Conductivity of interacting massless Dirac particles in graphene: Collisionless regime

Collisionless regime

Juricic, V.; Vafek, O.; Herbut, I.F.

Citation

Juricic, V., Vafek, O., & Herbut, I. F. (2010). Conductivity of interacting massless Dirac particles in graphene: Collisionless regime. Physical Review B, 82(23), 235402.

doi:10.1103/PhysRevB.82.235402

Version: Not Applicable (or Unknown)

License: Leiden University Non-exclusive license Downloaded from: https://hdl.handle.net/1887/51746

Note: To cite this publication please use the final published version (if applicable).

Conductivity of interacting massless Dirac particles in graphene: Collisionless regime

Vladimir Juričić,^1,2Oskar Vafek,³and Igor F. Herbut⁴

1Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

2Max-Planck-Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany

3National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahasse, Florida 32306, USA

4Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 共Received 16 September 2010; published 1 December 2010兲

We provide detailed calculation of the ac conductivity in the case of 1/r Coulomb interacting massless Dirac particles in graphene in the collisionless limit when␻ⰇT. The analysis of the electron self-energy, current vertex function, and polarization function, which enter into the calculation of physical quantities including the ac conductivity, is carried out by checking the Ward-Takahashi identities associated with the electrical charge conservation and making sure that they are satisfied at each step. We adopt a variant of the dimensional regularization of Veltman and ’t Hooft by taking the spatial dimension D = 2 −⑀ for ⑀⬎0. The procedure adopted here yields a result for the conductivity correction which, while explicitly preserving charge conser- vation laws, is nevertheless different from the results reported previously in literature.

DOI:10.1103/PhysRevB.82.235402 PACS number共s兲: 71.10.Pm, 73.63.⫺b

I. INTRODUCTION

The role of Coulomb electron-electron interactions in sys- tems described by massless two-dimensional Dirac fermi- ons has been a subject of interest for some time.^1–15Discov- ery of graphene, a single atomic layer of sp²hybridized car- bon, and more recently of topological insulators, both of which support such massless Dirac fermions, brought this issue into sharp focus. In particular, which physically mea- surable quantities are modified from their noninteracting val- ues, and by how much, would allow deeper understanding of the physics governed by electron-electron interactions in these systems.

When weak, the unscreened 1/r Coulomb interactions are expected to modify the velocity of the Dirac fermions as v_F→vF+^e₄²ln⌳/k, where k is the wavenumber measured from the Dirac point. This modification of the electronic dispersion is expected to lead to logarithmic suppression of the density of states near the Dirac point, an effect, in principle, observable in tunneling experiments. In addition, the low-temperature electronic contribution to the specific heat should be suppressed from T²to T²/log²T, as shown in Ref. 4, and the strength of this suppression is related to the strength of the Coulomb interaction.

The role of Coulomb interaction in ac electrical trans- port was investigated by Mishchenko in Ref. 7, who origi- nally concluded that the ac conductivity ␴共␻兲 vanishes as ␻→0 and the system is a 共weak兲 insulator. Were this the case, the interactions would have dramatic effect on transport since the ac conductivity of the noninteracting system is finite,¹⁶ i.e., ␴0共␻兲=␲^e2/2h for ␻ⰇT. This was later argued to be incorrect by Sheehy and Schmalian,⁸ and independently by the present authors⁹ using renormalization-group 共RG兲 scaling analysis. While the former presented only a scaling argument, without calculat- ing the correction to transport, the latter reported on an explicit calculation where

␴共␻兲 =␴0

冢

^{1 +CvF+ee422ln⌳␻}

冣

^共1兲

with the coefficient found to be C=共25−6␲兲/12⯝0.5125.

Note that, since e² does not renormalize,^9,17 any change in the cutoff in the above expression for the conductivity may be compensated by a redefinition of the Fermi velocity, vF. At small ␻ the correction vanishes, and the noninteracting value of ␴ is recovered. At small but finite frequencies, the correction scales as 1/兩log␻兩, with the interaction indepen- dent prefactor determined byC. The numerical value of this correction, which can be understood as correction to scaling near the Gaussian fixed point and which is expected to be universal, has since been a subject of debate. In subsequent work, Mishchenko¹⁰ recovered the functional form in Eq.

共1兲, which gives metallic conductivity at small␻, but argued for a different value of C=共19−6␲兲/12⯝0.01254 which happens to be much smaller than the one found by us. Tech- nically, the difference originated from different regulariza- tion adopted in the two approaches. The standard momentum space cutoff, motivated by the underlying discrete lattice structure and reported in Ref. 9 was questioned in Ref. 10, where the correction to conductivity was calculated using a cutoff on the 1/r interaction and argued to be the same re- gardless of whether it is calculated using Kubo formula 共current-current correlator兲 or continuity equation and density-density correlator. The same calculational procedure was later advocated by Sheehy and Schmalian,¹⁸who argued that unlike hard cutoff in momentum space, cutoff on the interaction leads to expressions obeying Ward-Takahashi identity. In addition, they claimed the result obtained in such way is consistent with the experimentally measured optical conductivity, where, surprisingly, no discernible correction to the noninteracting value was reported.¹⁹

In quantum-field theories, it seems reasonable that if two ultraviolet共UV兲 regularization schemes give different results for physical quantities, then the regularization that is typi-

cally chosen is the one which respects charge U共1兲 symme- try, as is the case for chiral anomaly in 共3+1兲-dimensional massless quantum electrodynamics 共QED兲, for instance.²⁰ Here we argue that the regularization of the electron-electron interaction alone is incomplete and cannot serve as a consis- tent regularization of the theory. We also show by explicit calculation that the dimensional regularization used here pre- serves the Ward-Takahashi identity, i.e., that it is consistent with U共1兲 gauge symmetry of the theory, and that, moreover, has the additional advantage of serving as an interaction- independent regularization scheme for the whole field theory.

The interaction correction to the conductivity within this regularization scheme is calculated independently using the current-current and the density-density correlators, which both yield the same number C=共11−3␲兲/6⯝0.2625 in Eq.

