Conductivity of interacting massless Dirac particles in graphene: Collisionless regime
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Conductivity of interacting massless Dirac particles in graphene: collisionless regime
Vladimir Juriˇci´c,
Oskar Vafek, and Igor F. Herbut Instituut-Lorentz for Theoretical Physics, Universiteit Leiden,P.O. Box 9506, 2300 RA Leiden, The Netherlands Max-Planck-Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany National High Magnetic Field Laboratory and Department of Physics,Florida State University, Tallahasse, Florida 32306, USA Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
We provide detailed calculation of the a.c. conductivity in the case of 1 /r Coulomb interactingmassless Dirac particles in graphene in the collisionless limit when ω ≫ T . The analysis of the elec-tron self-energy, current vertex function and polarization function, which enter into the calculationof physical quantities including the a.c. conductivity, is carried out by checking the Ward-Takahashiidentities associated with the electrical charge conservation and making sure that they are satisfiedat each step. We adopt a variant of the dimensional regularization of Veltman and ’t Hooft bytaking the spatial dimension D = 2 − ǫ for ǫ >
0. The procedure adopted here yields a result for theconductivity correction which, while explicitly preserving charge conservation laws, is neverthelessdifferent from the results reported previously in literature.
I. INTRODUCTION
The role of Coulomb electron-electron interactions insystems described by massless two-dimensional Diracfermions has been a subject of interest for some time.
Discovery of graphene, a single atomic layer of sp hy-bridized carbon, and more recently of topological insula-tors, both of which support such massless Dirac fermions,brought this issue into sharp focus. In particular, whichphysically measurable quantities are modified from theirnon-interacting values, and by how much, would al-low deeper understanding of the physics governed byelectron-electron interactions in these systems.When weak, the unscreened 1 /r Coulomb interactionsare expected to modify the velocity of the Dirac fermionsas v F → v F + e ln Λ /k , where k is the wavenumbermeasured from the Dirac point. This modification ofthe electronic dispersion is expected to lead to logarith-mic suppression of the density of states near the Diracpoint, an effect in principle observable in tunneling exper-iments. In addition, the low temperature electronic con-tribution to the specific heat should be suppressed from T to T / log T , as shown in Ref. 4, and the strength ofthis suppression is related to the strength of the Coulombinteraction.The role of Coulomb interaction in a.c. electrical trans-port was investigated by Mishchenko in Ref. 7, who orig-inally concluded that the a.c. conductivity σ ( ω ) van-ishes as ω → insulator .Were this the case, the interactions would have dra-matic effect on transport since the a.c. conductivityof the non-interacting system is finite, i.e., σ ( ω ) = πe / h for ω ≫ T . This was later argued to be in-correct by Sheehy and Schmalian , and independentlyby the present authors using Renormalization Group(RG) scaling analysis. While the former presented onlya scaling argument, without calculating the correction totransport, the latter reported on an explicit calculation where σ ( ω ) = σ C e v F + e ln Λ ω ! (1)with the coefficient found to be C = (25 − π ) / ≃ . e does not renormalize, anychange in the cutoff in the above expression for the con-ductivity may be compensated by a redefinition of theFermi velocity, v F . At small ω the correction vanishesand the non-interacting value of σ is recovered. At smallbut finite frequencies, the correction scales as 1 / | log ω | ,with the interaction independent prefactor determined by C . The numerical value of this correction, which canbe understood as correction to scaling near the Gaus-sian fixed point and which is expected to be universal,has since been a subject of debate. In subsequent work,Mishchenko recovered the functional form (1), whichgives metallic conductivity at small ω , but argued fora different value of C = (19 − π ) / ≃ . /r interac-tion, and argued to be the same regardless of whetherit is calculated using Kubo formula (current-current cor-relator) or continuity equation and density-density cor-relator. The same calculational procedure was later ad-vocated by Sheehy and Schmalian, who argued thatunlike hard-cutoff in momentum space, cutoff on the in-teraction leads to expressions obeying Ward-Takahashiidentity. In addition, they claimed the result obtained insuch way is consistent with the experimentally measuredoptical conductivity, where, surprisingly, no discerniblecorrection to the non-interacting value was reported. In quantum field theories, it seems reasonable that ifwo ultraviolet (UV) regularization schemes give differ-ent results for physical quantities, then the regulariza-tion that is typically chosen is the one which respectscharge U(1) symmetry, as is the case for chiral anomalyin (3+1)-dimensional massless quantum electrodynam-ics (QED), for instance. Here we argue that the reg-ularization of the electron-electron interaction alone isincomplete and cannot serve as a consistent regulariza-tion of the theory. We also show by explicit calcula-tion that the dimensional regularization used here pre-serves the Ward-Takahashi identity, i.e., that it is con-sistent with U (1) gauge symmetry of the theory, andthat, moreover, has the additional advantage of serving asan interaction-independent regularization scheme for thewhole field theory. The interaction correction to the con-ductivity within this regularization scheme is calculatedindependently using the current-current and the density-density correlators, which both yield the same number C = (11 − π ) / ≃ . U (1) gauge symmetry.Furthermore, we show that while the hard-cutoff regular-ization in principle violates the Ward-Takahashi identity,the original integral expression for the constant C is infact UV convergent, and when computed with necessarycare it unambiguously leads to the same value as quotedabove.A comparison with experiment which has been per-formed at high frequencies near the cutoff (see alsoRef. 22) may be misleading, since the result for the lead-ing logarithmic correction to the conductivity in Eq.(1)is valid only at frequencies of the order of 1meV, much smaller than the cutoff. As the Coulomb coupling con-stant in graphene e /v F is believed to be of order one, weexpect that in this region the interaction corrections todifferent observables, relative to the values in the nonin-teracting theory, should be significant. Why the interac-tion correction to the conductivity, in particular, appearsto be small even in the high-frequency regime is unclearat the moment.Whereas the results in the collisionless limit ( ω ≫ T )discussed here at least in principle follow from a straight-forward application of the perturbative renormalizationgroup, transport in the collision-dominated regime ( ω ≪ T ) requires re-summation of an infinite series of Feyn-man diagrams. This is easily seen in the non-interactinglimit where a finite temperature T produces a finite, lin-ear in T , ”Drude” δ − function response in conductivity, ∼ T δ ( ω ). Collisions due to the electron-electron scat-tering lead to broadening of the δ − function and clearlythe result must be non-analytic in e /v F as the interac-tion V ( r ) →
0. Alternative approach has been advancedin Refs. 23,24 where the leading correction is argued tobe captured by the solution to the quantum Boltzmanequation with the collision integral calculated perturba-tively in the interaction strength. In the clean limit, theconductivity in the collision dominated regime is found to increase with decreasing T and proportional to ln ( T /
Λ).Interestingly, experiments on suspended samples at the neutrality point find conductivity which decreases withdecreasing T . Finally, T − linear increase of the d.c. con-ductivity observed in small devices , has been argued toarise from purely ballistic transport where conductiv-ity grows with the sample size L and temperature T as σ ∼ T L/ ~ v F .The paper is organized as follows: in section II we in-troduce the Lagrangian and the response functions, andin section III we discuss different regularization schemesfor massless Dirac fermions. In section IV, we reviewsome well-known results regarding the polarization ten-sor and the conductivity. In section V, we explicitlyconstruct polarization tensor for the non-interacting the-ory, and in section VI we consider the same problem forthe contact interactions to first order in the interactionstrength and to O ( N ). We do not discuss the (RPA-like) contribution to the order N which, while simpleto calculate, does not contribute to transport. The mainresults of the paper are presented in section VII wherewe show that the Coulomb correction to the polarizationtensor is transversal, as well as that the Coulomb ver-tex function obeys the Ward-Takahashi identity withinthe dimensional regularization. In this section, we alsopresent calculations of the Coulomb correction to thea.c. conductivity using both the current-current corre-lator (Kubo formula) and the density-density correlator,within the dimensional regularization. Section VIII isreserved for further discussion of these results and com-parison with previous results reported in the literature.Various technical details of the calculations are presentedin the appendices. II. HAMILTONIAN, LAGRANGIAN AND THERESPONSE FUNCTIONS
We start with the Hamiltonianˆ H = Z d D r ψ † ( r ) v F σ a p a ψ ( r )+ Z d D r d D r ′ ψ † ( r ) ψ ( r ) V ( | r − r ′ | ) ψ † ( r ′ ) ψ ( r ′ ) , (2)where we consider N copies of two-component Fermifields ψ ( r , τ ) (which therefore has 2 N components), themomentum operator p a = − i ~ ∂ a and σ a are Pauli ma-trices. Operators in the interaction term are assumednormal ordered. Hereafter, the Latin letters a, b areused only for the spatial indices, while the Greek letters µ, ν are reserved for the spacetime ones, and summationover the repeated indices is assumed. V ( r ) is the two-body interaction potential, which is left unspecified atthe moment. Later we will consider two different cases: ashort-range contact interaction V ( r ) = uδ ( r ) and the 3DCoulomb potential V ( r ) = e /r with e /v F as the dimen-sionless Coulomb coupling constant. To simplify the no-tation, we will work in the natural units ~ = c = k B = 1.When the speed of light c does not appear, we will alsoet v F = 1. In our final results we will restore the physi-cal units.The corresponding imaginary time Lagrangian is L = L + L int (3)where L = Z d D r ψ † ( τ, r ) (cid:20) ∂∂τ + v F σ · p (cid:21) ψ ( τ, r ) (4)and L int = 12 Z d D r d D r ′ ρ ( τ, r ) V ( | r − r ′ | ) ρ ( τ, r ′ ) , (5)where ρ ( τ, r ) ≡ ψ † ( τ, r ) ψ ( τ, r ) is the density of fermions.The quantum partition function can then be written asthe imaginary time Grassman path integral Z = Z D ψ † D ψ exp − Z β dτ L ! (6)where the inverse temperature factor β = 1 / ( k B T ). Inthe sections which follow, the additional imaginary-timeindex on the Fermi field ψ ( τ, r ) inside a path integralautomatically means that they are considered to be co-herent state Grassman fields. We will take T → T → Specifically, for somebosonic operator ˆ O a ( t, r ) in the (real time) Heisenbergrepresentation, the retarded correlation function S retab ( t − t ′ , r , r ′ ) = − iθ ( t − t ′ ) h h ˆ O a ( t, r ) , ˆ O b ( t ′ , r ′ ) i i (7)can be related to the imaginary time ordered correlationfunction S ab ( τ − τ ′ , r , r ′ ) = −h T τ ˆ O a ( τ, r ) ˆ O b ( τ ′ , r ′ ) i , (8)where ˆ O a ( τ, r ) = e β ˆ H ˆ O a ( r ) e − β ˆ H . (9)In the Eqs.(7-8) the angular brackets denote thermal av-eraging h . . . i = 1 Z Tr (cid:16) e − β ˆ H . . . (cid:17) . (10)Specifically, the frequency Fourier transforms S retab ( ω ; r , r ′ ) = Z ∞−∞ dte i Ω t S retab ( t, r , r ′ ) , (11) S ab ( i Ω n ; r , r ′ ) = Z β dτ e i Ω n τ S ab ( τ, r , r ′ ) , (12) satisfy S retab (Ω; r , r ′ ) = S ab ( i Ω n → Ω + i + ; r , r ′ ) , (13)where the bosonic Matsubara frequency is Ω n = 2 πn/β for n = 0 , ± , ± , . . . . We will use the above relations inwhat follows when we focus on the electrical conductivity,in which case the bosonic operator ˆ O of interest will beeither charge density or charge current.For completeness we note that for V ( r ) = 0 the two-particle imaginary time Green’s function is h ψ ( iω, k ) ψ † ( iω ′ , k ′ ) i = βδ ω,ω ′ (2 π ) δ ( k − k ′ ) G k ( iω )(14)where G k ( iω ) = iω + σ · k ω + k , (15)which will be used extensively in the later sections.Strictly speaking, in any solid state system which sup-ports massless Dirac particles, the above propagator isvalid only for wavevectors smaller than some cutoff Λ,which depends on the physical situation. In the case ofelectrons on the honeycomb lattice, the order of magni-tude of the cutoff, ˚ A − , is determined by the requirementthat the true electronic dispersion does not deviate ap-preciably from the conical (Dirac). III. REGULARIZATION SCHEMES FORMASSLESS DIRAC FERMIONS
Since we are interested in the long distance (low fre-quency) behavior of physical quantities, we can use theabove low energy field theory, given by the above La-grangian with the corresponding propagators, providedthat divergent terms in the perturbation theory are prop-erly regularized. In the context of high energy physics itis also well known that a quantum field theory of Diracfermions needs to be regularized, and typically thereis no unique way of doing so. Additional requirements,usually based on the symmetries of the theory, determinewhat type of regularization should be employed.In case of the theory of the Coulomb interacting Diracfermions, we will require that the U (1) gauge symmetrymust be preserved, or equivalently, that the charge mustbe conserved. As we show below, dimensional regulariza-tion introduced by ’t Hooft and Veltman is consistentwith this requirement. Before discussing this regulariza-tion scheme, let us briefly review the hard cutoff and thePauli-Villars regularization schemes in the context of thefermionic field theory considered here. A. Hard cutoff
The idea of the hard cutoff regularization is to imposea cutoff in the upper limit of an otherwise divergent mo-mentum integral. Physically, this is due to the k − spaceestriction on the modes which appear in the theory, acondition which appears naturally within Wilson formu-lation of the RG. The singular part of the integral thenappears dependent on the cutoff scale. Although verysimple to implement, this regularization scheme is knownto violate U (1) gauge symmetry of QED, for instance. Terms that violate the gauge symmetry appear as apower of the cutoff scale and must be subtracted in or-der to insure that the gauge symmetry is preserved. Onthe other hand, the typical divergent terms appear as thelogarithm of the cutoff scale. Of course, the cutoff scalemust not appear explicitly in any observable quantity inorder for the theory to be physically meaningful. Thedisappearance of the cutoff scale Λ indeed occurs in thecalculation of the interaction correction to the a.c. con-ductivity within quantum field theory of the Coulombinteracting Dirac fermions, as discussed below Eq. (1).However, as we show in Appendix D, and as was antic-ipated in Ref.18, the hard-cutoff regularization violatesthe Ward-Takahashi identity. We are thus led to concludethat this regularization scheme is in principle not consis-tent with U (1) gauge symmetry of the theory. This con-clusion notwithstanding, the particular coefficient C fromthe introduction may be written as an integral which isunambiguous and perfectly convergent in the upper limit,provided the momentum cutoff is taken to infinity after all the integrals have been performed (see Appendix H). B. Pauli-Villars regularization
