ21 = −λ21 +3(4pi)2λ41 (C.5)Once we verify that Zφ, Zψ and Zλ1 don’t depend on M , it will be sufficientto calculate Zψ and ZM to two loop order, to determine βM .In the following, all Feynman diagrams have been drawn using the pack-age [31]. Solid lines correspond to fermion propagators, dashed lines corre-spond to boson propagators, and an arrow is used to indicate the directionof charge. This charge is +1 for the fermion propagator, and +2 for theboson propagator.C.1 One Loop DiagramsC.1.1 Fermion PropagatorAt one loop, the fermion propagator is renormalized by a single diagram,shown in Figure C.1. It equalskpFigure C.1: Fermion self energy in renormalized perturbation theory= (−2λ1)2∫ddp(2pi)dD(p)CGT (p− k)C = 4λ21∫ddp(2pi)d1p2i(p† − k†)−M(p− k)2(C.6)where we’ve usedC(ip +M)TC = ip† −M (C.7)51C.1. One Loop DiagramsIntroducing a Feynman parameter x according to1p2(p− k)2 =∫ 10dx1[(p− xk)2 + x(1− x)k2]2 (C.8)and changing variables to l = p− xk, we have the diagram equalling4λ21∫ 10dx∫ddl(2pi)di(l† + (x− 1)k†)−M[l2 + ∆]2(C.9)where∆ := x(1− x)k2 (C.10)Finally, using the integral formula∫ddl(2pi)d1[l2 + ∆]n=1(4pi)d/2Γ(n− d2)Γ(n)∆d2−n (C.11)we find that the diagram (C.1) equals4λ21∫ 10dx[i(x− 1)k† −M ] ∆d2−2(4pi)d/2Γ(2− d2)Γ(2)(C.12)Replacing d with 4 − , we find that (C.1) has the following divergingterm in the → 0 limit:− 4λ21(4pi)2[i2k† +M ]2(C.13)To cancel this divergence, we introduce renormalization constants δZψand δZM into the Lagrangian density, that produce the following terms tothis order in λ21:− iδZψψ¯k†ψ +MδZM ψ¯ψ (C.14)Note that the renormalization constants in (C.1) satisfyZx = 1 + δZx + two loop terms (C.15)To achieve cancellation, we requireδZψ = − 4λ21(4pi)2δZM =8λ21(4pi)2(C.16)52C.1. One Loop DiagramskpFigure C.2: Boson self energy in renormalized perturbation theoryC.1.2 Boson PropagatorAt one loop, the boson propagator is renormalized by the single diagram inFigure C.2. It equals= −12(−2λ21)∫ddp(2pi)dtrCG(p)CGT (k − p) (C.17)The factor of 12 is a symmetry factor, and the overall minus sign is determinedfrom Wick’s theorem, and is a common feature of fermion traces in Feynmandiagrams. Introducing the same Feynman parameter as in (C.8), this equals− 4λ212∫ddp(2pi)dddlp(2pi)dtr[(ip +M)(ik† − ip† −M)] 1p2(k − p)2 (C.18)= −4λ212∫ 10dx∫ddl(2pi)dtr[l2 − x(1− x)k2 + iM(1− x)k† − iMxk][l2 + ∆]2(C.19)While tr/p = 0trMp = −2p3M 6= 0 (C.20)However, this term is not Lorentz invariant, and can be dropped using themodified  expansion introduced in the previous section. Using a secondaryintegral formula∫ddl(2pi)dl2[l2 + ∆]n=∆d2+1−n(4pi)d/2d2Γ(n− d2 − 1)Γ(n)(C.21)we find the one loop integral (C.2) equals− 4λ21(4pi)d/2∫ 10dx[d2Γ(−1 + 2)− Γ( 2)]∆1−2 (C.22)53C.2. Two Loop DiagramsNow using [ 2− 1]Γ(2− 1) = Γ( 2) (C.23)andΓ(2− 1) = −2+O(1) (C.24)the divergent behaviour as → 0 in (C.22) is4λ21(4pi)2∫ 10dx6∆=λ21(4pi)2k2(C.25)To cancel this divergence, we introduce a counterterm δZφ that appearsin the Lagrangian density ask2δZφ|φ|2 (C.26)To achieve a cancellation, we requireδZφ = − 4λ21(4pi)2(C.27)C.1.3 Interaction VertexAt one loop there is no diagram