共1兲, precisely as a consequence of explicitly preserved U共1兲 gauge symmetry. Furthermore, we show that while the hard- cut-off regularization, in principle, violates the Ward- Takahashi identity, the original integral expression⁹ for the constant C is in fact UV convergent, and when computed with necessary care it unambiguously leads to the same value as quoted above.

A comparison with experiment which has been performed at high frequencies near the cutoff^19,21共see also Ref.22兲 may be misleading, since the result for the leading logarithmic correction to the conductivity in Eq.共1兲 is valid only at fre- quencies of the order of 1 meV, much smaller than the cutoff.

As the Coulomb coupling constant in graphene e²/vFis be- lieved to be of order 1, we expect that in this region the interaction corrections to different observables, relative to the values in the noninteracting theory, should be significant.

Why the interaction correction to the conductivity, in particu- lar, appears to be small even in the high-frequency regime is unclear at the moment.

Whereas the results in the collisionless limit共␻ⰇT兲 dis- cussed here at least, in principle, follow from a straightfor- ward application of the perturbative renormalization group, transport in the collision-dominated regime共␻ⰆT兲 requires resummation of an infinite series of Feynman diagrams. This is easily seen in the noninteracting limit where a finite tem- perature T produces a finite, linear in T, “Drude”␦^-function response in conductivity, ⬃T␦共␻兲. Collisions due to the electron-electron scattering lead to broadening of the␦^func- tion and clearly the result must be nonanalytic in e²/vFas the interaction V共r兲→0. Alternative approach has been ad- vanced in Refs. 23 and24, where the leading correction is argued to be captured by the solution to the quantum Boltz- man equation with the collision integral calculated perturba- tively in the interaction strength. In the clean limit, the con- ductivity in the collision dominated regime is found to increase with decreasing T and proportional to ln²共T/⌳兲. In- terestingly, experiments on suspended samples at the neutral- ity point²⁵find conductivity which decreases with decreasing T. Finally, T-linear increase of the dc conductivity observed in small devices,²⁶has been argued to arise from purely bal- listic transport,²⁷ where conductivity grows with the sample size L and temperature T as␴⬃TL/បvF.

The paper is organized as follows: in Sec.IIwe introduce the Lagrangian and the response functions, and in Sec.IIIwe discuss different regularization schemes for massless Dirac

fermions. In Sec. IV, we review some well-known results regarding the polarization tensor and the conductivity. In Sec. V, we explicitly construct polarization tensor for the noninteracting theory and in Sec. VI we consider the same problem for the contact interactions to first order in the in- teraction strength and to O共N兲. We do not discuss the 共random-phase approximation-like兲 contribution to the order N² which while simple to calculate, does not contribute to transport. The main results of the paper are presented in Sec.

VII, where we show that the Coulomb correction to the po- larization tensor is transversal, as well as that the Coulomb vertex function obeys the Ward-Takahashi identity within the dimensional regularization. In this section, we also present calculations of the Coulomb correction to the ac conductivity using both the current-current correlator共Kubo formula兲 and the density-density correlator, within the dimensional regu- larization. SectionVIII is reserved for further discussion of these results and comparison with previous results reported in the literature. Various technical details of the calculations are presented in the Appendices A–H.

II. HAMILTONIAN, LAGRANGIAN, AND THE RESPONSE FUNCTIONS We start with the Hamiltonian

Hˆ =

冕

^{dDr␺†共r兲vF␴apa␺共r兲}

+1

2

冕

^{dDrdDr⬘␺†共r兲␺}共r兲V共兩r − r⬘兩兲␺^†共r⬘兲␺共r⬘兲, 共2兲 where we consider N copies of two-component Fermi fields

␺共r,␶兲 共which therefore has 2N components兲, the momentum operator pa= −iប⳵a and␴a are Pauli matrices. Operators in the interaction term are assumed normal ordered. Hereafter, the Latin letters a , b are used only for the spatial indices while the Greek letters ␮,␯ are reserved for the spacetime ones and summation over the repeated indices is assumed.

V共r兲 is the two-body interaction potential, which is left un- specified at the moment. Later we will consider two different cases: a short-range contact interaction V共r兲=u␦共r兲 and the three-dimensional 共3D兲 Coulomb potential V共r兲=e²/r with e²/vF as the dimensionless Coulomb coupling constant. To simplify the notation, we will work in the natural units ប

= c = k_B= 1. When the speed of light c does not appear, we will also set vF= 1. In our final results we will restore the physical units.

The corresponding imaginary time Lagrangian is

L = L0+Lint, 共3兲 where

L0=

冕

^{dDr␺†共␶,r兲}

冋

^{⳵␶⳵ +vF␴· p}

册

^{␺共␶,r兲 共4兲}

and

Lint=1

2

冕

^{dDrdDr⬘␳共␶,r兲V共兩r − r⬘兩兲␳共␶,r⬘兲, 共5兲}

where␳共␶, r兲⬅␺^†共␶, r兲␺共␶, r兲 is the density of fermions. The quantum partition function can then be written as the imagi- nary time Grassman path integral²⁸

Z =

冕

^{D␺†D␺exp}

冉

⁻

^冕

^0␤d␶L

冊

^{, 共6兲}

where the inverse temperature factor␤^{= 1}/共kBT兲. In the sec- tions which follow, the additional imaginary-time index on the Fermi field ␺共␶, r兲 inside a path integral automatically means that they are considered to be coherent state Grassman fields. We will take T→0 first and then perform the calcula- tions. Note that in light of the discussion in the Introduction, taking T→0 first automatically sets the collisionless limit.