Another way to regularize divergent self-energy andvertex diagrams in QED is to introduce an additionalartificial ”heavy photon”. In Euclidean spacetime thisleads to the following replacement of the photon propa-gator 1Ω + k → + k − + k + M and the mass parameter M is sent to ∞ at the end of thecalculation. Since the additional fictitious particle cou-ples minimally to the fermions, the regularization pre-serves Ward-Takahashi identities which relate the self-energy to the current vertex. However, as such, this reg-ularization is unable to render photon polarization dia-grams finite. This can be avoided by introducing addi-tional Pauli-Villars fermions, at the expense of makingthe method complicated. In the context of the (2+1)D massless Dirac fermionsinteracting with static (non-retarded) 1 /r Coulomb in-teraction, the analog of the Pauli-Villars regularizationis 1 | k | → | k | − √ k + M . Physically, this corresponds to cutting-off the short-distance divergence of the 1 /r interaction, without af-fecting its long range tail. This modified interaction pre-serves Ward-Takahashi identities relating vertex and the self-energy, but, just as in the case of QED, it fails toregularize the polarization function without introducingadditional Pauli-Villars fermions. Therefore, as such itcannot serve as a complete regularization of the theory. C. Dimensional regularization
Originally introduced in the context of relativisticquantum field theory, the basic idea of the dimensionalregularization is to regularize four-momentum integralsby lowering the number of spacetime dimensions overwhich the integral is performed. This procedure was in-troduced by ’t Hooft and Veltman to preserve the sym-metries of gauge theories. It also bypasses the necessityto introduce Pauli-Villars fermions and bosons.Here we employ a variant of the dimensional regular-ization scheme in that the frequency integrals are per-formed from −∞ to + ∞ while the momentum integralsare analytically continued from D = 2 to D = 2 − ǫ di-mensions. Such separation of time from space is usedbecause in the case considered here the Lorentz invari-ance is violated by the interaction terms. A momentumintegral is therefore calculated for an arbitrary numberof dimensions D , and expanded in the parameter ǫ . Sin-gular parts of the integral then appear as the first-orderpoles in the Laurent expansion over the parameter ǫ , i.e.,as terms of the form 1 /ǫ , and the finite part is the termof order ǫ in this expansion.The following D -dimensional (Euclidean) integrals arefrequently encountered in this regularization scheme Z d D ℓ (2 π ) D ℓ + ∆) n = Γ (cid:0) n − D (cid:1) (4 π ) D/ Γ( n ) 1∆ n − D , (16)and Z d D ℓ (2 π ) D ℓ ( ℓ + ∆) n = 1(4 π ) D/ D (cid:0) n − D − (cid:1) Γ( n ) × n − D − , (17)where Γ( x ) is the Euler gamma function and ∆ ≥ D = 2 − ǫ -dimensional space. We thus use the follow-ing identity σ a σ µ σ a = Dδ µ + (2 − D ) σ a δ aµ , (18)where the sum over the Latin letters a, b , used only forthe spatial indices, is assumed. The Greek letters µ, ν are reserved for the spacetime indices. The last term onthe right-hand side turns out to be crucial for the proofof the Ward-Takahashi identity, guaranteed by the U (1)charge conservation. This is discussed in later sections.In short, the last term in Eq.(18) yields the last term inEq. (A13). If the latter were omitted the Ward-Takahashiidentity would be violated. As elaborated on in the dis-cussion section, the same term also accounts for the dis-crepancy between the results found in this work, Eqs.82), and the result we found previously [Eq. (G29)] forthe Coulomb interaction correction to the conductivitywhere the last term was omitted. Details of this calcula-tion can be found in Appendix E. IV. CONSERVATION LAWS, CONDUCTIVITY,AND THE STRUCTURE OF THEPOLARIZATION TENSOR
In the interest of self-containment, in this section wereview some well known results regarding response func-tions and U (1) conservation laws. Most of these resultscan be found (scattered) in many body – quantum-fieldtheory textbooks. In order to calculate the response functions to externalelectro-magnetic fields, it is useful to define the imaginarytime correlation functionΠ µν ( τ, r ) = h T τ j µ ( τ, r ) j ν (0 , i , (19)where the current ”three-vector” j µ is composed of theimaginary time density and current as j ( τ, r ) = ( ρ ( τ, r ) , j ( τ, r ))= (cid:0) ψ † ( τ, r ) ψ ( τ, r ) , v F ψ † ( τ, r ) ~σψ ( τ, r ) (cid:1) . (20)In this section we temporarily restore v F to clearly dis-tinguish it from the speed of light c used below.By fluctuation-dissipation theorem, the expectationvalue of the electrical current-density operator J ( t, r ), inreal time t , is related to the imaginary time correlatorΠ µν ( i Ω , q ). The latter is the Fourier transform (12) ofthe tensor defined in Eq.(19). The expectation value ofthe Fourier transform of the electrical current-density isthen h J a (Ω , q ) i = − e ~ Π a ( i Ω n → Ω + i , q )Φ(Ω , q )+ e ~ Π ab ( i Ω n → Ω + i , q ) A b (Ω , q ) c . (21)The Fourier components of the electric and magneticfields are related to the ones of the scalar and vectorpotentials as E a (Ω , q ) = i Ω c A a (Ω , q ) − iq a Φ(Ω , q ) , (22) B (Ω , q ) = iǫ ab q a A b (Ω , q ) , (23)where ǫ ab is completely antisymmetric (Levi-Civita) ranktwo tensor. Using Faraday’s law of induction we canfurther relate the Fourier components of the electric andmagnetic fields as ǫ ab q a E b (Ω , q ) = Ω c B (Ω , q ) . (24)In condensed matter systems with massless Dirac parti-cles, propagating with velocity v F , as the relevant low-energy degrees of freedom considered here, the (pseudo) Lorentz invariance is violated by interactions. If we wereto consider finite temperature T the (pseudo) Lorentzinvariance would be violated even in the non-interactinglimit. Nevertheless, when spatial O (2) rotational invari-ance is preserved, as is the case for problems studied here,the general structure of the imaginary time polarizationtensor is Π µν ( i Ω n , q ) = Π A ( i Ω n , | q | ) A µν + Π B ( i Ω n , | q | ) B µν (25)where the three-tensors are B µν = δ µa (cid:18) δ ab − q a q b q (cid:19) δ bν (26) A µν = g µν − q µ q ν q − B µν . (27)The Euclidean three-momenta appearing in the abovetensors are g µν = diag[ − , , µν , (28) q µ = g µν ( − i Ω n , q ) ν = ( i Ω n , q ) µ , (29) q = q µ g µν q ν = Ω n + q . (30)The real time continuity equation ∂∂t ρ + ∇ · J = 0 (31)requires that, with our choice of the imaginary time”three”-current j = ( ρ,~j ), the transversality of theΠ µν ( i Ω , q ) is equivalent to the condition( − i Ω , q ) µ Π µν ( i Ω , q ) = Π µν ( i Ω , q ) ( − i Ω , q ) ν = 0 . (32)Note that this is explicitly satisfied by the expression(25). If, in addition, the Lorenz invariance is satisfied,Π A = Π B , and there is no need to separate out the spa-tially transverse component of the polarization tensor. A. Ward-Takahashi identity and vertex functions
In addition to the condition (32), the continuity equa-tion (31) constrains the form of the vertex function. Ifwe define the four-point matrix function π µ ( r ′ − r , τ ′ − τ ; r − r ′′ , τ − τ ′′ ) = h T τ j µ ( τ, r ) ψ ( τ ′ , r ′ ) ψ † ( τ ′′ , r ′′ ) i (33)where the imaginary time ”three”-current was defined inEq.(20), then we must have (cid:18) ∂∂τ , ∇ i (cid:19) µ π µ ( τ ′ − τ, r ′ − r ; τ − τ ′′ , r − r ′′ ) = (cid:0) δ ( τ − τ ′′ ) δ D ( r − r ′′ ) − δ ( τ ′ − τ ) δ D ( r ′ − r ) (cid:1) ××G ( τ ′ − τ ′′ , r ′ − r ′′ ) . (34)The above expression relates the exact imaginary timefour-point function to the exact imaginary time Green’sfunction G ( τ, r ) = h T τ ψ ( τ, r ) ψ † (0 , i . (35)f we rewrite the Fourier transform of π µ in terms of thevertex function Λ µ as π µ ( k , iω ; k + q , iω + i Ω) = G k ( iω )Λ µ ( k , iω ; k + q , iω + i Ω) G k + q ( iω + i Ω)(36)then the Ward-Takahashi identity for the vertex functionΛ µ can be written as( − i Ω , q ) µ Λ µ ( k , iω ; k + q , iω + i Ω) = G − k + q ( iω + i Ω) − G − k ( iω ) = Σ k + q ( iω + i Ω) − Σ k ( iω ) . (37)This identity has to be satisfied order by order in per-turbation theory. In what follows, we show that this isindeed the case for the interacting theories studied herewhen we adopt the dimensional regularization. B. Electrical conductivity
To relate the polarization tensor to the electrical con-ductivity, we simply need to relate the expectation valueof the current to the electric field. Since we have theresponse to the electromagnetic scalar and vector poten-tials, we just need to relate those to the electric andmagnetic fields. Finally, magnetic field can be relatedto the electric field using Maxwell’s equations. As is wellknown, at finite wavevector q and frequency Ω, one candefine the logitudinal and transverse conductivity as theproportionality between the induced current and the lon-gitudinal or transverse component of the electric field.Using Eqs. (21,25-27), we find h J a (Ω , q ) i = e ~ Π A (Ω + i , | q | ) Ω q a q − Ω Φ(Ω , q ) − e ~ Π A (Ω + i , | q | ) Ω q a q b q ( q − Ω ) A b (Ω , q ) c + e ~ Π B (Ω + i , | q | ) (cid:18) δ ab − q a q b q (cid:19) A b (Ω , q ) c . (38)Furthermore, Eqs. (22)-(24) imply h J a (Ω , q ) i = e ~ Π A (Ω + i , | q | ) i Ω q − Ω q a q b q E b (Ω , q )+ e ~ Π B (Ω + i , | q | ) 1 i Ω (cid:18) δ ab − q a q b q (cid:19) E b (Ω , q ) . (39)From the above equations we can read off the longitudinaland transverse electrical conductivity σ k (Ω , | q | ) = e ~ i Ω q − Ω Π A (Ω + i , | q | ) , (40) σ ⊥ (Ω , | q | ) = e ~ Π B (Ω + i , | q | ) i Ω . (41) For q = 0 (and Ω = 0), σ k need not be equal σ ⊥ . How-ever, at q = 0, the a.c. conductivities σ k (Ω , q = 0) = σ ⊥ (Ω , q = 0) (42)due to the O (2) spatial rotational symmetry.In the following, we will work solely in the imaginarytime – Matsubara frequency space, and since we restrictourselves to T = 0, we will drop the subscript n on i Ω n . V. NON-INTERACTING LIMIT: V ( r ) = 0 For the sake of completeness, and in order to illustratehow the general results presented in the previous sectionappear in the specific solvable problem, we first examineΠ µν ( i Ω , q ) in the limit of vanishing V ( r ). The Fouriertransform (12) of the polarization function (19) in thenon-interacting limit is easily shown to beΠ (0) µν ( i Ω , q ) = − N Z ∞−∞ dω π Z d D k (2 π ) D Tr[ G k ( iω ) σ µ G k + q ( iω + i Ω) σ ν ](43)where σ is the 2 × P µ ( q , i Ω) = Z ∞−∞ dω π Z d D k (2 π ) D G k ( iω ) σ µ G k + q ( iω + i Ω) , (44)in terms of whichΠ (0) µν ( i Ω , q ) = − N Tr[ P µ ( q , i Ω) σ ν ] . (45)The above expression is divergent at large momenta (UVdivergent) as is easily seen by counting powers. Note thatthis appears even in the non-interacting theory when cal-culating the response functions. As is well known in thecontext of relativistic field theories, this UV divergence isunphysical and to obtain the correct answer a regulariza-tion is necessary. The regularization of choice here isdimensional regularization which leads to finite expres-sions and which is consistent with U (1) gauge symmetryof the theory.As shown in detail in the Appendix A, using dimen-sional regularization, we obtain P µ ( q , i Ω) = p Ω + q × (cid:20) σ µ − δ µ − ( i Ω + σ · q ) σ µ ( i Ω + σ · q )Ω + q (cid:21) . (46)Performing the trace in Eq. (45) we findΠ (0) µν ( i Ω , q ) = − N p Ω + q − q − i Ω q x − i Ω q y − i Ω q x q y + Ω − q x q y − i Ω q y − q x q y q x + Ω µν . (47)e can write the above matrix more compactly asΠ (0) µν ( i Ω , q ) = − N (cid:18) g µν − q µ q ν q (cid:19) p Ω + q (48)where we used definitions from Eqs.(28-30). The correla-tion function (47-48) is explicitly transverse, as it shouldbe, and( − i Ω , q ) µ Π (0) µν ( i Ω , q ) = Π (0) µν ( i Ω , q ) ( − i Ω , q ) ν = 0 . (49)From the above equations we findΠ (0) A ( i Ω , q ) = Π (0) B ( i Ω , q ) = − N p Ω + q . (50)Analytically continuing according to Eqs. (40-41), withthe branch-cut of the √ z -function lying along negativereal axis, we find the well-known expression for the(Gaussian) a.c. conductivity σ (0) k (Ω) = σ (0) ⊥ (Ω) = N e ~ . (51)As a side remark, if we were to define ˜Π (0) µν as a cor-relation function of a slightly different ”three”-current( − iρ,~j ), the result obtained directly from Eqs. (47-48)transforms as tensor under Euclidean O (3) transforma-tions. In real frequencies this is equivalent to relativis-tic Lorentz transformations, due to the invariance of thenon-interacting Lagrangian L .Therefore, regulating the UV divergences via dimen-sional regularization implemented here leads to finite ex-pressions which preserve the required U (1) conservationlaws. The necessary regularization of the ”integrationmeasure”, as done here via dimensional regularization,is independent of the electron-electron interaction V ( r ),as it must be if the non-interacting theory is to lead tofinite correlation functions. Therefore, as shown alreadyby this example, regulating only the ”momentum trans-fer” as advocated in Refs. 10,18 is clearly insufficient. VI. SHORT RANGE INTERACTIONS: V ( r ) = uδ ( r ) While the problem of (2+1)D massless Dirac fermionswith the contact interactions is not exactly solvable, one can calculate the interaction corrections to the polariza-tion tensor perturbatively in powers of the interactionstrength u . Such contact interactions certainly consti-tute an idealized special case. Nevertheless, this theoryhas the advantage that one can determine the first correc-tion in u to the non-interacting (Gaussian) polarizationtensor Π (0) µν , found in the previous section, explicitly forfinite q and Ω. We can then test the general symme-try requirements listed before. The technique of choiceis again the (variant of the) dimensional regularizationof Veltman and ’t Hooft introduced in section III. Sincethis interaction violates Lorentz invariance we can alsouse this example to study how the difference betweenΠ A and Π B arises in such theory.It is straightforward to use the Wick’s theorem to showthat in this case, the first order in u , and to O ( N ), cor-rection to the polarization tensor is δ Π µν ( i Ω , q ) = uN Z d D k (2 π ) D Z ∞−∞ dω π Z d D p (2 π ) D Z ∞−∞ dω ′ π { Tr[ G k ( iω ) σ µ G k + q ( iω + i Ω) G p ( iω ′ ) σ ν G p − q ( iω ′ − i Ω)]+ Tr[ G k ( iω ) σ µ G k + q ( iω + i Ω) G p ( iω ′ ) G k + q ( iω + i Ω) σ ν ]+ Tr[ G k ( iω ) σ µ G k + q ( iω + i Ω) σ ν G k ( iω ) G p ( iω ′ )] . } (52)The last two terms correspond to the self-energy correc-tion, while the first one is the vertex correction. Becausethe self-energy for the contact interaction vanishes, Z d D k (2 π ) D Z ∞−∞ dω π G k ( iω ) = 0 , (53)the last two terms in the Eq.(52) vanish as well. Theremaining term can be written rather succinctly in termsof P µ defined previously in Eq.(44) as δ Π µν ( i Ω , q ) = uN Tr [ P µ ( q , i Ω) P ν ( − q , − i Ω)] . (54)The above expression is manifestly transverse, i.e., it sat-isfies Eq.(32), as can be seen from Eq.(46).Namely, inserting Eq. (46) and performing the traces we find δ Π µν ( i Ω , q )= uN + q ) q ( q − Ω ) i Ω q x ( q − Ω ) i Ω q y ( q − Ω ) i Ω q x ( q − Ω ) q x ( q y − Ω ) + ( q y + Ω ) − q x q y (3Ω + q ) i Ω q y ( q − Ω ) − q x q y (cid:0) + q (cid:1) q x − Ω q y + Ω + q x ( q y + 2Ω ) µν . Finally, the above tensor can be factorized as given by Eqs. (25-26), and we find to first non-trivial order in theontact coupling u Π A ( i Ω , | q | ) = − N p Ω + q + uN