renormalizing the fermion-boson interac-tion, so that δZλ1 = 0. Moreover, the diagrams renormalizing the pureboson interaction involve only the field φ at one loop order. Therefore, allrenormalization constants are independent of mass, so that we can safelyuse the one loop result for βλ21 , (C.5), from [15]. Now, we must find the twoloop contributions to Zψ and ZM .C.2 Two Loop DiagramsThere are two types of diagrams contributing at two loops. Some are iden-tical to (C.1), but with either the internal boson or internal fermion aug-mented with a one loop counterterm δZφ or δZψ. We call these countertermdiagrams:C.2.1 Boson Counterterm DiagramThe diagram involving an inserting of δZφ is shown in Figure C.3. It equals= −δZφ4λ21∫dd(2pi)dD(p)CGT (p− k)C (C.28)54C.2. Two Loop DiagramskpFigure C.3: Boson counterterm diagram in renormalized perturbation theoryWe now wish to extract the diverging behaviour:− δZφ4λ21∫dd(2pi)dD(p)CGT (p− k)C (C.29)→ −δZφ4λ21∫ 10dx[i(x− 1)k† −M ]∆d/2−2(4pi)d/2Γ(2− d2)Γ(2)= − 4λ21(4pi)2δZφ∫ 10dx[i(x− 1)k† −M ](∆(4pi)2)− 2[2− γ +O()](C.30)Now, in the MS scheme (see Section 3.3), factors of γ and 4pi are takencare of by rescaling the coupling constants appropriately. Therefore, theonly diverging behaviour is8λ21(4pi)2δZφ[i2k† +M ] (C.31)C.2.2 Fermion Counterterm DiagramIn the fermion counterterm diagram, displayed in Figure C.4,we replace the fermion propagator according to1−ip +M →1−ip +M (−iδZψp− 2MδZψ)1−ip +M (C.32)where we used δZM = −2δZψ. This isδZψ−ip +M − 3MδZψ(ipp2)2(C.33)Replacing(ip)2 = −p(p† − 2ip3) = −p2 + 2ipp3 → −p2 (C.34)55C.2. Two Loop DiagramskpFigure C.4: Fermion counterterm diagram in renormalized perturbation the-oryin the modified -expansion, we obtainδZψ−ip +M +3MδZψp2. (C.35)The diagram (C.4) then equals− δZψ4λ21∫ddp(2pi)dD(p)i(p† − k†)− 4M(p− k)2 (C.36)Using the calculations of (C.1), we find the following diverging behaviour:δZψλ21(4pi)2[i2k† + 4M]8(C.37)There are two remaining diagrams that contribute to βM at two loops.One includes a fermion self energy bubble, and one includes a boson selfenergy bubble.C.2.3 Internal Boson Bubble DiagramThe diagram including a boson self energy bubble is shown in Figure C.5.It equals= −(−2λ1)2∫ddp(2pi)dCGT (p−k)C 1p4[−∫ 10dx4λ21(4pi)d/2(3− )[x(1− x)p2]1− 2 Γ(−1 + 2)](C.38)where we’ve used (C.22). Now, a generalized version of (C.8) lets us write1(p− k)2[p2]1+ 2 =∫ 10dy(1− y)/2[l2 + ∆(y)]2+2[1 +2](C.39)56C.2. Two Loop DiagramskpqFigure C.5: Two loop diagram with internal boson bubble in renormalizedperturbation theoryso that the diagram (C.5) equals= − 16λ41(4pi)d/2[1 +2]2(3−)Γ( 2)∫ 10dxdy[x(1−x)]1− 2∫ddl(2pi)di(y − 1)k† −M[l2 + ∆(y)]2+2(1−y)/2(C.40)= − 16λ41(4pi)d[1 +2]2 (3− )Γ( 2)Γ()Γ(2 + 2)∫ 10dxdy[x(1−x)]1− 2 (1−y)/2[i(y−1)k†−M ]∆(C.41)where in the second line we used (C.11). The diverging behaviour is:= − 16λ41(4pi)4[1 +2]2 (3− )Γ( 2)Γ()(1 + 2)2Γ(2)∫ 10dxdy[x(1−x)]1− 2 (1−y)/2[i(y−1)k†−M ](C.42)Now, using the expansions∫ 10dx[x(1− x)]1− 2 = 16+536+O(2) (C.43)∫ 10dy(y − 1)(1− y)/2 = −12+8+O(2) (C.44)∫ 10dy(1− y)/2 = 1− 2(C.45)The expression (C.42) simplifies to= −116λ41(4pi)4[1 + ] Γ()∫ 10dy(1− y)/2[−i(y − 1)k† −M ] (C.46)= −116λ41(4pi)4Γ()[− i2k†[1 +34]−M[1 +2]](C.47)57C.2. Two Loop DiagramsFinally, MS lets us replace Γ()→ 1 :=1216λ41(4pi)4[i2k†[1 +34]−M[1 +2]](C.48)We record this result and move to the final diagram.C.2.4 Internal Fermion Bubble DiagramThe final diagram is shown in Figure C.6. It equalskk − pqFigure C.6: Two loop diagram with internal fermion bubble in renormalizedperturbation theory= −4λ21∫ddp(2pi)dD(p)CGT (p− k)Σψ(p− k)TGT (p− k)C (C.49)where Σψ is the one loop fermion self energy diagram (C.1). The integrandisD(p)CGT (p− k)C2Σ(p− k)TC2GT (p− k)C (C.50)=4λ21Γ(2)(4pi)d/2∫ 10dxCGT (p− k)C[i(x− 1)(p− k) +M ]CGT (p− k)C∆− 2(C.51)(where we used Cp†TC = p). We can rearrange the propagator factorsaccording toCGT (p)C[i(x− 1)p +M ]CGT (p)C (C.52)=1−ip−M [i(x− 1)p +M ]1−ip−M =(x− 1)[ip +M ]−ip−M +(2− x)M(−ip−M)2(C.53)=i(1− x)p† − (3− 2x)Mp2+O(M2) (C.54)58C.2. Two Loop Diagramswhere in the last line we made the replacement (C.34). The diagram (C.6)then equals−16λ41Γ(2)(4pi)d/2∫ 10dx[x(1−x)]−/2∫ddp(2pi)d1p2[i(1− x)(p† − k†)− (3− 2x)M(p− k)2+](C.55)Introducing a Feynman parameter as in (C.39),1(p− k)2+p2 =∫ 10dyy/2[l2 + ∆(y)]2+2[1 +2](C.56)we have−[1 +2] 16λ41Γ( 2)(4pi)d/2∫ 10dydx[x(1− x)]−/2y/2 (C.57)×∫ddl(2pi)d1[l2 + ∆(y)]2+2[i(1− x)(y − 1)k†)− (3− 2x)M]= −2Γ()16λ41(4pi)d∫ 10dydx[x(1−x)]−/2y/2[i(1− x)(y − 1)k†)− (3− 2x)M]∆−(C.58)where in the last line we used (C.11). In the MS scheme, we replace Γ()→1 . Therefore, the divergent piece of (C.6) is= − 22λ41(4pi)d∫ 10dydx[x(1− x)]−/2y/2[i(1− x)(y − 1)k†)− (3− 2x)M](C.59)We now use the following expansions:∫ 10dxx−/2(1− x)1−/2 = Γ(1−2)Γ(2− 2)Γ(3− ) =12[1 + ] +O(2) (C.60)∫ 10dx[x(1− x)]−/2 = 1 + +O(2) (C.61)∫ 10dyy/2 = 1− 2+O(2) (C.62)∫ 10dyy/2+1 =12− 8+O(2) (C.63)We find the following divergent behaviour:ik16λ41(4pi)4[122(1 +14)]+M16λ41(4pi)442(1 +2) (C.64)59C.2. Two Loop DiagramsC.2.5 Fermion Mass Beta FunctionAdding up the divergences (C.31, C.37, C.48, C.64), we find=8λ21(4pi)2δZψ(ik)+40δZψλ21(4pi)2M+(ik)16λ41(4pi)4[12+12]+M16λ41(4pi)4[5 + ](C.65)Using the one loop resultδZψ =−4λ21(4pi)2(C.66)the total divergence at two loops is= − 16λ41(4pi)4(ik)[12− 12]− 16λ41(4pi)4M[52− 52](C.67)This determines the renormalization constants to this order:Zψ = 1− 4λ21(4pi)2− 16λ41(4pi)42+8λ41(4pi)4(C.68)ZM = 1 +8λ21(4pi)2+80λ41(4pi)42− 40λ41(4pi)42(C.69)Having obtained Zψ and ZM , we can now calculate their respectiveanomalous dimensions. Expanding:logZψ = − 4λ21(4pi)2− 24λ41(4pi)42+8λ41(4pi)4+O(λ61) (C.70)logZM =8λ21(4pi)2+48λ41(4pi)42− 40λ41(4pi)42+O(λ61) (C.71)Then differentiating and using the one loop beta function (C.5), we findγψ = βλ21d logZψdλ21= − 4λ21(4pi)2+16λ41(4pi)4(C.72)andγM =8λ21(4pi)2− 80λ41(4pi)4(C.73)Therefore, the beta function isβM = M [1 + γψ − γM ] = M − 12λ21M(4pi)2+96λ41M(4pi)4(C.74)60C.2. Two Loop DiagramsUsing the critical value of λ21 found in [15],λ21,∗(4pi)2=12+236(C.75)the beta function equalsβM (λ1,∗) =[1− 12[12+236]+ 962144]M =[1− + 23]M (C.76)which agrees with the relation (5.33) to O(2).61

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