By the standard spectral representation theorems we can first calculate the correlation functions as imaginary time- ordered products, Fourier transform over time and then ana- lytically continue to find the physical retarded共or advanced兲 response functions.²⁸Specifically, for some bosonic operator Oˆa共t,r兲 in the 共real time兲 Heisenberg representation, the re- tarded correlation function

S_ab^ret共t − t⬘^,r,r⬘^{兲 = − i}␪共t − t⬘^{兲具关Oˆa共t,r兲,Oˆb共t}⬘^,r⬘^{兲兴典 共7兲} can be related to the imaginary time-ordered correlation function

S_ab共␶⁻␶⬘^,r,r⬘兲 = − 具T_␶Oˆa共␶^,r兲Oˆb共␶⬘^,r⬘兲典, 共8兲 where

Oˆa共␶^,r兲 = e^␤HˆOˆa共r兲e^−␤Hˆ. 共9兲 In the Eqs. 共7兲 and 共8兲 the angular brackets denote thermal averaging

具 ¯ 典 =1

ZTr共e^−␤Hˆ¯兲. 共10兲 Specifically, the frequency Fourier transforms

S_ab^ret共␻^;r,r⬘兲 =

冕

−⬁

⬁

dte^i⍀tS_ab^ret共t,r,r⬘兲, 共11兲

S_ab共i⍀n;r,r⬘兲 =

冕

0

␤

d␶^ei⍀n␶Sab共␶^,r,r⬘兲 共12兲 satisfy

S_ab^ret共⍀;r,r⬘兲 = Sab共i⍀n→ ⍀ + i0⁺;r,r⬘兲, 共13兲 where the bosonic Matsubara frequency is ⍀n= 2␲ⁿ/␤ ^for n = 0 ,⫾1, ⫾2,.... We will use the above relations in what follows when we focus on the electrical conductivity, in which case the bosonic operator Oˆ of interest will be either charge density or charge current.

For completeness we note that for V共r兲=0 the two- particle imaginary time Green’s function is

具␺共i␻,k兲␺^†共i␻⬘^,k⬘兲典 =␤␦␻,␻⬘共2␲兲²␦共k − k⬘兲Gk共i␻兲, 共14兲 where

Gk共i␻兲 =i␻⁺␴^{· k}

␻^{2+ k2 , 共15兲} which will be used extensively in the later sections. Strictly speaking, in any solid state system which supports massless Dirac particles, the above propagator is valid only for wavevectors smaller than some cutoff ⌳, which depends on the physical situation. In the case of electrons on the honey- comb lattice, the order of magnitude of the cutoff, Å⁻¹, is determined by the requirement that the true electronic disper- sion does not deviate appreciably from the conical共Dirac兲.

III. REGULARIZATION SCHEMES FOR MASSLESS DIRAC FERMIONS

Since we are interested in the long distance 共low- frequency兲 behavior of physical quantities, we can use the above low-energy field theory, given by the above Lagrang- ian with the corresponding propagators, provided that diver- gent terms in the perturbation theory are properly regular- ized. In the context of high energy physics it is also well known that a quantum-field theory of Dirac fermions needs to be regularized²⁰ and typically there is no unique way of doing so. Additional requirements, usually based on the sym- metries of the theory, determine what type of regularization should be employed.

In case of the theory of the Coulomb interacting Dirac fermions, we will require that the U共1兲 gauge symmetry must be preserved, or equivalently, that the charge must be conserved. As we show below, dimensional regularization introduced by ’t Hooft and Veltman²⁹is consistent with this requirement. Before discussing this regularization scheme, let us briefly review the hard cutoff and the Pauli-Villars regularization schemes in the context of the fermionic field theory considered here.

A. Hard cutoff

The idea of the hard-cut-off regularization is to impose a cutoff in the upper limit of an otherwise divergent momen- tum integral. Physically, this is due to the k-space restriction on the modes which appear in the theory, a condition which appears naturally within Wilson formulation of the RG.³⁰ The singular part of the integral then appears dependent on the cut-off scale. Although very simple to implement, this regularization scheme is known to violate U共1兲 gauge sym- metry of QED, for instance.²⁰ Terms that violate the gauge symmetry appear as a power of the cut-off scale and must be subtracted in order to ensure that the gauge symmetry is preserved. On the other hand, the typical divergent terms appear as the logarithm of the cut-off scale. Of course, the cut-off scale must not appear explicitly in any observable quantity in order for the theory to be physically meaningful.

The disappearance of the cut-off scale⌳ indeed occurs in the calculation of the interaction correction to the ac conductiv-

ity within quantum-field theory of the Coulomb interacting Dirac fermions, as discussed below Eq.共1兲. However, as we show in Appendix D, and as was anticipated in Ref.18, the hard-cut-off regularization violates the Ward-Takahashi iden- tity. We are thus led to conclude that this regularization scheme is, in principle, not consistent with U共1兲 gauge sym- metry of the theory. This conclusion notwithstanding, the particular coefficientC from the introduction may be written as an integral which is unambiguous and perfectly conver- gent in the upper limit, provided the momentum cutoff is taken to infinity after all the integrals have been performed 共see Appendix H兲.

B. Pauli-Villars regularization

Another way to regularize divergent self-energy and ver- tex diagrams in QED is to introduce an additional artificial

“heavy photon.”²⁰ In Euclidean spacetime this leads to the following replacement of the photon propagator:

1

⍀²+ k²→ 1

⍀²+ k²− 1

⍀²+ k²+ M²

and the mass parameter M is sent to ⬁ at the end of the calculation. Since the additional fictitious particle couples minimally to the fermions, the regularization preserves Ward-Takahashi identities which relate the self-energy to the current vertex. However, as such, this regularization is unable to render photon polarization diagrams finite.

This can be avoided by introducing additional Pauli-Villars fermions,³¹ at the expense of making the method complicated.²⁰

In the context of the 共2+1兲D massless Dirac fermions interacting with static 共nonretarded兲 1/r Coulomb interac- tion, the analog of the Pauli-Villars regularization is

1 兩k兩→ 1

兩k兩− 1

冑

k²+ M².

Physically, this corresponds to cutting off the short-distance divergence of the 1/r interaction, without affecting its long range tail. This modified interaction preserves Ward- Takahashi identities relating vertex and the self-energy,¹⁸but, just as in the case of QED, it fails to regularize the polariza- tion function without introducing additional Pauli-Villars fer- mions. Therefore, as such it cannot serve as a complete regu- larization of the theory.

C. Dimensional regularization

Originally introduced in the context of relativistic quantum-field theory, the basic idea of the dimensional regu- larization is to regularize four-momentum integrals by low- ering the number of spacetime dimensions over which the integral is performed. This procedure was introduced by ’t Hooft and Veltman²⁹ to preserve the symmetries of gauge theories. It also bypasses the necessity to introduce Pauli- Villars fermions and bosons.