512 (Ω − q ) + O ( u ) , Π B ( i Ω , | q | ) = − N p Ω + q + uN
512 (Ω + q ) + O ( u ) . (55)Expectedly, the above expression shows that the inter-action correction to the polarization functions Π A andΠ B are different (note the sign difference in front of q ).As stated above, the reason for the difference is that thecontact density-density interaction term u ( ψ † ( r ) ψ ( r )) breaks the Lorentz invariance of the non-interacting partof the Lagrangian. Lorentz transformations in generalrotate between density and current and we have purpose-fully omitted any current-current interaction.We can further test the Ward-Takahashi identity (37)for the vertex function (36) in this example with the shortrange interactions. It can be readily seen that the firstorder in u correction to the vertex vector is δ Λ µ ( k , iω ; k + q , iω + i Ω) = − u P µ ( q , i Ω) . (56)It follows from the Eq. (46) that − i Ω P ( q , i Ω) + q a P a ( q , i Ω) = 0 . (57)Therefore the Ward-Takahashi identities (37) are satis-fied, since, as mentioned previously in this section, theself-energy correction vanishes to first order in u for theshort-range interactions.Finally, from Eqs. (40-41), we can infer that the aboveterms correct only the imaginary part of the a.c. conduc-tivity, but not the real part. At q = 0, correction is thesame for the longitudinal and the transverse components,and to this order in u we have σ k , ⊥ (Ω) = e ~ N (cid:16) − i u
32 Ω (cid:17) . (58)Again, the equality between σ k (Ω) and σ ⊥ (Ω) is guaran-teed due to the O (2) rotational invariance of this theory.Note also that the fact that the interaction correctionis proportional to the frequency is implied by the powercounting at the Gaussian fixed point of the theory, andis characteristic for any finite-range interaction. VII. COULOMB INTERACTION: V ( r ) = e / | r | Armed with the above results we now focus on themain part of the paper where we study the effects ofthe Coulomb interaction. Unlike in the previous cases,we have been unable to find the explicit expression forthe first order correction to the polarization tensor at fi-nite q and Ω. Nevertheless, we have been able to showexplicitly that the first order correction to the polariza-tion tensor is transverse, i.e., it satisfies Eq.(32). Thisis shown using dimensional regularization in D = 2 − ǫ introduced in section III. Next, we study the first cor-rection to the Coulomb vertex function which must alsosatisfy the Ward-Takahashi identity (37). Since in thiscase the first order self-energy is known to diverge loga-rithmically, the first order correction to the vertex func-tion should also diverge as ǫ →
0. This can be foundexplicitly in terms of elliptic integrals to order ǫ − and ǫ and the identity (37) is also explicitly confirmed. Fi-nally, we proceed with the calculation of the electricalconductivity, first by using the spatial component of thepolarization tensor at q = 0 but finite Ω (current-currentcorrelation function), and then by using time componentof the polarization tensor at finite but small q and finiteΩ. The final results for the conductivity calculated inboth ways are found to be the same. Specifically, we find C = (11 − π ) / V ( r ) = e /r the effect of screening due to dielectric medium is easilytaken into account by rescaling e in the above formula.The O ( e ) and O ( N ) correction to the polarization func-tion is then δ Π ( c ) µν ( i Ω , q ) = N Z d D k (2 π ) D Z ∞−∞ dω π Z d D p (2 π ) D Z ∞−∞ dω ′ π { V p − k × Tr [ G k ( iω ) σ µ G k + q ( iω + i Ω) G p + q ( iω ′ + i Ω) σ ν G p ( iω ′ )]+ V k − p × Tr [ G k ( iω ) σ µ G k + q ( iω + i Ω) G p + q ( iω ′ + i Ω) × G k + q ( iω + i Ω) σ ν ]+ V k − p × Tr [ G k ( iω ) σ µ G k + q ( iω + i Ω) σ ν G k ( iω ) G p ( iω ′ )] } (59)where V k = Z d r e i k · r V ( r ) = 2 πe | k | . (60)Just as in the case of contact interactions, the first termin the expression for δ Π ( c ) µν corresponds to the vertex cor-rection and the last two terms to the self-energy correc-tions. Unlike in the case of contact interactions, however,the self energy correction does not vanish. The expres-sion (59) will be used in later sections as a starting pointin the calculation of the Coulomb interaction correctionto the a.c. conductivity in the collisionless regime. A. Proof of the transversality of δ Π ( c ) µν withindimensional regularization Because, as mentioned above, the explicit evaluationof (59) at finite q and Ω yields intractable expressions,we proceed by first showing that (59) is transverse, i.e.,that it satisfies the condition (32), when dimensional reg-ularization employed in this paper is used. As such ittherefore does not lead to any violation of the chargeonservation, a virtue questioned in Ref. 18. By 2D rota-tional invariance, this in turn implies that the Coulombpolarization tensor can be written in the form (25).To prove (32) we follow Ref. 18 and use − i Ω σ + q · σ = G − k + q ( iω + i Ω) − G − k ( iω ) (61)to find( − i Ω , q ) µ δ Π ( c ) µν ( i Ω , q ) = N Z d D k (2 π ) D d D p (2 π ) D Z ∞−∞ dω π dω ′ πV p − k { Tr [ G k ( iω ) σ ν G k ( iω ) G p ( iω ′ )] − Tr [ G k + q ( iω + i Ω) σ ν G k + q ( iω + i Ω) G p + q ( iω ′ + i Ω)] } . At this point it is not immediately obvious that we canshift the integration variables k and p in the second termby q , which if true would readily yield the desired rela-tion (32), since the frequency integral can be shifted. Wetherefore define a function of frequency and two momen-tum variablesΣ p , q ( i Ω) = Z d D k (2 π ) D Z ∞−∞ dω ′ π V k − p G k + q ( iω ′ + i Ω) (62)in terms of which we have unambiguously( − i Ω , q ) µ δ Π ( c ) µν ( i Ω , q )= N Z d D k (2 π ) D Z ∞−∞ dω π { Tr [ G k ( iω ) σ ν G k ( iω )Σ k , (0)] − Tr [ G k + q ( iω + i Ω) σ ν G k + q ( iω + i Ω)Σ k , q ( i Ω)] } . To continue, we need to find an explicit expression forΣ p , q ( i Ω). Using the identity (16), Feynman parametriza-tion 1 A α B β = Γ( α + β )Γ( α )Γ( β ) Z dy y α − (1 − y ) β − [ yA + (1 − y ) B ] α + β , (63)for α = β = 1 /
2, and Z dyy α − (1 − y ) β − = Γ( α )Γ( β )Γ( α + β ) , (64)we findΣ p , q ( i Ω) = e (4 π ) D σ · ( p + q ) | p + q | − D Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) Γ( D ) , (65)which agrees with Eq.(12) of Ref. 35. Note that thisidentity shows that within dimensional regularization,Σ p , q ( i Ω) = Σ p + q , ( i Ω) . Moreover, in what follows, thereis no need to shift the integration variable. Rather, sincethe commutator of the self-energy and the Green’s func-tion vanishes,[ G k + q ( iω + i Ω) , Σ p , q ( i Ω)] = 0 , (66)after a straightforward use of the cyclic property of thetrace and the identity Z ∞−∞ dω π G k + q ( iω + i Ω) G k + q ( iω + i Ω) = 0 , (67) we prove that( − i Ω , q ) µ δ Π ( c ) µν ( i Ω , q ) = 0 . (68)The same procedure as the one used above also leads to δ Π ( c ) µν ( i Ω , q )( − i Ω , q ) ν = 0 . (69)This proof holds to all orders of ǫ . The regularizationtechnique implemented here is therefore perfectly ade-quate and does not lead to violation of the charge con-servation. B. Coulomb vertex and the proof of theWard-Takahashi identity
Next, we will demonstrate that the dimensional regu-larization used here preserves the Ward-Takahashi iden-tity for the Coulomb vertex function. This proof is tech-nically more involved than the proof in the previous sec-tion, but nevertheless, we find it important to presentits details since our technique is not widely used in thecommunity. We show the desired identity to order ǫ − and ǫ . Most of the technical details are presented in theappendices, and in this section we just present the mainsteps of the derivation.The Coulomb vertex function to the first order in thecoupling constant is δ Λ µ ( p , iν ; p + q , iν + i Ω) = P cµ ( q , p , i Ω) = − Z d D k (2 π ) D Z ∞−∞ dω π V p − k G k ( iω ) σ µ G k + q ( iω + i Ω) . (70)The integrals on the right are logarithmically divergentin D = 2 as can be easily seen by powercounting. Thisdivergence is related to the divergence of the electronself-energy, calculated in the previous section,Σ k ( iω ) ≡ Σ k , ( iω ) = e σ · k × (cid:18) ǫ − γ + ln 64 π − ln k + O ( ǫ ) (cid:19) , (71)where the Euler-Mascheroni constant γ = 0 .
577 and, asbefore, ǫ = 2 − D . Indeed, if the Ward-Takahashi identity,( − i Ω , q ) µ P cµ ( q , p , i Ω) = Σ p + q ( iν + i Ω) − Σ p ( iν ) , (72)is to be satisfied, the vertex function must diverge log-arithmically, which manisfests in the dimensional regu-larization as the first-order pole in Laurent expansion inthe parameter ǫ .In the second part of the Appendix A we use dimen-sional regularization to determine P c ( q , p , i Ω) to orders ǫ − and ǫ . Our final expression for finite q , p and i Ω,Eq. (A18), is left as an integral over a Feynman param-eter x . We wish to stress that all of the integrals in thisquation can be performed in the closed form in terms ofelliptic integrals. However, we found that doing so leadsto intractable and unrevealing expressions. We thereforechose to work with the expression (A18) and in effectmanipulate the integral representation of the elliptic in-tegrals. In the limiting case of q = 0, the vertex functionis determined in the closed form in Appendix A up to,and including, ǫ .In Appendix B we in turn find that the vertex function(A18) satisfies( − i Ω , q ) µ P cµ ( q , p , i Ω) = N ( p , q ) − e p Ω + q × σ · (cid:0) p (Ω + q ) L ( p , q , Ω) + q M ( p , q , Ω) (cid:1) . (73)Using the dimensional regularization, we then show thatthe function N ( p , q ) = e (cid:18) ǫ σ · q + (ln 64 π − γ ) σ · q − σ · ( p + q ) ln( p + q ) + σ · p ln p (cid:1) = Σ p + q ( iν + i Ω) − Σ p ( iν ) , (74)and, in Appendix C, that L ( p , q , Ω) = M ( p , q , Ω) =0. This proves the Ward-Takahashi identity to the firstorder in perturbation theory.
C. Calculation of the a.c. conductivity from thecurrent-current correlator (Kubo formula)
In this section we calculate the diagonal spatial com-ponent of the Coulomb interaction correction of the po-larization tensor, δ Π ( c ) xx , at q = 0 and finite i Ω. We thenuse this result to calculate the corresponding correctionto the electrical conductivity. Given the decomposition(25), one should in principle specify the direction in the q -plane along which the limit q → q x is taken to 0 before q y , δ Π ( c ) xx ( i Ω ,
0) is propor-tional to Π B ( i Ω , q y is taken to 0before q x , then δ Π ( c ) xx ( i Ω ,
0) is proportional to Π A ( i Ω , θ with the q x axis, then δ Π ( c ) xx ( i Ω) is proportionalto cos θ Π A ( i Ω ,
0) + sin θ Π B ( i Ω , O (2) rotational invariance, Π A ( i Ω ,
0) = Π B ( i Ω , θ . We can therefore useeither Eq. (40) or Eq. (41) along with diagonal spatialpart of the polarization tensor (59) to calculate the a.c.conductivity.We start by showing that δ Π ( c ) µν ( q = 0 , i Ω = 0) = 0 . (75)This is expected, since a space and time independentvector and scalar potential correspond to a pure gauge,and as such have no effect on the physics of the problem.Within our formalism, this identity can be shown to thefirst order in the Coulomb interaction by first performing the integral over the frequencies in Eq. (59), which, ascan be easily seen, yields δ Π ( c ) µν ( i Ω = 0 , q = 0) = N Z d D p (2 π ) D | p |× Tr (cid:20) P ( c ) µ (0 , p , (cid:18) σ ν − σ · p σ ν σ · pp (cid:19)(cid:21) + N × Z d D k (2 π ) D | k | Tr [( σ · k σ µ − σ µ σ · k ) (Σ k σ ν − σ ν Σ k )] . Substituting the expression for the self-energy (71) andthe static vertex (A20), performing the traces and using k a k b → δ ab k /D , we conclude that Eq.(75) holds, as itshould. We are therefore free to subtract it from theexpression for δ Π ( c ) µν at either finite Ω and/or finite q .Next, we set q = 0 and consider finite Ω in Eq. (59).The polarization tensor can be written as sum of the con-tributions from the self-energy correction and the vertexcorrection δ Π ( c ) µν ( i Ω ,
0) = δ Π ( a ) µν ( i Ω ,
0) + δ Π ( b ) µν ( i Ω , , (76)where the self-energy is given by δ Π ( a ) µν ( i Ω ,
0) = 2 N Z ∞−∞ dω π dω ′ π Z d D k (2 π ) D d D p (2 π ) D V k − p Tr [ G k ( iω ) σ µ G k ( iω + i Ω) σ ν G k ( iω ) G p ( iω ′ )] , (77)and the vertex correction reads δ Π ( b ) µν ( i Ω ,
0) = N Z ∞−∞ dω π dω ′ π Z d D k (2 π ) D d D p (2 π ) D V k − p Tr [ G k ( iω ) σ µ G k ( iω + i Ω) G p ( iω ′ + i Ω) σ ν G p ( iω ′ )] . (78)Both of these expressions need to be regulated due to theUV divergence.In appendix E we calculate both of these contributionsto the electrical conductivity. The contribution to theconductivity coming from the self-energy part expandedup to the order ǫ is found to be σ a = σ e (cid:18) − ǫ + 32 + γ − ln(64 π ) (cid:19) . (79)The corresponding vertex part is found to be σ b = σ e (cid:20)(cid:18) ǫ − − γ + ln 64 π (cid:19) + 8 − π (cid:21) . (80)Adding these two terms we obtain the first order correc-tion to the a.c. conductivity due to the Coulomb inter-action δσ ( c ) = σ a + σ b = 11 − π σ e , (81)which corresponds to the value C = 11 − π . Calculation of the a.c. conductivity using thedensity-density correlator To show that our previous result for the conductivityis consistent, we now calculate the longitudinal conduc-tivity given by Eq. (40) and show that it yields the samevalue of the constant C as in Eq. (82). This must bethe case if the Ward-Takahashi identity and the O (2) ro-tational invariance hold. The longitudinal correction tothe conductivity can be calculated by focusing on δ Π ( c )00 ,since B = 0. Unlike for Kubo formula, this componentof the polarization tensor must be calculated at finite Ωand finite q , since at q = 0 it vanishes. Fortunately,we need only the leading order term in the expansion insmall q , from which we can extract the conductivity.According to Eq. (59), the Coulomb interaction cor-rection to the density-density correlator reads δ Π ( c )00 ( i Ω , q ) = δ Π ( a )00 ( i Ω , q ) + δ Π ( b )00 ( i Ω , q ) , (83)where, just as before, we have separated the self-energycontribution δ Π ( a )00 ( i Ω , q ) = N Z ∞−∞ dω π dω ′ π Z d k (2 π ) d p (2 π ) V k − p × { Tr [ G k ( iω ) G p ( iω ′ ) G k ( iω ) G k + q ( iω + i Ω)]+ Tr [ G k ( iω ) G p ( iω ′ ) G k ( iω ) G k − q ( iω − i Ω)] } , (84)and the vertex contribution δ Π ( b )00 ( i Ω , q ) = N Z ∞−∞ dω π dω ′ π Z d k (2 π ) d p (2 π ) V k − p × Tr [ G k ( iω ) G p ( iω ′ ) G p − q ( iω ′ − i Ω) G k − q ( iω − i Ω)] . (85)The details of the calculations are presented in the Ap-pendix F. Here we just state the final result δσ ( c ) k = 11 − π σ e , (86)in agreement with the result (81) obtained from thecurrent-current correlator. Such agreement is expectedsince, as we have shown to this order in the Coulomb in-teraction, the dimensional regularization explicitly pre-serves the U (1) gauge symmetry of the theory of theCoulomb interacting Dirac fermions. VIII. DISCUSSION AND CONNECTION WITHPREVIOUS WORK
Let us now discuss the result (82) for the correctionto the a.c. conductivity due to the long-range Coulombinteraction in light of the ones previously reported in theliterature.
In Ref. 9, the Coulomb correction to the conductivityis shown to have the form given by Eq. (1), consistent with the renormalizability of the quantum field theoryof the Coulomb interacting Dirac fermions. Moreover,the value of the constant C = (25 − π ) /
12 has beencalculated from the current-current correlator, and us-ing hard-cutoff regularization. In Appendix D we showthat in general hard cutoff violates the Ward-Takahashiidentity. Nevertheless, it can be shown that the inte-gral for C is, despite appearances, in fact UV convergent,but sensible to the order of integration. A correct wayof performing the integral is to integrate both momentaup to finite cutoffs, and take the cutoff to infinity afterall the integrals are done first. This, as shown in theAppendix H, corrects the value of the constant preciselydown to the C = (11 − π ) /
6. As we showed in AppendixG, the previuos result C = (25 − π ) /
12 is also obtainedwhen using a version of the dimensional regularizationin which the Pauli matrices are treated as embedded instrictly two spatial dimensions, and which also violatesthe Ward-Takahashi identity, as we argued in AppendixD. Technically, the origin of the discrepancy between theresults for the Coulomb correction to conductivity withinthe two versions of the dimensional regularization maybe traced if we consider the self-energy correction to theconductivity in Eq. (G2) and its counterpart with Paulimatrices in D = 2 − ǫ , given by Eq. (E5). The dif-ference arises from the factor D − − ǫ which is aconsequence of the different treatment of Pauli matriceswithin the two schemes. The self-energy piece has a sin-gular part proportional to 1 /ǫ , and when multiplied bya term linear in ǫ coming from D −
1, it gives rise to afinite contribution to the self-energy correction. Analo-gous situation occurs in the vertex part, and in that casethe last three terms in the integrand of Eq. (E13) accountfor the difference. Namely, when the trace over spatialindices of Pauli matrices is taken in D = 2, these threeterms cancel out, as it may be seen from the term pro-portional to ( k · p ) in the integrand in Eq. (G6), but,in fact, when Pauli matrices are embedded in D = 2 − ǫ ,these terms yield a finite contribution to the conductiv-ity, which may be directly checked following the steps inEqs. (E18)-(E23).On the other hand, in Ref. 10, the result for the con-stant C = (19 − π ) /
12 has been calculated using threedifferent methods, namely, the density-density correlator,the current-current correlator and the kinetic equation,and it has been argued that in order to obtain the uniquevalue for the constant C a short-distance cutoff on thelong-range Coulomb interaction has to be imposed. Thisregularization is an analogue of the Pauli-Villars regular-ization in QED, but without the additional Pauli-Villarsfermions introduced, that are, in fact, necessary to ren-der it consistent. This value for the constant C has alsobeen obtained in Ref. 18 by regulating the short-distancebehavior of the Coulomb interaction in the same man-ner as in Ref. 10. Although it has been shown that thesame regularization preserves the Ward-Takahashi iden-tity, besides lacking the Pauli-Villars fermions, this reg-ularization cannot be applied to the theory of free Diracermions. Namely, the latter needs to be regularizedwhen calculating the polarization bubble. Clearly, thiscannot be achieved by imposing a short-distance cutoffon the long-range Coulomb interaction. Therefore, thisregularization cannot serve as a consistent regularizationof the entire field theory. IX. ACKNOWLEDGEMENTS
V. J. wishes to acknowledge the support of the Nether-lands Organisation for Scientific Research (NWO). O. V.was supported in part by NSF grant DMR-00-84173 andNSF CAREER award grant DMR-0955561. I. F. H. issupported by the NSERC of Canada. The authors alsowish to thank the Aspen Center for Physics where a partof this work was preformed.
Appendix A: Vertex integrals
The quantity of interest, which enters into the evalua-tion of the bare bubble and the leading order correctionto the polarization tensor for short range interactions u is P µ ( q , i Ω) = Z d D k (2 π ) D Z ∞−∞ dω π G k ( iω ) σ µ G k + q ( iω + i Ω) . (A1)Substituting (15), using Feynman parametrization (63)for α = β = 1, and interchanging the order of integra-tions, we obtain P µ ( q , i Ω) = Z dx × Z d D k (2 π ) D Z ∞−∞ dω π ω + x Ω) + ( k + x q ) + ∆] × [ iω + σ · k ] σ µ [ iω + i Ω + σ · ( k + q )] (A2)where ∆ = x (1 − x )(Ω + q ) . (A3) The standard next step when working in dimensional reg-ularization is to define new integration variables ℓ ω = ω + x Ω and ℓ = k + x q . We then perform the integralover ℓ ω to obtain P µ ( q , i Ω) = Z dx Z d D ℓ (2 π ) D (cid:20) − σ µ √ ℓ + ∆ +( σ · ℓ − xS ) σ µ ( σ · ℓ + (1 − x ) S )4( ℓ + ∆) (cid:21) , (A4)where we defined S ≡ i Ω + σ · q . (A5)Since the integration measure is O (2)-symmetric, onlythe terms even in ℓ in numerator give a non-trivial con-tribution, and we find P µ ( q , i Ω) = 14 Z dx Z d D ℓ (2 π ) D (cid:20) − σ µ √ ℓ + ∆+ σ a σ µ σ a ℓ D ( ℓ + ∆) − x (1 − x ) Sσ µ S ( ℓ + ∆) (cid:21) . (A6)We next use the dimensional regularization integrals (16)and (17), as well as the identity (18) to find P µ ( q , i Ω) = 18 π Z dx h σ µ √ ∆ − δ µ √ ∆ − ( i Ω + σ · q ) σ µ ( i Ω + σ · q ) x (1 − x ) √ ∆ (cid:21) . (A7)Using Eq.(A3), we finally have P µ ( q , i Ω) = p Ω + q × (cid:20) σ µ − δ µ − ( i Ω + σ · q ) σ µ ( i Ω + σ · q )Ω + q (cid:21) . (A8)
1. Coulomb vertex
The Coulomb vertex function to the first order in the coupling constant is P cµ ( q , p , i Ω) = − Z d D k (2 π ) D Z ∞−∞ dω π πe | p − k | G k ( iω ) σ µ G k + q ( iω + i Ω) , (A9)with the free fermion Green’s function given by Eq. (15). After introducing Feynman parameters using Eq. (63), weobtain P cµ ( q , p , i Ω) = − Z dx Z d D ℓ (2 π ) D πe | p + x q − ℓ | (cid:20) − σ µ √ ℓ + ∆ + ( σ · ℓ − x ( i Ω + σ · q )) σ µ ( σ · ℓ + (1 − x )( i Ω + σ · q ))4( ℓ + ∆) (cid:21) . (A10)ow, we consider two terms in the above form of the Coulomb vertex separately. Using the Feynman parametrization(63) and the D -dimensional integral (16), the first term in the last equation acquires the form Z d D ℓ (2 π ) D | p + x q − ℓ | √ ℓ + ∆ = 1 π Z dy p y (1 − y ) Z d D ℓ (2 π ) D ℓ + (1 − y ) ( y ( p + x q ) + ∆)= 1 π Γ[1 − D ](4 π ) D Z dy √ y (1 − y ) − D y ( p + x q ) + ∆) − D . (A11)After expanding the integrand to the first order in the parameter ǫ = 2 − D and integrating over y , we have Z d D ℓ (2 π ) D | p + x q − ℓ | √ ℓ + ∆ = Γ[1 − D ](4 π ) D − ǫ p + x q ) − ǫ tanh − s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) ! . (A12)The second term in Eq. (A10), after introducing the Feynman parameter, shifting the variable ℓ − y ( p + x q ) → ℓ ,and integrating over ℓ , becomes Z d D ℓ (2 π ) D ( σ · ℓ − xS ) σ µ ( σ · ℓ + (1 − x ) S ) | p + x q − ℓ | ( ℓ + ∆) = 2 π Γ[1 − D ](4 π ) D Z dy (1 − y ) D − √ y D δ µ ( y ( p + x q ) + ∆) − D + 2 π Γ[2 − D ](4 π ) D Z dy (1 − y ) D − √ y ( yσ · ( p + x q ) − xS ) σ µ ( yσ · ( p + x q ) + (1 − x ) S )( y ( p + x q ) + ∆) − D + 2 π Γ[2 − D ](4 π ) D Z dy (1 − y ) D − √ y δ µa σ a ( y ( p + x q ) + ∆) − D , (A13)where S = i Ω + σ · q , and we also used the identity (18) for the trace of the Pauli matrices over spatial indices.Note that the last term in the previous equation arises from the term proportional to ǫ = 2 − D in Eq. (18). If wetreated Pauli matrices strictly in D = 2, this term would be omitted what would thus lead to the violation of theWard identity, as it may be directly checked in Eq. (B16). Therefore, in order to preserve the gauge invariance of thetheory, it is crucial to treat Pauli matrices, as well as the momentum integrals, in a general spatial dimension, andonly at the end of the calculation to take D = 2 − ǫ , and expand in ǫ . After expanding in ǫ , keeping terms up to order ǫ , and performing the integral over y in the first term, we have Z d D ℓ (2 π ) D ( σ · ℓ − xS ) σ µ ( σ · ℓ + (1 − x ) S ) | p + x q − ℓ | ( ℓ + ∆) = Γ[1 − D ](4 π ) D δ µ − ǫ p + x q ) − ǫ tanh − s ∆( p + x q ) + ∆ − ǫ p ∆(( p + x q ) + ∆) − ∆( p + x q ) − ǫ ! + 14 π δ µa σ a + 12 π Z dy p y (1 − y ) ( yσ · ( p + x q ) − xS ) σ µ ( yσ · ( p + x q ) + (1 − x ) S ) y ( p + x q ) + ∆= Γ[1 − D ](4 π ) D δ µ − ǫ p + x q ) − ǫ tanh − s ∆( p + x q ) + ∆ − ǫ p ∆(( p + x q ) + ∆) − ∆( p + x q ) − ǫ ! + 14 π δ µa σ a + 12 π (cid:0) − x (1 − x ) Sσ µ S I [( p + x q ) , ∆]+ ( − xSσ µ σ · ( p + x q ) + (1 − x )( p + x q ) · σσ µ S ) I [( p + x q ) , ∆] + σ · ( p + x q ) σ µ σ · ( p + x q ) I [( p + x q ) , ∆] (cid:1) (A14)ith ∆ defined in Eq. (A3). The remaining integrals over y read I ( a, ∆) = Z dy p y (1 − y ) 1 ya + ∆ = π p ∆( a + ∆) (A15) I ( a, ∆) = Z dy p y (1 − y ) yya + ∆ = πa − r ∆ a + ∆ ! (A16) I ( a, ∆) = Z dy p y (1 − y ) y ya + ∆ = π a a − − r ∆ a + ∆ !! . (A17)Therefore, using Eqs. (A11) and (A14), we can write the Coulomb vertex in the form P cµ ( q , p , i Ω) = − πe Z dx ( Γ[1 − D ](4 π ) D/ ǫδ µ ∆ − p ∆(∆ + ( p + x q ) )( p + x q ) − ! + 14 π σ a δ µa − σ a δ µa Γ[1 − D ](4 π ) D/ − ǫ p + x q ) − ǫ tanh − s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) ! + 12 π " − x (1 − x )( i Ω + σ · q ) σ µ ( i Ω + σ · q ) p ∆(( p + x q ) + ∆)+ ( − x ( i Ω + σ · q ) σ µ σ · ( p + x q ) + (1 − x )( p + x q ) · σσ µ ( i Ω + σ · q ))( p + x q ) − s ∆( p + x q ) + ∆ ! + σ · ( p + x q ) σ µ σ · ( p + x q )2( p + x q ) ( p + x q ) − − s ∆( p + x q ) + ∆ !! , (A18)where again ∆ = x (1 − x )(Ω + q ), and we explicitly wrote the functions I , I , and I , defined in Eqs.(A15)-(A17).The above expression diverges as D → ǫ → + . As discussed in the main text, thisdivergence is tied to the divergence of the self-energy and is a consequence of the Ward-Takahashi identity provedbelow. All the integrals over x in the above expression can be performed in the closed form in terms of ellipticintegrals. (The expressions involving tanh − need to be integrated by parts first to bring them to the form easilyexpressible in terms of the elliptic integrals.) However, we found that doing so leads to intractable expressions andwe thus chose to work with the above form of the Coulomb vertex, in which the integrals over the variable x can bethought of as the integral representation of the elliptic integrals.At q = 0 the above expressions simplify significantly and we have P cµ (0 , p , i Ω) = − δ µa e " σ a − (cid:18) ǫ − γ + ln 64 π − ln p (cid:19) + 2 | Ω | p Ω + 4 p K ! + [ p · σ, σ a ] i Ω p − | Ω | p Ω + 4 p K − p Ω + 4 p | Ω | ( E − K ) ! + σ · p σ a σ · pp − Ω p + (Ω − p ) E + (16 p − p ) K | Ω | p p Ω + 4 p ! , (A19)where the arguments of the complete elliptic integrals of the first and second kind, respectively, are K ≡ K | Ω | p Ω + 4 p ! ; E ≡ E | Ω | p Ω + 4 p ! . Note that to this order, at q = 0 (and at any Ω), only the spatial components of P cµ are finite. This is a consequenceof the Ward-Takahashi identity, since to this order the self-energy is frequency-independent.Finally, at q = 0 and Ω = 0 the integrals in Eq.(A10) can be performed for arbitrary D without the necessity ofexpanding in powers of ǫ . The Coulomb vertex then becomes P cµ (0 , p ,