Here we employ a variant of the dimensional regulariza- tion scheme in that the frequency integrals are performed from −⬁ to +⬁ while the momentum integrals are analyti-

cally continued from D = 2 to D = 2 −⑀ dimensions. Such separation of time from space is used because in the case considered here the Lorentz invariance is violated by the interaction terms. A momentum integral is therefore calcu- lated for an arbitrary number of dimensions D, and expanded in the parameter⑀. Singular parts of the integral then appear as the first-order poles in the Laurent expansion over the parameter⑀, i.e., as terms of the form 1/⑀, and the finite part is the term of order⑀⁰in this expansion.

The following D-dimensional 共Euclidean兲 integrals are frequently encountered in this regularization scheme²⁰

冕

共2^d␲^Dᐉ兲^D 1 共ᐉ²+⌬兲ⁿ=

⌫

冉

^{n −D2}

冊

共4␲兲^D/2⌫共n兲 1

⌬^{n−共D/2兲} 共16兲 and

冕

共2^d␲^Dᐉ兲^D ᐉ²

共ᐉ²+⌬兲ⁿ= 1 共4␲兲^D/2

D 2

⌫

冉

^{n −D}2 − 1

冊

⌫共n兲

1

⌬^{n−共D/2兲−1}, 共17兲 where⌫共x兲 is the Euler gamma function and ⌬ⱖ0.

Furthermore, Pauli matrices are also embedded in D = 2

−⑀-dimensional space. We thus use the following identity:

␴a␴␮␴a= D␦0␮+共2 − D兲␴a␦a␮, 共18兲 where the sum over the Latin letters a , b, used only for the spatial indices, is assumed. The Greek letters ␮^,␯ ^{are re-} served for the spacetime indices. The last term on the right- hand side turns out to be crucial for the proof of the Ward- Takahashi identity, guaranteed by the U共1兲 charge conservation. This is discussed in later sections. In short, the last term in Eq.共18兲 yields the last term in Eq. 共A13兲. If the latter were omitted the Ward-Takahashi identity would be violated. As elaborated on in the discussion section, the same term also accounts for the discrepancy between the results found in this work, Eqs.共82兲, and the result we found previ- ously 关Eq. 共G29兲兴 for the Coulomb interaction correction to the conductivity, where the last term was omitted. Details of this calculation can be found in Appendix E.

IV. CONSERVATION LAWS, CONDUCTIVITY, AND THE STRUCTURE OF THE POLARIZATION TENSOR In the interest of self-containment, in this section we re- view some well-known results regarding response functions and U共1兲 conservation laws. Most of these results can be found 共scattered兲 in many body—quantum-field theory textbooks.^20,28

In order to calculate the response functions to external electromagnetic fields, it is useful to define the imaginary time-correlation function

⌸_␮␯共␶^,r兲 = 具T_␶j_␮共␶^,r兲j_␯共0,0兲典, 共19兲 where the current “three-vector” j_␮ is composed of the imaginary time density and current as

j共␶,r兲 = 关␳共␶,r兲,j共␶,r兲兴 = 关␺^†共␶,r兲␺共␶,r兲,vF␺^†共␶,r兲␴ជ␺^共␶,r兲兴.

共20兲 In this section we temporarily restore vF to clearly distin- guish it from the speed of light c used below.

By fluctuation-dissipation theorem,²⁸ the expectation value of the electrical current-density operator J共t,r兲, in real time t, is related to the imaginary time correlator⌸_␮␯共i⍀,q兲.

The latter is the Fourier transform in Eq. 共12兲 of the tensor defined in Eq. 共19兲. The expectation value of the Fourier transform of the electrical current-density is then

具Ja共⍀,q兲典 = −e²

ប⌸a0共i⍀n→ ⍀ + i0,q兲⌽共⍀,q兲 +e²

ប⌸ab共i⍀n→ ⍀ + i0,q兲A_b共⍀,q兲 c . 共21兲 The Fourier components of the electric and magnetic fields are related to the ones of the scalar and vector potentials as

Ea共⍀,q兲 = i⍀

cAa共⍀,q兲 − iqa⌽共⍀,q兲, 共22兲 B共⍀,q兲 = i⑀abqaAb共⍀,q兲, 共23兲 where ⑀ab is completely antisymmetric 共Levi-Civita兲 rank two tensor. Using Faraday’s law of induction we can further relate the Fourier components of the electric and magnetic fields as

⑀abqaEb共⍀,q兲 = ⍀

cB共⍀,q兲. 共24兲

In condensed matter systems with massless Dirac particles, propagating with velocityvF, as the relevant low-energy de- grees of freedom considered here, the 共pseudo兲 Lorentz in- variance is violated by interactions. If we were to consider finite temperature T the 共pseudo兲 Lorentz invariance would be violated even in the noninteracting limit. Nevertheless, when spatial O共2兲 rotational invariance is preserved, as is the case for problems studied here, the general structure of the imaginary time polarization tensor is³²

⌸_␮␯共i⍀n,q兲 = ⌸A共i⍀n,兩q兩兲A_␮␯+⌸B共i⍀n,兩q兩兲B_␮␯, 共25兲 where the three tensors are

B_␮␯=␦␮a

冉

^{␦ab−qqaq2b}

冊

^{␦b␯, 共26兲}

A_␮␯= g_␮␯−q_␮q_␯

q² − B_␮␯. 共27兲 The Euclidean three-momenta appearing in the above tensors are

g_␮␯= diag关− 1,1,1兴_␮␯, 共28兲 q_␮= g_␮␯共− i⍀n,q兲_␯=共i⍀n,q兲_␮, 共29兲

q²= q_␮g_␮␯q_␯=⍀n2+ q². 共30兲 The real-time continuity equation

⳵

⳵^t␳⁺ⵜ · J = 0 共31兲

requires that, with our choice of the imaginary time “three”

current j =共␳^{, j}ជ兲, the transversality of the ⌸_␮␯共i⍀,q兲 is equivalent to the condition

共− i⍀,q兲_␮⌸_␮␯共i⍀,q兲 = ⌸_␮␯共i⍀,q兲共− i⍀,q兲_␯= 0. 共32兲 Note that this is explicitly satisfied by the expression共25兲. If, in addition, the Lorenz invariance is satisfied, ⌸A=⌸B, and there is no need to separate out the spatially transverse com- ponent of the polarization tensor.