0) = e (4 π ) D δ µa | p | − D Γ[ D + ]Γ[ D − ]Γ[ D ] (cid:18) σ a Γ (cid:20) − D (cid:21) − (cid:18) σ a + σ · p σ a σ · pp (cid:19) Γ (cid:20) − D (cid:21)(cid:19) . (A20)his form of the vertex function is used to show that δ Π ( c ) µν (0 ,
0) = 0 for any D , a fact which is in turn used in thecalculation of the electrical conductivity. Appendix B: Coulomb vertex and the Ward-Takahashi identities in dimensional regularization
In this appendix we show in detail that to the leading order in the Coulomb interaction coupling constant e andto O ( N ), the Ward-Takahashi identity, questioned to hold in Ref. 18, is satisfied. As a first step, we define thecontraction q µ P cµ ≡ − i Ω P c + q a P ca . Using Eq. (A18), a straightforward calculation shows that all the terms in thecontraction proportional to i Ω cancel out, and the contraction simplifies to q µ P cµ ( q , p , i Ω) = − πe Z dx ( − σ · q Γ[1 − D ](4 π ) D/ − ǫ p + x q ) − ǫ tanh − s ∆( p + x q ) + ∆ ! + 14 π σ · q + 12 π " − σ · q s ∆( p + x q ) + ∆ + σ · ( p + x q ) (1 − x )(Ω + q )( p + x q ) − s ∆( p + x q ) + ∆ ! − ∆( p + x q ) σ · ( p + x q ) σ · q σ · ( p + x q ) + σ · ( p + x q ) σ · q σ · ( p + x q )2( p + x q ) + σ · ( p + x q ) σ · q σ · ( p + x q )( p + x q ) s ∆ ( p + x q ) + ∆ . (B1)In order to show the Ward-Takahashi identity, we first note that the self-energy to the first-order in the Coulombcoupling is independent of the frequency. Thus all the terms in the contraction (B1) that contain frequency have tovanish if the Ward-Takahashi identity holds. In fact, as we will show in what follows, the contraction can be writtenin the form q µ P cµ ( q , p , i Ω) = N ( p , q ) − e + q ) W ( p , q ) − e σ · p p (Ω + q ) L ( p , q , Ω) − e σ · q p Ω + q M ( p , q , Ω) , (B2)where the functions N ( p , q ), W ( p , q ), L ( p , q , Ω), and M ( p , q , Ω) are defined in Eqs. (B16), (B3), (B17), and (B18),respectively. This condition is, therefore, satisfied if W ( p , q ) = 0, M ( p , q , Ω) = 0, and L ( p , q , Ω) = 0. Finally, tocomplete the proof of the identity, we will show that N ( p , q ) = Σ p + q ( iν + iω ) − Σ p ( iν ).Let us first show that term proportional to Ω + q vanishes, i.e., that W ( p , q ) ≡ Z dx (cid:18) σ · ( p + x q ) (1 − x )( p + x q ) − x (1 − x )( p + x q ) σ · ( p + x q ) σ · q σ · ( p + x q ) (cid:19) = 0 . (B3)Using the identity σ · p σ · q σ · p = 2 p · q σ · p − p σ · q , (B4)we can rewrite the above integral as W ( p , q ) = σ · p Z dx p (1 − x ) − (2 p · q + q ) x ( p + x q ) + σ · qq Z dx (cid:18) x [2 q p · q − p q + 4( p · q ) ] + 2 x p [ q + 2 p · q ] + p ( p + x q ) − (cid:19) ≡ σ · p W ( p , q ) + σ · qq W ( p , q ) . (B5)When p = q , it is easy to show that both W and W vanish, and we thus concentrate on the case D ≡ p q − ( p · q ) >
0. In order to calculate the integrals W and W , we use the following identities K ≡ Z dx ( p + x q ) = p q − p · q ) − ( p · q ) q p ( p + q ) D + q D Z dx ( p + x q ) , (B6) K ≡ Z dx x ( p + x q ) = q + p · q p + q ) D − p · q D Z dx ( p + x q ) , (B7) K ≡ Z dx x ( p + x q ) = − p + p · q p + q ) D + p D Z dx ( p + x q ) . (B8)traightforward calculation yields W ( p , q ) = 0 and W ( p , q ) = 0, and thus W ( p , q ) = 0. The contraction given byEq. (B1) then simplifies to q µ P cµ ( q , p , i Ω) = πe σ · q J − e Z dx (cid:20) σ · q + σ · ( p + x q ) σ · q σ · ( p + x q )( p + x q ) + 2 (cid:18) − σ · q − σ · ( p + x q ) (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) σ · ( p + x q ) σ · q σ · ( p + x q )( p + x q ) (cid:19) × s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) , (B9)where J ≡ Γ[1 − D ](4 π ) D/ Z dx − ǫ p + x q ) − ǫ tanh − s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) ! . (B10)After a partial integration in the last term, J becomes J = Γ[1 − D ](4 π ) D/ Z dx (cid:16) ǫ ln 4 − ǫ ln hp ( p + x q ) + x (1 − x )(Ω + q ) + p x (1 − x )(Ω + q ) i(cid:17) . (B11)Finally, after performing another partial integration in the last term of the previous equation, we obtain J = Γ[1 − D ](4 π ) D/ Z dx (cid:26) ǫ ln 4 − ǫ p + q ) ] + ǫ x ( x q + p · q )( p + x q ) + ǫ p + x q ) (cid:20) − x − x ( p + x q ) − x ( x q + p · q ) (cid:21) s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) ) . (B12)Expansion of the prefactor in ǫ has the formΓ[1 − D ](4 π ) D/ = 12 π (cid:18) ǫ + 12 [ − γ + ln 4 π ] (cid:19) + O ( ǫ ) , (B13)which, after keeping the terms up to the order ǫ in Eq. (B12), yields J = 12 π Z dx (cid:26) ǫ + 12 [ − γ + ln 64 π ] −
12 ln[( p + q ) ] + x ( x q + p · q )( p + x q ) + 12( p + x q ) (cid:20) − x − x ( p + x q ) − x ( x q + p · q ) (cid:21) s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) ) . (B14)Now, after substituting Eq. (B14) into Eq. (B9), the contraction reads q µ P cµ ( q , p , i Ω) = N ( p , q ) − e σ · p p (Ω + q ) L ( p , q , Ω) − e σ · q p Ω + q M ( p , q , Ω) , (B15)which is, in fact, the form (B2) of the contraction, since we have already shown that W ( p , q ) = 0. The frequency-independent part in the above equation reads N ( p , q ) = e Z dx (cid:26) σ · q (cid:20) ǫ − γ + ln 64 π − ln[( p + q ) ] + 2 x ( x q + p · q )( p + x q ) − (cid:21) − σ · ( p + x q ) σ · q σ · ( p + x q )( p + x q ) (cid:27) , (B16)while the remaining terms are L ( p , q , Ω) = Z dx x ( p + q ) − (1 − x ) p ( p + x q ) s x (1 − x )( p + x q ) + x (1 − x )(Ω + q ) , (B17)nd M ( p , q , Ω) = 12 Z dx (cid:18) x ( p + q ) − (1 − x ) p (1 − x )( p + x q ) − (cid:19) s x (1 − x )( p + x q ) + x (1 − x )(Ω + q )+ Z dx (cid:18) x (Ω + q )[(3 x − p + 2 x (2 x − p · q + x q ]( p + x q ) (cid:19) s x (1 − x )( p + x q ) + x (1 − x )(Ω + q ) . (B18)The frequency-independent term (B16), after using the identity (B4), and performing the remaining integral, has theform N ( p , q ) = e (cid:18) ǫ σ · q + ( − γ + ln 64 π ) σ · q − σ · ( p + q ) ln[( p + q ) ] + σ · p ln p (cid:19) = Σ p + q ( iν + i Ω) − Σ p ( iν ) . (B19)Here, Σ p ( iν ) is the self-energy defined in Eq. (71) in the main text. Appendix C: Evaluation of the functions L ( p , q , Ω) and M ( p , q , Ω) In this Appendix we show that the functions L ( p , q , Ω), Eq. (B17), and M ( p , q , Ω), Eq. (B18), vanish identically,and therefore, the Ward-Takahashi identity is, indeed, satisfied within the dimensional regularization used here.In order to evaluate integrals in Eqs. (B17) and (B18), we introduce new variables p ′ = pq e − iϕ , w = Ω q , (C1)with p = | p | , q = | q | , and cos ϕ = p · q / ( pq ), and thus we can write( p + x q ) = q ( x + p ′ )( x + p ′∗ ) , (C2)with p ′∗ denoting complex conjugate of p ′ . The integral (B17) can now be rewritten in terms of the new variables as L = 1 q r q Ω Z dx (1 + p ′ )(1 + p ′∗ ) x (1 − x ) − x (1 − x ) p ′ p ′∗ ( x + p ′ ) ( x + p ′∗ ) p x ( x − x − x + )( x − x − ) , (C3)where x ± are the roots of the quadratic equation ( p + x q ) + x (1 − x )(Ω + q ) = 0, x ± = ( p + q ) − p + Ω ± p (( p + q ) − p + Ω ) + 4Ω p , (C4)or in terms of the variables p ′ and wx ± = 1 + 2 Re ( p ′ ) + w ± p (1 + 2 Re ( p ′ ) + w ) + 4 w p ′ p ′∗ w . (C5)Assuming that x + >
1, we have the folowing sequence x + > > > x − , (C6)which is important for expressing the function L ( p , q , Ω) in terms of the elliptic integrals, as we will see below. Weuse partial fractions to calculate the integral in Eq. (C3). The first term reads x (1 − x )( x + p ′ ) ( x + p ′∗ ) = − (cid:18) ip ′ [ p ′ (1 + 2 p ′ ) − (3 + 4 p ′ ) p ′∗ ]8[ Im ( p ′ )] ( x + p ′ ) + p ′ (1 + p ′ )4[ Im ( p ′ )] ( x + p ′ ) + C.c. (cid:19) ≡ − (cid:18) Ax + p ′ + B ( x + p ′ ) + C.c. (cid:19) , (C7)hile the second term is x (1 − x ) ( x + p ′ ) ( x + p ′∗ ) = − (cid:18) i (1 + p ′ ) [ p ′ − p ′ p ′∗ − Re ( p ′ )]4[ Im ( p ′ )] ( x + p ′ ) + p ′ (1 + p ′ ) Im ( p ′ )] ( x + p ′ ) + C.c. (cid:19) ≡ − (cid:18) A x + p ′ + B ( x + p ′ ) + C.c. (cid:19) . (C8)We therefore reduced the problem of evaluating the integral (C3) to the calculation of the following integrals I m = Z dx x + p ) m p x ( x − x − x + )( x − x − ) , (C9)with m = 0 , ,
2, and x + > > > x − . In terms of the integrals I m , the integral L reads L = − [1 + 2 Re ( p ′ )] I + [((1 + p ′ )(1 + p ′∗ ) A − p ′ p ′∗ A ) I + ((1 + p ′ )(1 + p ′∗ ) B − p ′ p ′∗ B ) I + C.c. ]= − [1 + 2 Re ( p ′ )] I + (cid:20) − ip ′ (1 + p ′ )(1 + 2 p ′ )2 Im ( p ′ ) I + ip ′ (1 + p ′ ) Im ( p ′ ) I + C.c. (cid:21) . (C10)The integrals I m with m = 0 , , x + > > > x − have the form (Eqs. (255.00), (255.38), (340.01), and (340.02)in Ref. 37) I = gF ( k ) , (C11) I = g (1 + p ′ ) α (cid:2) ( α − α )Π( α , k ) + α F ( k ) (cid:3) , (C12) I = g (1 + p ′ ) α (cid:20) α F ( k ) + 2 α ( α − α )Π( α , k ) + ( α − α ) α − k − α ) (cid:0) α E ( k ) + ( k − α ) F ( k )+ (2 α k + 2 α − α − k )Π( α , k ) (cid:1)(cid:3) , (C13)where F ( k ), E ( k ), and Π( α , k ) are the complete elliptic integrals of the first, second, and the third kind, respectively,defined in terms of the corresponding incomplete integrals as F ( k ) ≡ F ( π/ , k ), E ( k ) ≡ E ( π/ , k ), and Π( α , k ) ≡ Π( π/ , α , k ), with incomplete elliptic integrals defined as in Ref. 37. Here, k = x + − x − x + (1 − x − ) , g = 1 p x + (1 − x − ) , (C14)and α = 1 x + , α = x + + p ′ x + (1 + p ′ ) . (C15)Therefore, the integral L has the form L = R ( p ′ , w ) F ( k ) + P ( p ′ , w ) E ( k ) + Im [ G ( p ′ , w )Π( α , k )] . (C16)Note that imaginary part of α is non-vanishing. In the following, we will show that the coefficient P ( p ′ , w ) = 0.Then, by expressing Im [ G ( p ′ , w )Π( p ′ , w )] in terms of the functions F ( k ) and Π( α , k ), with α defined below purelyreal, we obtain that the function L vanishes. The coefficient P ( p ′ , w ) reads P ( p ′ , w ) = ip ′ ( α − α ) Im ( p ′ )( α − α ( k − α ) + C.c = − ip ′ x + (1 − x + )(1 + p ′ )4 Im ( p ′ )( p ′ + x + )( p ′ + x − ) + C.c = 12 Im ( p ′ ) Im (cid:20) p ′ x + (1 − x + )(1 + p ′ )( x + + p ′ )( x − + p ′ ) (cid:21) , (C17)but p ′ x + (1 − x + )(1 + p ′ )( x + + p ′ )( x − + p ) = − w (1 + 4 w ) h Re ( p ′ ) + w + p (1 + 2 Re ( p ′ ) + w ) + 4 p ′ p ′∗ w i × h Re ( p ′ ) − w + p (1 + 2 Re ( p ′ ) + w ) + 4 p ′ p ′∗ w i (C18)s purely real, and thus P ( p ′ , w ) vanishes identically. The coefficient R ( p ′ , w ) reads R ( p ′ , w ) = − g (1 + 2 Re ( p ′ )) + g (cid:20)(cid:18) − ip ′ (1 + p ′ )(1 + 2 p ′ )2 Im ( p ′ )( x + + p ′ ) + ip ′ (1 + p ′ )4 Im ( p ′ )( x + + p ′ ) [2 p ′ (1 + p ′ ) + x + (1 − x + )] (cid:19) + C.c. (cid:21) , (C19)while G ( p ′ , w ) = g p ′ ( α − α ) Im ( p ′ ) α (cid:20) p ′ + p ′ α (cid:18) α (3 α + k − α (1 + k )) + α ( α + 3 k − α (1 + k ))2( α − k − α ) (cid:19)(cid:21) . (C20)Imaginary part of the product G ( p ′ , w )Π( α , k ) can be expressed in terms of the complete elliptic function of the firstkind and the third kind, as given by Eq. (419.00) in Ref. 37, Im [ G ( p ′ , w )Π( α , k )] = gIm ( p ′ ) 1 m r ( s t − s t ) (cid:8) [ a ( s r − k s m ) + b ( k t m − t r )] F ( k )+ n m r ( a s − b t )Π( α , k ) (cid:9) , (C21)where G = a + ib , α = − γ − iγ , r = γ + γ ,m = − γ + k γ , α = k m r ,s = 1 − k r , n = m [ α − (2 + m ) α + (1 − α ) k ] − r m ( γ + ( γ + α ) ) t = 2( k + 2 γ + γ ) γ (2 γ + k ) , t = m + ( γ + 2 − α ) m + n m ( γ + α ) m γ , s = 1 − n − m , (C22)since r + 2 γ + k = 0 in our case, with k given by Eq. (C14), and a , b , γ , and γ real. Thus, the integral L acquires the form L = [ R ( p ′ , w ) + R ( p ′ , w )] F ( k ) + S ( p ′ , w )Π( α , k ) , (C23)where R ( p ′ , w ) = gIm ( p ′ ) a ( s r − k s m ) + b ( k t m − t r ) m r ( s t − s t ) (C24)and S ( p ′ , w ) = gIm ( p ′ ) n ( a s − b t ) s t − s t , (C25)while R ( p ′ , w ) is given by Eq. (C19).Straightforward calculation shows that a s − b t = 0 and s t − s t = 0, and thus the coefficient S ( p ′ , w )vanishes identically. Furthermore, using that a s − b t = 0, the form of the coefficient R ( p ′ , w ), given by Eq. (C24),can be simplified to R ( p ′ , w ) = − gIm ( p ′ ) a k r t . (C26)Finally, the previous equation together with Eqs. (C19) and (C22), since w is purely real, yields R ( p ′ , w )+ R ( p ′ , w ) =0, and, therefore, the integral L ( p , q , Ω) given by Eq. (B17) vanishes. When the two vectors are (anti)collinear, i.e.,when Im ( p ′ ) = 0, the continuity implies L ( p , q , Ω) = 0. When x + <
1, the analogous calculation shows that L ( p , q , Ω) = 0, as well.Let us now turn to the integral M ( p , q , Ω), and consider the case Im ( p ′ ) = 0, and x + >
1. In terms of partialfractions, this integral reads M = 1 + w w Z dx (cid:26)(cid:20) p ′ (1 + p ′ ) iIm ( p ′ )( x + p ′ ) + ip ′ (1 + p ′ )(1 + 2 p ′ )2 Im ( p ′ )( x + p ′ ) + C.c. (cid:21) + p ′∗ + 2(1 + p ′ ) Re ( p ′ ) + x − x (cid:27) × p x ( x − x − x + )( x − x − ) + 12 w Z dx (cid:26)(cid:20) p ′ (1 + p ′ ) x + p ′ + C.c. (cid:21) − Re [ p ′ (1 + p ′ )] + [2 Re ( p ′ ) − x + 2 x (cid:27) × p x ( x − x − x + )( x − x − ) . (C27)t follows from Eq. (255.17) in Ref. 37 that J ≡ Z xdx p x ( x − x − x + )( x − x − ) = gα (cid:2) ( α − α , k ) + F ( k ) (cid:3) (C28) J ≡ Z x dx p x ( x − x − x + )( x − x − ) = gα (cid:8) F ( k ) + 2( α − α , k )+ α − k − α ) (cid:2) α E ( k ) + ( k − α ) F ( k ) + (2 α k + 2 α − α − k )Π( α , k ) (cid:3)(cid:27) , (C29)which together with Eqs. (C11), (C12) and (C13) allows us to express the function M in terms of the elliptic integralsas M = gw (cid:2) A ( p ′ , w )Π( α , k ) + A ( p ′ , w ) ∗ Π( α , k ) ∗ + B ( p ′ , w ) F ( k ) + X ( p ′ , w )Π( α , k ) + Y ( p ′ , w ) E ( k ) (cid:3) , (C30)with the coefficients in the above equation of the form A ( p ′ , w ) = p ′ (cid:18) − α α (cid:19) (cid:26) i (1 + 2 p ′ )(1 + w ) Im ( p ′ ) − ip ′ (1 + w ) α Im ( p ′ ) (cid:20) α + ( α − α )(2 α k + 2 α − α − k )2( α − k − α ) (cid:21)(cid:27) , (C31) B ( p ′ , w ) = 1 + w Im ( p ′ ) Im (cid:26) p ′ α (cid:20) α + ( α − α ) α − (cid:21)(cid:27) + Re (cid:20) α p ′ α (cid:18) i (1 + 2 p ′ )(1 + w ) Im ( p ′ ) (cid:19)(cid:21) + (1 + w )[ p ′∗ + 2(1 + p ′ ) Re ( p ′ )] − Re [ p ′ (1 + p ′ )] + 1 α (cid:20)
12 + w + Re ( p ′ ) (cid:21) (C32) − w (1 + α )2 α , (C33) X ( p ′ , w ) = (cid:18) − α (cid:19) (cid:20)
12 + Re ( p ′ ) + w − w (2 α k − α + k − α )2 α ( k − α ) (cid:21) ,Y ( p ′ , w ) = 1 + w Im ( p ′ ) Im (cid:20) p ′ ( α − α ) α ( α − k − α ) (cid:21) − w ( α − α ( k − α ) . (C34)Here, k , α , and α are defined by Eqs. (C14) and (C15). Since w is purely real, the functions A ( p ′ , w ), B ( p ′ , w ), X ( p ′ , w ), and Y ( p ′ , w ) vanish, and thus the integral M ( p , q , Ω) given by Eq. (B18) is identically equal to zero. Thisresult also implies that, because of the continuity, in the case of (anti)collinear vectors p and q , i.e., when Im ( p ′ ) = 0,the function M ( p , q , Ω) also vanishes. When the root x + in Eq, (C4) is less than one, the analogous calculation showsthat the function M = 0, as well. Appendix D: Violation of the Ward-Takahashiidentity within hard cutoff regularization
In this Appendix we show that Ward-Takahashi iden-tity does not hold within the hard cutoff regularization.In order to show that, we will follow the same steps asin the previous two appendices. Let us first calculate theself-energy, given by Eq. (62),Σ p ( iω ) = Z dω ′ π Z d k (2 π ) πe | p − k | iω ′ + σ · k ω ′ + k = πe Z d k (2 π ) σ · k p | p − k | , (D1)which after using the Feynman parametrization (63),shifting the momentum variable, and performing angular integration becomesΣ p ( iω ) = e π σ · p Z dy r y − y × Z Λ0 dk kk + y (1 − y ) p , (D2)where Λ is the momentum cutoff regulating the ultravi-olet divergence of the integral. Remaining integrationsthen yieldΣ p ( iω ) = e σ · p (ln Λ − ln p + 2 ln 2) . (D3)Therefore, the divergent part of the momentum integralappears as the logarithm of the cutoff which correspondsto 1 /ǫ pole in the dimensional regularization, see Eq.71). Note that the divergent parts appear with preciselythe same coefficients, but the finite parts are differentwithin the two regularizations.We now consider the Coulomb vertex function definedin Eq. (70) which after introducing the Feynman parame-ters may be written in the form given by Eq. (A10) withthe integral performed in D = 2. After following thesame steps as in Appendix A, and performing straight-forward integrations, we obtain Z d ℓ (2 π ) | p + x q − ℓ | p ℓ + x (1 − x )(Ω + q )= 12 π (cid:18) ln Λ −
12 ln ( p + x q ) − tanh − s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) ! . (D4) Analogously, we have Z d ℓ (2 π ) ( σ · ℓ − xS ) σ µ ( σ · ℓ + (1 − x ) S ) | p + x q − ℓ | ( ℓ + ∆) = δ µ π ln Λ −
12 ln ( p + x q ) − tanh − s x (1 − x )(Ω + q )( p + x q ) + x (1 − x )(Ω + q ) − p ∆(( p + x q ) + ∆) − ∆( p + x q ) ! + 12 π (cid:8) − x (1 − x ) Sσ µ S I [( p + x q ) , ∆] + [ − xSσ µ σ · ( p + x q )+ (1 − x )( p + x q ) · σσ µ S ] I [( p + x q ) , ∆] + σ · ( p + x q ) σ µ σ · ( p + x q ) I [( p + x q ) , ∆] (cid:9) , (D5)with I , I , and I defined in Eqs. (A15)-(A17). Notethat the term ∼ δ µa σ a is not present in the above equa-tion, since the Pauli matrices here are treated strictly in D = 2, and thus σ a σ µ σ a = 2 δ µ . Taking the contrac-tion q µ P cµ , and following the steps in Appendix B, itsfrequency-independent part reads˜ N ( p , q ) = e Z dx { σ · q [ln Λ + ln 4 − ln[ | p + q | ]+ x ( x q + p · q )( p + x q ) (cid:21) − σ · ( p + x q ) σ · q σ · ( p + x q )2( p + x q ) (cid:27) , (D6)which after performing straightforward integrals yields˜ N ( p , q ) = Σ p + q ( iν + i Ω) − Σ p ( iν ) + e σ · q , (D7)and therefore the Ward-Takahashi identity is violatedwithin the hard-cutoff regularization scheme. The otherfrequency-dependent terms, actually, vanish, since theyhave the same form as within the dimensional regulariza-tion, but even if this were not the case they could not can-cel purely momentum-dependent term on the right-handside of Eq. (D7) that spoils the Ward-Takahashi identity.In fact, using exactly the same procedure, one can showthat within dimensional regularization with Pauli matri-ces treated in strictly D = 2 the Ward-Takahashi identity is also violated precisely because of the last term on theright-hand side in the frequency-independent part of thecontraction q µ P cµ . Appendix E: Kubo formula and the a.c.conductivity within dimensional regularization withPauli matrices in D = 2 − ǫ In this Appendix we perform explicit calculation of theCoulomb correction to the conductivity within the di-mensional regularization, in which both the momentumintegrals and the Pauli matrices are treated in D = 2 − ǫ ,which, as we demonstrated, is consistent with the U (1)gauge symmetry of the theory, and show that this regu-larization yields C = (11 − π ) / σ x σ · k σ x = 2 σ x k x − σ · k , (E1)the self-energy part of δ Π µν ( i Ω ,
0) for µ = ν = x , Π ( a ) xx ( i Ω , δ Π ( a ) xx ( i Ω ,
0) = 2
N e Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D/ Γ ( D ) × Z dω π Z d D k (2 π ) D k D − ( ω + k ) [( ω + Ω) + k ] × Tr { ( iω + σ · k )[ i ( ω + Ω) + 2 σ x k x − σ · k ]( iω + σ · k ) σ · k } . (E2)Performing the trace and the frequency integral, we ob-tain δ Π ( a ) xx ( i Ω ,
0) = − N e (cid:0) − D (cid:1) Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) (4 π ) D/ Γ ( D ) × Γ (cid:18) D − (cid:19) Z d D k (2 π ) D k D − (4 k − Ω )(Ω + 4 k ) . (E3)After subtracting the zero-frequency part of the Coulombcorrection to the polarization tensor, and using Eq. (41),the self-energy part of the Coulomb interaction correctionto the conductivity reads σ a = − σ Ω e (cid:0) − D (cid:1) Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D Γ ( D ) Γ (cid:0) D (cid:1) × Z ∞ dk k D − Ω + 12 k (Ω + 4 k ) , (E4)where σ is the Gaussian conductivity of the Diracfermions given by Eq. (51). The remaining integral thenyields σ a = − σ e Ω D − (cid:0) − D (cid:1) Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D Γ ( D ) Γ (cid:0) D (cid:1) × − D ( D − π cos( Dπ ) . (E5)Taking D = 2 − ǫ and expanding up to the order ǫ , weobtain the self-energy part of the Coulomb contributionto the conductivity σ a = 12 σ e (cid:18) − ǫ + 32 + γ − ln(64 π ) + O ( ǫ ) (cid:19) . (E6)Let us now concentrate on the vertex part of theCoulomb correction to the conductivity. Taking the tracein δ Π ( b ) xx given by Eq. (78), we have δ Π ( b ) xx ( i Ω ,
0) = 2 N Z ∞−∞ dω π Z d D k (2 π ) D Z ∞−∞ dω ′ π × Z d D p (2 π ) D V k − p ω + Ω) + k ][( ω ′ + Ω) + p ] × ω + k )( ω ′ + p ) { ω ( ω + Ω) ω ′ ( ω ′ + Ω) − [ ωω ′ + ( ω + Ω)( ω ′ + Ω)] k · p + 4 k · p k x p x − k x p − p x k + p k − [ ω ( ω ′ + Ω) + ω ′ ( ω + Ω)](2 k x p x − k · p ) − ω ( ω + Ω)(2 p x − p ) − ω ′ ( ω ′ + Ω)(2 k x − k ) } . (E7) Integration over the frequencies then yields δ Π ( b ) xx ( i Ω ,
0) = 2 N Z d D k (2 π ) D Z d D p (2 π ) D V k − p × kp (Ω + 4 k )(Ω + 4 p ) [Ω ( k x p x − k · p )+ 4( k · p k x p x + k p − p x k − p k x )] . (E8)After subtracting the zero-frequency part of δ Π xx ( i Ω , ),we obtain the vertex part of the Coulomb correction tothe conductivity σ b = 8 σ Ω Z d D k (2 π ) D Z d D p (2 π ) D V k − p k p (Ω + 4 k ) × + 4 p (cid:8) ( k · p k x p x + k p − p x k − p k x )[Ω + 4( k + p )] + 4 k p ( k · p − k x p x ) (cid:9) = σ b + σ b (E9)where σ b = 8 σ Ω Z d D k (2 π ) D Z d D p (2 π ) D V k − p Ω + 4( k + p ) k p (Ω + 4 k ) × + 4 p (cid:8) ( k · p k x p x + k p − p x k − p k x ) (cid:9) , (E10)and σ b = 32 σ Ω Z d D k (2 π ) D Z d D p (2 π ) D V k − p × k · p − k x p x kp (Ω + 4 k )(Ω + 4 p ) . (E11)The contribution σ b , by adding and subtracting Ω inthe first term in the numerator, may be rewritten as σ b = σ (1) b + σ (2) b + σ (3) b , (E12)where (1) b = 2 σ Ω Z d D k (2 π ) D Z d D p (2 π ) D V k − p k · p k x p x + k p − p x k − p k x k p h k + (cid:0) Ω2 (cid:1) i h p + (cid:0) Ω2 (cid:1) i " k + (cid:18) Ω2 (cid:19) + p + (cid:18) Ω2 (cid:19) = 4 σ Ω Z d D k (2 π ) D Z d D p (2 π ) D V k − p k · p k x p x + k p − p x k − p k x k p h k + (cid:0) Ω2 (cid:1) i , (E13) σ (2) b = − σ Ω Z d D k (2 π ) D Z d D p (2 π ) D V k − p kp h k + (cid:0) Ω2 (cid:1) i h p + (cid:0) Ω2 (cid:1) i , (E14)and σ (3) b = − σ Ω Z d D k (2 π ) D Z d D p (2 π ) D V k − p k · p k x p x − p k x k p h k + (cid:0) Ω2 (cid:1) i h p + (cid:0) Ω2 (cid:1) i . (E15)The advantage of this decomposition of the integral σ b isthat its diverging part is now isolated, and it is containedin the integral σ (1) b , whereas all the other integrals arefinite in D = 2.We first consider the term σ (1) b . Using the Feynmanparametrization1 A α B β C γ = Γ[ α + β + γ ]Γ[ α ]Γ[ β ]Γ[ γ ] Z dx Z − x dy × (1 − x − y ) α − x β − y γ − { (1 − x − y ) A + xB + yC } α + β + γ , (E16)we write 1 k | k − p | (cid:2) ( Ω2 ) + k (cid:3) = 4 π Z dx Z − x dy (1 − x − y ) / x − / h ( k − x p ) + x (1 − x ) p + y (cid:0) Ω2 (cid:1) i . (E17)Shifting the momentum k − x p → k , and retaining termseven in k as these are the only non-vanishing ones dueto the rotational invariance of the integrand, we obtain σ (1) b = 32 σ Ω e Z dx Z − x dy (1 − x − y ) − / x / × Z d D p (2 π ) D p − p x p Z d D k (2 π ) D (cid:20)(cid:18) − D (cid:19) k + x p (cid:21) × h k + x (1 − x ) p + y (cid:0) Ω2 (cid:1) i . (E18) After performing the integral over k , we have σ (1) b = 32 σ Ω e (4 π ) D Γ (cid:0) D (cid:1) (cid:18) − D (cid:19) Z dx Z − x dy x − / × (1 − x − y ) / Z ∞ dp p D − h x (1 − x ) p + y (cid:0) Ω2 (cid:1) i − D × (