A. Ward-Takahashi identity and vertex functions In addition to the condition 共32兲, the continuity Eq. 共31兲 constrains the form of the vertex function. If we define the four-point matrix function

␲␮共r⬘^{− r,}␶⬘⁻␶;r − r⬙^,␶−␶⬙兲

=具T_␶j_␮共␶^,r兲␺共␶⬘^,r⬘兲␺^†共␶⬙^,r⬙兲典, 共33兲 where the imaginary time three current was defined in Eq.

共20兲, then we must have²⁰

冉

⳵␶^{⳵ ,} ⵜ

i

冊

_␮^{␲␮共␶⬘−␶,r⬘− r;␶−␶⬙,r − r⬙兲}

=关␦共␶⁻␶⬙兲␦^D共r − r⬙兲 −␦共␶⬘⁻␶兲␦^D共r⬘^{− r}兲兴

⫻G共␶⬘⁻␶⬙^,r⬘^{− r}⬙^{兲. 共34兲} The above expression relates the exact imaginary time four- point function to the exact imaginary time Green’s function G共␶,r兲 = 具T_␶␺共␶,r兲␺^†共0,0兲典. 共35兲 If we rewrite the Fourier transform of ␲␮ in terms of the vertex function ⌳_␮as

␲_␮共k,i␻^{;k + q,i}␻^{+ i}⍀兲 = Gk共i␻兲⌳_␮共k,i␻^{;k + q,i}␻ + i⍀兲Gk+q共i␻+ i⍀兲 共36兲 then the Ward-Takahashi identity for the vertex function⌳_␮ can be written as

共− i⍀,q兲_␮⌳_␮共k,i␻^{;k + q,i}␻+ i⍀兲 = G_k+q⁻¹ 共i␻+ i⍀兲 − Gk

−1共i␻兲

=⌺k+q共i␻+ i⍀兲 − ⌺k共i␻兲. 共37兲 This identity has to be satisfied order by order in perturbation theory. In what follows, we show that this is indeed the case for the interacting theories studied here when we adopt the dimensional regularization.

B. Electrical conductivity

To relate the polarization tensor to the electrical conduc- tivity, we simply need to relate the expectation value of the current to the electric field. Since we have the response to the

electromagnetic scalar and vector potentials, we just need to relate those to the electric and magnetic fields. Finally, mag- netic field can be related to the electric field using Maxwell’s equations. As is well known, at finite wave vector q and frequency⍀, one can define the longitudinal and transverse conductivity as the proportionality between the induced cur- rent and the longitudinal or transverse component of the electric field.

Using Eqs.共21兲 and 共25兲–共27兲, we find

具Ja共⍀,q兲典 =e²

ប⌸A共⍀ + i0,兩q兩兲 ⍀qa

q²−⍀²⌽共⍀,q兲

−e²

ប⌸A共⍀ + i0,兩q兩兲 ⍀²qaqb

q²共q²−⍀²兲

Ab共⍀,q兲 c +e²

ប⌸B共⍀ + i0,兩q兩兲

冉

^␦ab−qq^aq2b

冊

^{Ab共⍀,q兲}c . 共38兲 Furthermore, Eqs.共22兲–共24兲 imply

具Ja共⍀,q兲典 =e²

ប⌸A共⍀ + i0,兩q兩兲 i⍀

q²−⍀² qaqb

q² Eb共⍀,q兲 +e²

ប⌸B共⍀ + i0,兩q兩兲 1

i⍀

冉

^{␦ab−qqaq2b}

冊

^{Eb共⍀,q兲.}

共39兲 From the above equations we can read off the longitudinal and transverse electrical conductivity

␴储共⍀,兩q兩兲 =e² ប

i⍀

q²−⍀²⌸A共⍀ + i0,兩q兩兲, 共40兲

␴⬜共⍀,兩q兩兲 =e² ប

⌸B共⍀ + i0,兩q兩兲

i⍀ . 共41兲

For q⫽0 共and ⍀⫽0兲,␴储need not be equal␴_⬜. However, at q = 0, the ac conductivities

␴储共⍀,q = 0兲 =␴⬜共⍀,q = 0兲 共42兲 due to the O共2兲 spatial rotational symmetry.

In the following, we will work solely in the imaginary time—Matsubara frequency space, and since we restrict our- selves to T = 0, we will drop the subscript n on i⍀n.

V. NONINTERACTING LIMIT: V(r) = 0

For the sake of completeness, and in order to illustrate how the general results presented in the previous section ap- pear in the specific solvable problem, we first examine

⌸_␮␯共i⍀,q兲 in the limit of vanishing V共r兲. The Fourier trans- form in Eq. 共12兲 of the polarization function in Eq. 共19兲 in the noninteracting limit is easily shown to be

⌸_␮␯^共0兲共i⍀,q兲 = − N

冕

_−⬁^{⬁ 2d␻}␲

冕

共2^d␲^Dk兲^D

⫻Tr关Gk共i␻兲␴_␮^Gk+q共i␻^{+ i}⍀兲␴_␯兴, 共43兲 where␴0 is the 2⫻2 unit matrix. To this end it is useful to define the vertex function

P_␮共q,i⍀兲 =

冕

_−⬁^{⬁ d2␻}␲

冕

共2^d␲^Dk兲^DG_k共i␻兲␴_␮^Gk+q共i␻^{+ i}⍀兲 共44兲 in terms of which

⌸_␮␯^共0兲共i⍀,q兲 = − N Tr关P_␮共q,i⍀兲␴_␯兴. 共45兲 The above expression is divergent at large momenta 共UV divergent兲 as is easily seen by counting powers. Note that this appears even in the noninteracting theory when calcu- lating the response functions. As is well known in the context of relativistic field theories, this UV divergence is unphysical and to obtain the correct answer a regularization is necessary.^20,33 The regularization of choice here is dimen- sional regularization which leads to finite expressions and which is consistent with U共1兲 gauge symmetry of the theory.