12 ( D − (cid:18) − D (cid:19) + x p Γ (cid:0) − D (cid:1) x (1 − x ) p + y (cid:0) Ω2 (cid:1) ) . (E19)Integration over p then yields σ (1) b = σ e Ω D − − D Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) (cid:0) − D (cid:1) (4 π ) D Γ (cid:0) D (cid:1) × Z dx x − D/ (1 − x ) − D +12 Z − x dy y D − × (1 − x − y ) / . (E20)Using Eq. (64) and the identity Z − x dy (1 − x − y ) / y D − = √ π (cid:0) D − (cid:1) Γ( D ) (1 − x ) D − , (E21)after integration over y and x , σ (1) b acquires the form σ (1) b = σ e Ω D − − D (cid:0) − D (cid:1) Γ (cid:0) − D (cid:1) Γ (cid:0) − D (cid:1) (4 π ) D Γ (cid:0) D (cid:1) Γ( D ) × Γ (cid:18) D − (cid:19) Γ (cid:18) D + 12 (cid:19) Γ (cid:18) D − (cid:19) . (E22)Finally, expanding the previous result in the parameter ǫ , we obtain σ (1) b = 12 σ e (cid:20) ǫ −
12 (1 + 2 γ −
12 ln 2 − π ) + O ( ǫ ) (cid:21) . (E23)herefore, poles coming from the self-energy and the ver-tex parts cancel out, as it should be, since the theory ofCoulomb interacting Dirac fermions is renormalizable, atleast to the second order in the Coulomb coupling. Let us turn to the remaining contributions which areall finite in D = 2. We first consider the term σ (2) b inEq. (E14). Using the Feynman parametrization (E16),we have 1 p | k − p | (cid:2) ( Ω2 ) + p (cid:3) = 1 π Z dx Z − x dy × (1 − x − y ) − / x − / h ( p − x k ) + x (1 − x ) k + y (cid:0) Ω2 (cid:1) i . (E24)After shifting the momentum p − x k → p , and integrat-ing over p , the term σ (2) b in Eq. (E14) becomes σ (2) b = − π σ e Ω Z dx Z − x dy (1 − x − y ) − / × Z d k (2 π ) x − / k h x (1 − x ) k + y (cid:0) Ω2 (cid:1) i h k + (cid:0) Ω2 (cid:1) i . (E25)Integration over the remaining momentum variable thenyields σ (2) b = − π σ e Z dx Z − x dy p x (1 − x − y ) × y + p xy (1 − x ) . (E26)After integrating out the Feynman parameter y , the term σ (2) b is σ (2) b = iπ σ e Z dx sec − √ x √ x (1 − x ) = − π σ e . (E27)We now evaluate the term σ (3) b in Eq. (E15). UsingEq. (E17), after shifting the momentum variable, andretaining only terms even in p , we have σ (3) b = − σ Ω e Z dx Z − x dy (1 − x − y ) / x − / × Z d D k (2 π ) D k x k h k + (cid:0) Ω2 (cid:1) i × Z d D p (2 π ) D (cid:0) D − (cid:1) p − x k h p + x (1 − x ) k + y (cid:0) Ω2 (cid:1) i . (E28) After integrating over p and setting D = 2, we obtain σ (3) b = σ Ω e π Z dx Z − x dy (1 − x − y ) / x − / × Z ∞ dk h k + (cid:0) Ω2 (cid:1) i h x (1 − x ) k + y (cid:0) Ω2 (cid:1) i × "
32 + x k x (1 − x ) k + y (cid:0) Ω2 (cid:1) . (E29)Integration over k then gives σ (3) b = − π σ e Z dx Z − x dy (1 − x − y ) / × (1 − x ) − / x − q yx (1 − x ) + 2 x − √ y [ √ y + p x (1 − x )] , (E30)which, after integrating over the remaining variables,yields σ (3) b = 112 (4 + 3 π ) σ e . (E31)Let us now calculate the term σ b given by Eq. (E11).Using Eq. (E24), shifting the momentum p − x k → p ,and retaining only terms even in p , we have σ b = 4 σ e Ω Z dx Z − x dy (1 − x − y ) − / x − / × Z d D k (2 π ) D x ( k − k x ) k h k + (cid:0) Ω2 (cid:1) i × Z d D p (2 π ) D h p + x (1 − x ) k + y (cid:0) Ω2 (cid:1) i . (E32)After the integration over p and setting D = 2, we obtain σ b = σ e Ω(2 π ) Z dx Z − x dy (1 − x − y ) − / x / × Z ∞ dk k h k + (cid:0) Ω2 (cid:1) i h x (1 − x ) k + y (cid:0) Ω2 (cid:1) i . (E33)Integration over k then gives σ b = σ e π Z dx Z − x dy [(1 − x )(1 − x − y )] − √ y + p x (1 − x )= σ e π Z dx √ − x (cid:20) π + 2 i r x − x sec − √ x (cid:21) , (E34)where sec − x is the inverse function of sec x ≡ / cos x .Finally, integration over x yields σ b = 4 − π σ e . (E35)herefore, using Eqs. (E6), (E23), (E27), (E31), and(E35) we obtain the first order correction to the a.c. con-ductivity due to the Coulomb interaction δσ ( c ) = σ a + σ (1) b + σ (2) b + σ (3) b + σ b = 11 − π σ e , (E36)which corresponds to the value C = 11 − π Appendix F: Longitudinal conductivity usingdensity-density correlator
In order to obtain the longitudinal conductivity, weexpand δ Π ( c )00 ( i Ω , q ) in Eq. (83), which is the time com-ponent ( µ = ν = 0) of the Coulomb correction to thepolarization tensor (59), to the order q .Let us first consider the self-energy part (84) whichmay be written as δ Π ( a )00 ( i Ω , q ) = δ Π ( a ( i Ω , q ) + δ Π ( a ( − i Ω , − q ) , (F1)where δ Π ( a ( i Ω , q ) = N Z d k (2 π ) Z dω π Z d p (2 π ) Z dω ′ πV k − p Tr [ G k ( iω ) G p ( iω ′ ) G k ( iω ) G k + q ( iω + i Ω)] . (F2)Using Eq. (65) to integrate over the momentum p , andtaking the trace over Pauli matrices, we obtain δ Π ( a ( i Ω , q ) = 2 N e Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D/ Γ ( D ) × Z dω π Z d D k (2 π ) D k D − ( ω + k ) [( ω + Ω) + ( k + q ) ] × [( − ω + k ) k · ( k + q ) − ω ( ω + Ω) k ] . (F3)Note that above expression contains a part divergent in D = 2 that arises from the self-energy (65), and whenmultiplied with the remaining terms of the order ǫ givesa finite result, i.e., the final result does not have a polein ǫ . Therefore, in order not to overlook this subtlecancellation, we have to perform the integrations in D -dimensions first, and only at the end of the calculation totake D = 2 − ǫ , with ǫ →
0. Expanding the q -dependentterm in the denominator to the quadratic order in q , andkeeping only the terms quadratic in q in the expression for δ Π ( a ( i Ω , q ), we find δ Π ( a ( i Ω , q ) = 2 N e Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D/ Γ ( D ) × Z dω π Z d D k (2 π ) D k D − ( ω + k ) [( ω + Ω) + k ] × [( − ω + k ) k · ( k + q ) − ω ( ω + Ω) k ] × (cid:18) − k · q − q + 4( k · q ) ( ω + Ω) + k (cid:19) = δ Π ( a ( i Ω , q ) + δ Π ( a ( i Ω , q ) , (F4)where δ Π ( a ( i Ω , q ) = − N e Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D/ Γ ( D ) × Z dω π Z d D k (2 π ) D ( − ω + k )( k · q ) k − D ( ω + k ) [( ω + Ω) + k ] , (F5)and δ Π ( a ( i Ω , q ) = 2 N e Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D/ Γ ( D ) × Z dω π Z d D k (2 π ) D ( − ω + k ) k − ωk ( ω + Ω) k − D ( ω + k ) [( ω + Ω) + k ] × (cid:18) − q + 4( k · q ) ( ω + Ω) + k (cid:19) . (F6)We first consider the term δ Π ( a in Eq. (F5). Afterintegrating over ω and using the rotational symmetry ofthe integrand, we have δ Π ( a ( i Ω , q ) = − N e Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D/ Γ ( D ) × q D Z d D k (2 π ) D k D − k − k Ω − Ω (Ω + k ) . (F7)After integrating over k , and expanding the result in ǫ ,we obtain to the order ǫ δ Π ( a ( i Ω , q ) = 364 N e q ω = 34 σ e q | Ω | . (F8)The term δ Π ( a ( i Ω , q ) given by Eq. (F6), after integra-tion over the frequency and using the rotational symme-try of the integrand, acquires the form δ Π ( a ( i Ω , q ) = − N e Γ (cid:0) − D (cid:1) Γ (cid:0) D +12 (cid:1) Γ (cid:0) D − (cid:1) (4 π ) D/ Γ ( D ) × q D Z d D k (2 π ) D k D − (Ω + 4 k ) × [16( D − k + 24 k Ω + (3 − D )Ω ] . (F9)Integration over k in the last expression, and expansionin ǫ then yield δ Π ( a ( i Ω , q ) = − N e q | Ω | = − σ e q | Ω | . (F10)herefore, using Eq. (F4), we have δ Π ( a ( i Ω , q ) = 164 N e q | Ω | = 14 σ e q | Ω | , (F11)which together with Eq. (F1) gives for the self-energypart δ Π ( a )00 ( i Ω , q ) = 12 σ e q | Ω | . (F12) Let us now concentrate on the vertex part of thedensity-density correlator given by Eq. (85). In order tocalculate this contribution, we need the following integralover the frequency Z ∞−∞ dω ′ π G p ( iω ′ ) G p − q ( iω ′ − i Ω) = 12 1( | p − q | + p ) + Ω × (cid:18) −| p − q | − p + i Ω σ · p − q | p − q | + ( − i Ω σ · p + p − p · q − iσ p × q ) | p − q | + p | p − q | p (cid:19) . (F13)Since the frequency integrals in Eq. (85) are decoupled, we use the above equation to perform them separately, andthen take the trace using the identity Tr[( a + b · σ )( c + d · σ )] = 2 ac + 2 b · d . (F14)Integrations over ω and ω ′ in the vertex part, given by Eq. (85), yield the coefficients a = − ( | p − q | + p ) (cid:18) − p · ( p − q ) p | p − q | (cid:19) , c = − ( | k − q | + k ) (cid:18) − k · ( k − q ) k | k − q | (cid:19) , b = (cid:26) i Ω (cid:18) p − q | p − q | − p p (cid:19) , − i ( p × q ) z | p − q | + p | p − q | p (cid:27) , d = (cid:26) i Ω (cid:18) k − q | k − q | − k k (cid:19) , i ( k × q ) z | k − q | + k | k − q | k (cid:27) . (F15)in Eq. (F14), where ( p × q ) z = p x q y − p y q x . Notice that b and d are three-dimensional vectors. Expanding theexpressions in Eq. (F15) to the order q , we obtain a = − q p − ( q · p ) p ,c = − q k − ( q · k ) k . b = (cid:26) i Ω p ( p · q ) − q p p , − i ( p × q ) z p (cid:27) , d = (cid:26) i Ω k ( k · q ) − q k k , i ( k × q ) z k (cid:27) , (F16)thus ac = O ( q ), and therefore does not contribute tothe conductivity, whereas b · d = − Ω [ p ( p · q ) − q p ] · [ k ( k · q ) − q k ] p k + 4 ( p × q ) z ( k × q ) z pk . (F17)Setting D = 2 in the momentum integrals, since thereare no divergent subintegrals in the vertex part as it wasthe case in the self-energy term, the vertex part of the density-density correlator to the order q has the form δ Π ( b )00 ( i Ω , q ) = δ Π ( b ( i Ω , q ) + δ Π ( b ( i Ω , q ) , where δ Π ( b ( i Ω , q ) = − N Ω Z d k (2 π ) Z d p (2 π ) V k − p × (cid:0) p ( p · q ) − q p (cid:1) · (cid:0) k ( k · q ) − q k (cid:1) p k h (2 p ) + Ω i h (2 k ) + Ω i ≡ N Z d k (2 π ) (cid:0) k ( k · q ) − q k (cid:1) · I ( k , q , Ω) k ((2 k ) + Ω ) , (F18)and δ Π ( b ( i Ω , q ) = 2 N Z d k (2 π ) Z d p (2 π ) V k − p × p · k q − p · q q · k pk h (2 k ) + Ω i [(2 p ) + Ω ] ≡ N Z d k (2 π ) I ( k , q , Ω) k [(2 k ) + Ω ] . (F19)ere, we defined I ( k , q , Ω) ≡ − Ω Z d p (2 π ) V k − p p ( p · q ) − q p p h p + (cid:0) Ω2 (cid:1) i , (F20)and I ( k , q , Ω) ≡ Z d p (2 π ) V k − p p · k q − p · q q · k p h p + (cid:0) Ω2 (cid:1) i . (F21)We consider the term δ Π ( b ( i Ω , q ), given by Eq. (F18),and, as a first step, calculate the integral I ( k ). Usingthe Feynman parametrization (E16), this integral can bewritten as I ( k , q , Ω) = − e Ω Z dx Z − x dy √ − x − y √ x × Z d p (2 π ) p ( p · q ) − q p (cid:2) ( p − x k ) + x (1 − x ) k + y Ω (cid:3) , (F22)which after shifting the momentum variable p − x k → p ,and integrating over p , acquires the form I ( k , q , Ω) = − e Ω π Z dx Z − x dy √ − x − y √ x × (cid:18) − q x (1 − x ) k + y Ω + x k ( k · q ) − q x k ( x (1 − x ) k + y Ω ) (cid:19) . (F23)Using the previous result for the integral I ( k , q , Ω), afterintegration over k , we obtain δ Π ( b ( i Ω , q ) = − N e q π | Ω | Z dx Z − x dy √ − x − y √ xy × √ y + p (1 − x ) x + x √ − x (cid:16)p (1 − x ) x + √ y (cid:17) = − N e q π | Ω | (cid:20) π π (cid:18) π − (cid:19)(cid:21) = N e q | Ω | (cid:18) − π (cid:19) = e q | Ω | σ (cid:18) − π (cid:19) . (F24)We now evaluate the term δ Π ( b ( i Ω , q ) in Eq. (F19).First, we compute the integral I ( k , q , Ω) in Eq. (F21)using the Feynman parametrization (E16) I ( k , q , Ω) = e Z dx Z − x dy √ x √ − x − y × Z d p (2 π ) p · k q − p · qq · k (( p − x k ) + x (1 − x ) k + y Ω ) = e q π Z dx Z − x dy √ x √ − x − y k [ x (1 − x ) k + y Ω ] . (F25) We use this result to calculate integral over k in Eq.