As shown in detail in the Appendix A, using dimensional regularization, we obtain

P_␮共q,i⍀兲 =

冑

⍀²+ q² 64

⫻

冋

^{␴␮− 2␦␮0−共i⍀ +␴· q⍀兲2␴+ q␮共i⍀ +2 ␴· q兲}

册

^.

共46兲 Performing the trace in Eq. 共45兲 we find

⌸_␮␯^共0兲共i⍀,q兲 = − N

16

冑

_⍀²+ q²

冢

^{− i⍀q− i⍀q− q2xy q− i⍀q− qy2+x⍀qyx2 q− i⍀q− qx2+x⍀qyy2}

冣

^␮␯.

共47兲 We can write the above matrix more compactly as

⌸_␮␯^共0兲共i⍀,q兲 = − N

16

冉

^g␮␯−qq^␮q2␯

冊 ^冑

^{⍀2+ q2, 共48兲}

where we used definitions from Eqs. 共28兲–共30兲. The correla- tion function in Eqs.共47兲 and 共48兲 is explicitly transverse, as it should be, and

共− i⍀,q兲_␮⌸_␮␯^共0兲共i⍀,q兲 = ⌸_␮␯^共0兲共i⍀,q兲共− i⍀,q兲_␯= 0. 共49兲 From the above equations we find

⌸A共0兲共i⍀,q兲 = ⌸B共0兲共i⍀,q兲 = − N

16

冑

⍀²+ q². 共50兲 Analytically continuing according to Eqs.共40兲 and 共41兲, with the branch cut of the

冑

z-function lying along negative real axis, we find the well-known expression for the 共Gaussian兲 ac conductivity

␴储共0兲共⍀兲 =␴_⬜^共0兲共⍀兲 = N 16

e²

ប. 共51兲

As a side remark, if we were to define⌸˜

␮␯共0兲as a correlation function of a slightly different three current共−i␳^{, j}ជ兲, the re- sult obtained directly from Eqs.共47兲 and 共48兲 transforms as tensor under Euclidean O共3兲 transformations. In real fre- quencies this is equivalent to relativistic Lorentz transforma- tions due to the invariance of the noninteracting Lagrangian L0.

Therefore, regulating the UV divergences via dimensional regularization implemented here leads to finite expressions which preserve the required U共1兲 conservation laws. The necessary regularization of the “integration measure,” as done here via dimensional regularization, is independent of the electron-electron interaction V共r兲, as it must be if the noninteracting theory is to lead to finite-correlation func- tions. Therefore, as shown already by this example, regulat- ing only the “momentum transfer” as advocated in Refs.10 and18is clearly insufficient.

VI. SHORT-RANGE INTERACTIONS: V(r) = u␦^(r) While the problem of 共2+1兲D massless Dirac fermions with the contact interactions is not exactly solvable, one can calculate the interaction corrections to the polarization tensor perturbatively in powers of the interaction strength u. Such contact interactions certainly constitute an idealized special case.³⁴ Nevertheless, this theory has the advantage that one can determine the first correction in u to the noninteracting 共Gaussian兲 polarization tensor ⌸_␮␯^共0兲, found in the previous section, explicitly for finite q and ⍀. We can then test the general symmetry requirements listed before. The technique of choice is again the 共variant of the兲 dimensional regular- ization of Veltman and ’t Hooft introduced in Sec.III. Since

this interaction violates Lorentz invariance we can also use this example to study how the difference between⌸Aand⌸B

arises in such theory.

It is straightforward to use the Wick’s theorem to show that in this case, the first order in u, and toO共N兲, correction to the polarization tensor is

␦⌸_␮␯共i⍀,q兲 = uN

冕

共2^d␲^Dk兲^D

冕

_−⬁^{⬁ d2␻}␲

冕

⫻ d^Dp

共2␲兲^D

冕

_−⬁^{⬁ d2␻}␲^{⬘兵Tr关Gk共i}␻兲␴␮G_k+q共i␻ + i⍀兲Gp共i␻⬘^兲␴_␯^Gp−q共i␻⬘^{− i⍀兲兴}

+ Tr关Gk共i␻兲␴␮G_k+q共i␻+ i⍀兲Gp共i␻⬘兲Gk+q共i␻ + i⍀兲␴␯兴 + Tr关Gk共i␻兲␴␮G_k+q共i␻

+ i⍀兲␴␯Gk共i␻兲Gp共i␻⬘兲兴.其 共52兲 The last two terms correspond to the self-energy correction while the first one is the vertex correction. Because the self- energy for the contact interaction vanishes

冕

共2^dD␲^k兲^D

冕

_−⬁^{⬁ d2␻}␲^Gk共i␻兲 = 0 共53兲 the last two terms in the Eq.共52兲 vanish as well. The remain- ing term can be written rather succinctly in terms of P_␮ defined previously in Eq. 共44兲 as

␦⌸_␮␯共i⍀,q兲 = uN Tr关P_␮共q,i⍀兲P_␯共− q,− i⍀兲兴. 共54兲 The above expression is manifestly transverse, i.e., it satis- fies Eq.共32兲, as can be seen from Eq. 共46兲.

Namely, inserting Eq.共46兲 and performing the traces we find

␦⌸_␮␯共i⍀,q兲 = uN

512共⍀²+ q²兲

冤

^{i⍀qi⍀qq2共qxy共q共q222−−−⍀⍀⍀2兲22兲 q兲 x2共q− q2yi⍀q−xq⍀xy2共3⍀共q兲 + 共q2−2+ q⍀2y+2兲2⍀兲 2兲2 qx4−⍀2− qqi⍀qy2x+qyy⍀共3⍀共q42+ q−2+ q⍀x2共q2兲2y2兲+ 2⍀2兲}

冥

^␮␯.

Finally, the above tensor can be factorized as given by Eqs.

共25兲 and 共26兲, and we find to first nontrivial order in the contact coupling u

⌸A共i⍀,兩q兩兲 = − N

16

冑

⍀²+ q²+ uN

512共⍀²− q²兲 + O共u²兲,

⌸B共i⍀,兩q兩兲 = − N

16

冑

⍀²+ q²+ uN

512共⍀²+ q²兲 + O共u²兲.

共55兲 Expectedly, the above expression shows that the interaction

correction to the polarization functions ⌸A and⌸B are dif- ferent 共note the sign difference in front of q²兲. As stated above, the reason for the difference is that the contact density-density interaction term u关␺^†共r兲␺共r兲兴² breaks the Lorentz invariance of the noninteracting part of the Lagrang- ian. Lorentz transformations, in general, rotate between den- sity and current, and we have purposefully omitted any current-current interaction.

We can further test the Ward-Takahashi identity in Eq.

共37兲 for the vertex function in Eq. 共36兲 in this example with the short-range interactions. It can be readily seen that the first order in u correction to the vertex vector is

␦⌳_␮共k,i␻^{;k + q,i}␻+ i⍀兲 = − uP_␮共q,i⍀兲. 共56兲 It follows from the Eq.共46兲 that

− i⍀P0共q,i⍀兲 + qaPa共q,i⍀兲 = 0. 共57兲 Therefore the Ward-Takahashi identities in Eq.共37兲 are sat- isfied, since, as mentioned previously in this section, the self- energy correction vanishes to first order in u for the short- range interactions.

Finally, from Eqs. 共40兲 and 共41兲, we can infer that the above terms correct only the imaginary part of the ac con- ductivity but not the real part. At q = 0, correction is the same for the longitudinal and the transverse components, and to this order in u we have

␴储,⬜共⍀兲 =e² ប

N

16

冉

^{1 − i}32^u ⍀

冊

^{. 共58兲}

Again, the equality between␴储共⍀兲 and␴⬜共⍀兲 is guaranteed due to the O共2兲 rotational invariance of this theory. Note also that the fact that the interaction correction is proportional to the frequency is implied by the power counting at the Gauss- ian fixed point of the theory and is characteristic for any finite-range interaction.³⁵

VII. COULOMB INTERACTION: V(r) = e²Õ円r円 Armed with the above results we now focus on the main part of the paper where we study the effects of the Coulomb interaction. Unlike in the previous cases, we have been un- able to find the explicit expression for the first-order correc- tion to the polarization tensor at finite q and⍀. Nevertheless, we have been able to show explicitly that the first-order cor- rection to the polarization tensor is transverse, i.e., it satisfies Eq. 共32兲. This is shown using dimensional regularization in D = 2 −⑀introduced in Sec. III. Next, we study the first cor- rection to the Coulomb vertex function which must also sat- isfy the Ward-Takahashi identity in Eq. 共37兲. Since in this case the first order self-energy is known to diverge logarith- mically, the first-order correction to the vertex function should also diverge as⑀→0. This can be found explicitly in terms of elliptic integrals to order⑀^−1and⑀⁰and the identity 共37兲 is also explicitly confirmed. Finally, we proceed with the calculation of the electrical conductivity, first by using the spatial component of the polarization tensor at q = 0 but finite ⍀ 共current-current correlation function兲 and then by using time component of the polarization tensor at finite but small q and finite ⍀. The final results for the conductivity calculated in both ways are found to be the same. Specifi- cally, we findC=共11−3␲兲/6 in Eq. 共1兲.

For unscreened 3D Coulomb interactions V共r兲=e²/r the effect of screening due to dielectric medium is easily taken into account by rescaling e²in the above formula. TheO共e²兲 andO共N兲 correction to the polarization function is then

␦⌸_␮␯^共c兲共i⍀,q兲 = N

冕

共2^d␲^Dk兲^D

冕

_−⬁^{⬁ 2d}␲^␻

冕

共2^d␲^Dp兲^D

冕

_−⬁^{⬁ d2␻}␲^⬘兵Vp−k

⫻ Tr关Gk共i␻兲␴␮G_k+q共i␻+ i⍀兲Gp+q共i␻⬘ + i⍀兲␴␯Gp共i␻⬘兲兴 + Vk−p

⫻ Tr关Gk共i␻兲␴␮G_k+q共i␻+ i⍀兲Gp+q共i␻⬘+ i⍀兲

⫻ Gk+q共i␻^{+ i}⍀兲␴_␯兴 + Vk−p

⫻ Tr关Gk共i␻兲␴␮G_k+q共i␻

+ i⍀兲␴_␯^Gk共i␻兲Gp共i␻⬘^{兲兴其 共59兲} where

Vk=

冕

^{d2reik·rV共r兲 =2}兩k兩^␲e2. 共60兲 Just as in the case of contact interactions, the first term in the expression for␦⌸_␮␯^共c兲corresponds to the vertex correction and the last two terms to the self-energy corrections. Unlike in the case of contact interactions, however, the self-energy cor- rection does not vanish. The expression 共59兲 will be used in later sections as a starting point in the calculation of the Coulomb interaction correction to the ac conductivity in the collisionless regime.

A. Proof of the transversality of␦⌸_␮^(c)_␯within dimensional regularization

Because, as mentioned above, the explicit evaluation of Eq. 共59兲 at finite q and ⍀ yields intractable expressions, we proceed by first showing that Eq.共59兲 is transverse, i.e., that it satisfies the condition 共32兲, when dimensional regulariza- tion employed in this paper is used. As such it therefore does not lead to any violation of the charge conservation, a virtue questioned in Ref.18. By two-dimensional rotational invari- ance, this in turn implies that the Coulomb polarization ten- sor can be written in the form of Eq. 共25兲.

To prove Eq.共32兲 we follow Ref.18and use

− i⍀␴0+ q ·␴^{= G}k+q

−1 共i␻+ i⍀兲 − Gk

−1共i␻兲 共61兲 to find

共− i⍀,q兲_␮␦⌸_␮␯^共c兲共i⍀,q兲 = N

冕

共2^d␲^Dk兲^D d^Dp 共2␲兲^D

冕

−⬁

⬁

⫻d␻ 2␲

d␻⬘

2␲^{Vp−k兵Tr关Gk共i}␻兲␴_␯

⫻Gk共i␻兲Gp共i␻⬘^{兲兴 − Tr关G}k+q共i␻ + i⍀兲␴␯G_k+q共i␻+ i⍀兲Gp+q共i␻⬘ + i⍀兲兴其.