(F19). After performing straightforward integrations, wefind δ Π ( b ( i Ω , q ) = N e q π | Ω | Z dx Z − x dy √ x √ − x − y × x (1 − x ) + p x (1 − x ) y = N e q | Ω | (4 − π )= e q | Ω | − π . (F26)Using Eqs. (F24) and (F26), the vertex part of δ Π ( c )00 reads δ Π ( b )00 ( i Ω , q ) = e q | Ω | − π , (F27)which together with the self-energy part (F12) yields upto the order q δ Π ( c )00 ( i Ω , q ) = e q | Ω | − π . (F28)Finally, using Eq. (40), we obtain the Coulomb interac-tion correction to the longitudinal conductivity σ ( c ) k = 11 − π σ e , (F29)in agreement with the result (81) obtained from thecurrent-current correlator, which is expected, since thedimensional regularization explicitly preserves the U (1)gauge symmetry of the theory of the Coulomb interactingDirac fermions. Appendix G: Kubo formula and the a.c.conductivity within dimensional regularization withPauli matrices in D = 2 In this Appendix we present the calculation of theCoulomb correction to the a.c. conductivity using di-mensional regularization, but treating the Pauli matricesin D = 2 spatial dimensions strictly . As we commentedearlier, this leads to the violation of the Ward identity,and thus it is incompatible with the U (1) gauge sym-metry of the theory, but yields the number obtained inRef. 9. We use Eq. (41) with the Coulomb correctionto the polarization tensor given by Eq. (76). Since thesystem of Dirac fermions interacting only via the long-range Coulomb interaction is isotropic, translationallyand time-reversal invariant, the trace over spatial indicesof the polarization tensor at zero momentum and a finitefrequency is Π aa ( i Ω ,
0) = D Π B ( i Ω , D = 2, σ a σ µ σ a = 2 δ µ , takingthe trace over the Pauli matrices, integrating over therequencies, and subtracting the zero-frequency part, theself-energy contribution reads˜ σ a = − D σ Ω Z d D k (2 π ) D Z d D p (2 π ) D πe | k − p |× k · p kp Ω + 12 k k (Ω + 4 k ) . (G1)Performing the momentum integrals, we have˜ σ a = − σ e Ω D − − D Γ( D +12 )(1 − D ) π D − Γ( D/ D )Cos( πD ) × Γ (cid:18) − D (cid:19) Γ (cid:18) D − (cid:19) = 1 D − σ a , (G2) with σ a given by Eq. (E5). This result, after expandingin ǫ , reads˜ σ a = 12 σ e (cid:18) ǫ + 12 + γ − ln 64 π + O ( ǫ ) (cid:19) . (G3)The vertex part of the Coulomb correction to the con-ductivity is obtained from Eq. (78). The trace over spa-tial indices of δ Π ( b ) µν in Eq. (78) at the momentum q = 0,using the standard anticommutation and trace relationsfor the Pauli matrices in D = 2, assumes the form δ Π ( b ) aa ( i Ω , ) = − N Z ∞−∞ dω π Z d D k (2 π ) D Z ∞−∞ dν π Z d D p (2 π ) D × V k − p (( ω + Ω)( ν + Ω) − k · p )( − ων + k · p ) − ων k · p + k p ( ω + k )(( ω + Ω) + k )( ν + p )(( ν + Ω) + p ) . (G4)Performing the integrals over the frequencies ω and ν in the above equation, we have δ Π ( b ) aa ( i Ω , ) = 2 N Z d D k (2 π ) D Z d D p (2 π ) D V k − p k · p (4 k · p − Ω ) kp (Ω + 4 k )(Ω + 4 p ) . (G5)After subtracting the zero-frequency part of δ Π aa ( i Ω , σ b = 8 D σ Ω Z d D k (2 π ) D Z d D p (2 π ) D πe | k − p | k · p (cid:2) k · p (Ω + 4 k + 4 p ) + 4 k p (cid:3) k p (Ω + 4 k )(Ω + 4 p ) = ˜ σ b + ˜ σ b + ˜ σ b , (G6)where ˜ σ b = 4 D σ Ω Z d D k (2 π ) D Z d D p (2 π ) D πe | k − p | Ω + 4( k + p ) kp (Ω + 4 k )(Ω + 4 p ) , (G7)diverges in D = 2, as we will show in the following, and the remaining integrals˜ σ b = 32 D σ Ω Z d D k (2 π ) D Z d D p (2 π ) D πe | k − p | k · p kp (Ω + 4 k )(Ω + 4 p ) , (G8)and ˜ σ b = 4 D σ Ω Z d D k (2 π ) D Z d D p (2 π ) D πe | k − p | [2( k · p ) − k p ][Ω + 8 k ] k p (Ω + 4 k )(Ω + 4 p ) (G9)are finite in two dimensions. The above decomposition is obtained from Eq. (G6) by adding and subtracting Ω inthe term in the numerator multiplying ( k · p ) . We now further decompose the integral ˜ σ b in order to isolate itsdiverging part ˜ σ b = ˜ σ (1) b + ˜ σ (2) b . (G10)Here, the term ˜ σ (1) b = 8 D σ Ω Z d D k (2 π ) D Z d D p (2 π ) D πe | k − p | kp (Ω + 4 k ) , (G11)as the pole in Laurent expansion in ǫ , while the remaining one˜ σ (2) b = − D σ Ω Z d D k (2 π ) D Z d D p (2 π ) D πe | k − p | kp (Ω + 4 k )(Ω + 4 p ) (G12)is finite in D = 2.We first consider the term divergent in D = 2, namely, ˜ σ (1) b . Integration over p in Eq. (G11), after performing thestandard steps, yields Z d D p (2 π ) D p | k − p | = k D − π Γ (cid:0) − D (cid:1) (cid:2) Γ (cid:0) D − (cid:1)(cid:3) (4 π ) D/ Γ( D − , (G13)while after integrating over k , we find˜ σ (1) b = σ e Ω D − Γ (cid:0) − D (cid:1) (cid:2) Γ (cid:0) D − (cid:1)(cid:3) Γ (cid:0) D − (cid:1) Γ (cid:0) − D (cid:1) D − π D D Γ( D − (cid:0) D (cid:1) . (G14)The previous expression, after expanding in ǫ , reads˜ σ (1) b = 12 σ e (cid:20) ǫ + 12 − γ + ln(64 π ) + O ( ǫ ) (cid:21) . (G15)Thus the poles in ˜ σ a , given by Eq. (G3), and ˜ σ (1) b cancel out, and˜ σ (1) b + ˜ σ a = 12 σ e + O ( ǫ ) . (G16)The remaining integrals are finite in D = 2, and can be calculated as follows. Consider the term ˜ σ b given by Eq.(G8). Using Feynman parametrization (E16), we write1 p | k − p | (cid:2) ( Ω2 ) + p (cid:3) = 1 π Z dx Z − x dy (1 − x − y ) − / x − / h ( p − x k ) + x (1 − x ) k + y (cid:0) Ω2 (cid:1) i . (G17)As usual, we now shift the momentum p − x k → p , and integrate over p , to obtain˜ σ b = 12 π σ e Ω Z dx Z − x dy (1 − x − y ) − / x / × Z d k (2 π ) k h x (1 − x ) k + y (cid:0) Ω2 (cid:1) i h k + (cid:0) Ω2 (cid:1) i . (G18)The integral over k in the previous equation then yields˜ σ b = 14 π σ e Z dx Z − x dy p (1 − x )(1 − x − y ) (cid:16)p x (1 − x ) + √ y (cid:17) , (G19)while the integration over the variable y gives˜ σ b = 14 π σ e Z dx √ − x (cid:20) π + 2 i r x − x sec − √ x (cid:21) . (G20)Finally, after performing the remaining integral over x , we obtain˜ σ b = 4 − π σ e . (G21)Similarly, using Eq. (G17) after shifting p − x k → p , and integrating over p , the term ˜ σ (2) b in Eq. (G12) becomes˜ σ (2) b = − π σ e Ω Z dx Z − x dy (1 − x − y ) − / x − / × Z d k (2 π ) k h x (1 − x ) k + y (cid:0) Ω2 (cid:1) i h k + (cid:0) Ω2 (cid:1) i . (G22)ntegration over the remaining momentum variable yields˜ σ (2) b = − π σ e Z dx Z − x dy p x (1 − x − y ) (cid:16) y + p xy (1 − x ) (cid:17) . (G23)After integrating out the Feynman parameter y , the term ˜ σ (2) b is˜ σ (2) b = i π σ e Z dx sec − √ x √ x (1 − x ) = − π σ e . (G24)Let us now calculate the term σ b in Eq. (G9). Using Feynman parametrization (E16), we can write1 p | k − p | (cid:2) ( Ω2 ) + p (cid:3) = 4 π Z dx Z − x dy (1 − x − y ) / x − / h ( p − x k ) + x (1 − x ) k + y (cid:0) Ω2 (cid:1) i . (G25)After shifting p − x k → p , and integrating over p , we have˜ σ b = 18 π σ e Ω Z dx Z − x dy (1 − x − y ) / x / × Z d k (2 π ) k (Ω + 8 k ) h x (1 − x ) k + y (cid:0) Ω2 (cid:1) i h k + (cid:0) Ω2 (cid:1) i , (G26)while integration over k then gives˜ σ b = 18 π σ e Z dx Z − x dy √ − x − y h x (1 − x ) + 4 p xy (1 − x ) + 2 y i (1 − x ) / (cid:16) y + p xy (1 − x ) (cid:17) (cid:16) √ y + p x (1 − x ) (cid:17) , (G27)which after calculating the integral over the variable y becomes˜ σ b = 18 π σ e Z dx (1 − x ) − / h x − √ x + π (1 − x ) + 2 i p x (1 − x ) sec − √ x i = 14 − π σ e . (G28)Therefore, we calculated all the terms needed to obtainthe Coulomb correction to the conductivity within thisregularization scheme. Using Eqs. (G16), (G21), (G24),and (G28), we obtain the final result as found in Ref. 9˜ σ ( c ) = ˜ σ a +˜ σ b = ˜ σ a +˜ σ (1) b +˜ σ (2) b +˜ σ b +˜ σ b = 25 − π σ e , (G29)which thus yields C = (25 − π ) /
12, different than one inEq. (82) obtained using dimensional regularization withPauli matrices in D = 2 − ǫ spatial dimensions. Appendix H: Direct evaluation of the a. c.conductivity from the Kubo formula in twodimensions
In this section we show that the value of the coefficient C may also be unambiguously computed directly in two spatial dimensions, provided extra care is taken in eval-uations of the integral that defines it. The result is thenin agreement with the general dimensional-regularizationscheme used in the rest of the paper.As suggested by Mishchenko , the expression for thefirst-order correction to conductivity may also be conve-niently written as a sum of three terms σ ′ = σ ′ a + σ ′ b + I (H1)where σ ′ a = e ω Z d p d k (2 π ) V p − k cos θ pk ω − p p ( ω + 4 p ) , (H2) σ ′ b = e ω Z d p d k (2 π ) V p − k cos θ pk − ( ω /pk ) cos θ ( ω + 4 k )( ω + 4 p ) , (H3) = − e ω Z d p d k (2 π ) V p − k cos θ pk k − p cos θp k ( ω + 4 p ) . (H4)Here we have set the fermi velocity to unity for simplicity,and taken ω to be the Matsubara frequency.Now we show that all three terms are UV convergent,and when summed yield the same result as the dimen-sional regularization. First, σ + σ ′ a = 2 e ω Z d p (2 π ) v p p (1 + 4 v p p ) + O ( V ) , (H5)where v p = v F (cid:18) e p (cid:19) (H6)is the renormalized velocity. A simple change of variablesthen gives σ ′ a σ = β v ( e ) , (H7)where β v ( e ) = − dv p v p d ln( p ) = e O ( e ) (H8) is the beta-function for the velocity. The coefficients inthe series expansion of β v ( e ) are universal numbers, andtherefore, σ ′ a σ = e , with the result σ ′ b σ = (cid:18) − π (cid:19) e . (H10)We have also reproduced this value numerically within atenth of a percent.Finally, the third integral may be written as Iσ = − e π lim ω → Z Λ /ω dpp (1 + 4 p ) Z Λ /ω dk ∂∂k Z π dθ cos θ ( p + k − pk cos θ ) / . (H11)where we have carefully retained finite upper cutoffs onthe momentum integrals. One finds Iσ = − e π lim ω → Z Λ /ω dpp (1 + 4 p ) F (cid:18) p, Λ ω (cid:19) , (H12)where F ( x, y ) = 2 | x − y | xy (cid:20) ( x + y ) K (cid:18) − xy ( x − y ) (cid:19) (H13) − ( x + y ) E (cid:18) − xy ( x − y ) (cid:19)(cid:21) , and K ( z ) and E ( z ) are the elliptic functions. The sin-gle remaining numerical integration quickly converges toa value quite independent of the ratio Λ / Λ , and weobtain Iσ = 0 . e → e , (H14)where the last equality is the conjectured exact result.All put together gives σ ′ σ = (cid:20)
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