At this point it is not immediately obvious that we can shift the integration variables k and p in the second term by q, which if true would readily yield the desired relation 共32兲 since the frequency integral can be shifted. We therefore de- fine a function of frequency and two momentum variables

⌺p,q共i⍀兲 =

冕

共2^d␲^Dk兲^D

冕

_−⬁^{⬁ d2␻}␲^{⬘Vk−pGk+q共i}␻⬘^{+ i⍀兲 共62兲} in terms of which we have unambiguously

共− i⍀,q兲_␮␦⌸_␮␯^共c兲共i⍀,q兲

= N

冕

共2^d␲^Dk兲^D

冕

_−⬁^{⬁ 2d}␲^{␻兵Tr关Gk共i}␻兲␴_␯^Gk共i␻兲⌺k,0共0兲兴

− Tr关Gk+q共i␻+ i⍀兲␴_␯^Gk+q共i␻+ i⍀兲⌺k,q共i⍀兲兴其.

To continue, we need to find an explicit expression for

⌺p,q共i⍀兲. Using the identity 共16兲, Feynman parametrization 1

A^␣B^␤= ⌫共␣⁺␤兲

⌫共␣兲⌫共␤兲

冕

0 1

dy y^␣−1共1 − y兲^␤−1

关yA + 共1 − y兲B兴^␣+␤ 共63兲 for ␣⁼␤^{= 1}/2 and

冕

0 1

dyy^␣−1共1 − y兲^␤−1= ⌫共␣兲⌫共␤兲

⌫共␣⁺␤兲 共64兲 we find

⌺p,q共i⍀兲

= e² 共4␲兲^D/2

␴^·共p + q兲 兩p + q兩^2−D

⌫

冉

^{1 −D}2

冊

^⌫

冉

^{D + 1}2

冊

^⌫

冉

^{D − 1}2

冊

⌫共D兲 ,

共65兲 which agrees with Eq.共12兲 of Ref.36. Note that this identity shows that within dimensional regularization, ⌺p,q共i⍀兲

=⌺p+q,0共i⍀兲. Moreover, in what follows, there is no need to shift the integration variable. Rather, since the commutator of the self-energy and the Green’s function vanishes

关Gk+q共i␻^{+ i}⍀兲,⌺p,q共i⍀兲兴 = 0 共66兲 after a straightforward use of the cyclic property of the trace and the identity

冕

−⬁

⬁ d␻

2␲^Gk+q共i␻+ i⍀兲Gk+q共i␻+ i⍀兲 = 0 共67兲 we prove that

共− i⍀,q兲_␮␦⌸_␮␯^共c兲共i⍀,q兲 = 0. 共68兲 The same procedure as the one used above also leads to

␦⌸_␮␯^共c兲共i⍀,q兲共− i⍀,q兲_␯= 0. 共69兲 This proof holds to all orders of ⑀. The regularization tech- nique implemented here is therefore perfectly adequate and does not lead to violation of the charge conservation.

B. Coulomb vertex and the proof of the Ward-Takahashi identity

Next, we will demonstrate that the dimensional regular- ization used here preserves the Ward-Takahashi identity for the Coulomb vertex function. This proof is technically more involved than the proof in the previous section, but neverthe- less, we find it important to present its details since our tech- nique is not widely used in the community. We show the desired identity to order ⑀⁻¹ and ⑀⁰. Most of the technical details are presented in the Appendices A–C and in this sec-

tion we just present the main steps of the derivation.

The Coulomb vertex function to the first order in the cou- pling constant is

␦⌳_␮共p,i␯^{;p + q,i}␯+ i⍀兲

=P_␮^c共q,p,i⍀兲

= −

冕

共2^d␲^Dk兲^D

冕

−⬁

⬁ d␻

2␲^Vp−kGk共i␻兲␴␮G_k+q共i␻+ i⍀兲.

共70兲 The integrals on the right are logarithmically divergent in D = 2 as can be easily seen by powercounting. This diver- gence is related to the divergence of the electron self-energy, calculated in the previous section

⌺k共i␻兲 ⬅ ⌺k,0共i␻兲

=e²

8␴^{· k}

冋

²⑀⁻␥^{+ ln 64}␲^{− ln k2+}O共⑀兲

册

^,

共71兲 where the Euler-Mascheroni constant ␥= 0.577 and, as be- fore,⑀= 2 − D. Indeed, if the Ward-Takahashi identity

共− i⍀,q兲_␮P_␮^c共q,p,i⍀兲 = ⌺p+q共i␯+ i⍀兲 − ⌺p共i␯兲 共72兲 is to be satisfied, the vertex function must diverge logarith- mically, which manifests in the dimensional regularization as the first-order pole in Laurent expansion in the parameter ⑀^. In the second part of the Appendix A we use dimensional regularization to determineP^c共q,p,i⍀兲 to orders⑀⁻¹and⑀⁰. Our final expression for finite q, p, and i⍀, Eq. 共A18兲, is left as an integral over a Feynman parameter x. We wish to stress that all of the integrals in this equation can be performed in the closed form in terms of elliptic integrals. However, we found that doing so leads to intractable and unrevealing ex- pressions. We therefore chose to work with the expression 共A18兲 and in effect manipulate the integral representation of the elliptic integrals. In the limiting case of q = 0, the vertex function is determined in the closed form in Appendix A up to, and including, ⑀⁰.

In Appendix B we in turn find that the vertex function in Eq. 共A18兲 satisfies

共− i⍀,q兲_␮P_␮^c共q,p,i⍀兲 = N共p,q兲 −e²

4

冑

⍀²+ q²

⫻␴^·关p共⍀²+ q²兲L共p,q,⍀兲

+ qM共p,q,⍀兲兴. 共73兲

Using the dimensional regularization, we then show that the function

N共p,q兲 =e²

8

冉

²⑀␴^{· q +}共ln 64␲⁻␥兲␴^{· q}

−␴^·共p + q兲ln共p + q兲²+␴^{· p ln p2}

冊

=⌺p+q共i␯+ i⍀兲 − ⌺p共i␯兲 共74兲 and, in Appendix C, that L共p,q,⍀兲=M共p,q,⍀兲=0. This