On the convergence of the polarization tensor in space-time of three dimensions

In this paper, we consider the convergence properties of the polarization tensor of graphene obtained in the framework of thermal quantum field theory in three-dimensional space-time. During the last years, this problem attracted much attention in connection with calculation of the Casimir force in graphene systems and investigation of the electrical conductivity and reflectance of graphene sheets. There are contradictory statements in the literature, especially on whether this tensor has an ultraviolet divergence in three dimensions. Here, we analyze this problem using the well known method of dimensional regularization. It is shown that the thermal correction to the polarization tensor is finite at any $D$, whereas its zero-temperature part behaves differently for $D=3$ and 4. For $D=3$, it is obtained by the analytic continuation with no subtracting infinitely large terms. As to the space-time of $D=4$, the finite result for the polarization tensor at zero temperature is found after subtracting the pole term. Our results are in agreement with previous calculations of the polarization tensor at both zero and nonzero temperature. This opens possibility for a wider application of the quantum field theoretical approach in investigations of graphene and other two-dimensional novel materials.


I. INTRODUCTION
The term polarization tensor has many different meanings and was used for theoretical description of diverse physical phenomena.Here, we reconsider the problem of convergence of the vacuum photon polarization tensor of graphene in quantum electrodynamics (QED) at nonzero temperature in three-dimensional space-time.Independently of an entirely theoretical interest to calculation of the polarization tensor at both zero and nonzero temperature for the case of D = (2 + 1) dimensions [1][2][3][4][5], this problem attracted special attention [6][7][8][9][10] in connection with the advent of two-dimensional hexagonal structure of carbon atoms called graphene [11].
At energies below a few eV, the electronic properties of graphene are well-described by a set of massless or very light quasiparticles with spin 1/2 obeying the Dirac equation, where the speed of light c is replaced with the Fermi velocity v F ≈ c/300 [12][13][14][15] (in the following text, we use the system of units where = c = 1).This has opened an attractive opportunity of describing the reaction of graphene to the electromagnetic field using the well established methods of QED in (2+1) dimensions, especially the concept of the polarization tensor, i.e., restricting to the one-loop radiative correction in the language of QED.Taking into account that the properties of graphene strongly depend on temperature, this well may be done in the framework of thermal quantum field theory.
In spite of big progress in application of thermal QFT for obtaining the polarization tensor of graphene and describing its properties on this basis, the more phenomenological theoretical approach using the Kubo formula is often used in the literature for the same purpose (see, e.g., Refs.[53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72]).There are, however, significant conceptual differences between the quantum field theoretical and Kubo approaches.For instance, in the framework of the Kubo approach, dissipation is introduced by means of the phenomenological relaxation parameter treated as the imaginary part of complex frequency.Alternatively, the QFT does not use phenomenological parameters and describes dissipation by means of the imaginary part of the polarization tensor which arises for the scaled 3-momentum magnitudes exceeding the energy gap in graphene.
In the spatially local approximation, there is an agreement between the results obtained using different theoretical approaches [21,30,[42][43][44][46][47][48].As to the spatially nonlocal case, the quantum field theoretical approach predicts the presence of a double pole at zero frequency in the transverse dielectric permittivity of graphene [41], which is not obtainable in the Kubo approach.It was stated [73] that the presence of a double pole might be connected with an improper regularization of the polarization tensor obtained within thermal QFT.
In this regard, it should be noted that there are contradictory statements in recent literature concerning the convergence of this tensor.Thus, Ref. [17] found by power counting that the polarization tensor of graphene diverges and made it finite by a Pauli-Villars subtraction, whereas Refs.[27,28,30] conclude that in 2+1 dimensions it is finite because the ultraviolet divergence is not present due to the gauge invariance.As to Ref. [73], it states that the polarization tensor of graphene obtained by means of the quantum field theory is divergent and suggests an alternative regularization procedure, which brings it to exact coincidence with that obtained by means of the Kubo approach.
In view of the above, we feel that it is necessary to clarify the situation.We demonstrate the calculation of the polarization tensor of graphene using the methods of QED at nonzero temperature in detail and in such a way, that the calculation can be followed with a minimum of knowledge of the field theoretical methods.Thereby it must be underlined that in the standard QED at zero temperature the polarization tensor was calculated long ago both in (3+1) dimensions (see, e.g., the textbooks [74,75]) and in (2+1) dimensions [1,2].For the latter case, the key moments, gauge invariance and ultraviolet finiteness were mentioned explicitly in Sec. 2 of Ref. [2].
In the present paper, we reconsider the polarization tensor appearing in the quantum field theoretical approach to graphene at nonzero temperature in detail.We use dimensional regularization.First we demonstrate how the transversality of the polarization tensor can be seen before the momentum integration.Next, we demonstrate that this tensor consists of the zero-temperature part and a thermal correction to it.The immediate analytic calculation shows that the thermal correction to the polarization tensor is finite, so that the ultraviolet divergence, if any, might be contained only in its zerotemperature part.Then, we use the exponential representation for the propagators and carry out the momentum integrations.Finally, after carrying out the next-tolast integration, the transversality becomes evident also in this representation as well as the ultraviolet properties.
Specifically, by considering the polarization tensor in the space-time of complex D dimensions, we demonstrate that in the case of D = 3 the finite result is obtained using the regularization by means of analytic continuation from the case ReD < 2. In so doing, no pole terms need to be subtracted, i.e., no renormalization is needed.Ap-plying the same procedure to the polarization tensor in the space-time of D = 4 dimensions, we show that for obtaining the finite result it is necessary to subtract the pole term, i.e., regularization should be followed by the renormalization.Generally speaking, such behavior in the ultraviolet region is well known in QFT as a consequence of, together, power counting, gauge invariance, and parity.However, an active discussion for graphene, which exists in two spatial dimensions but interacts with the electromagnetic field existing in three-dimensional space, revealed the necessity to demonstrate this behavior in detail.The performed analysis is in confirmation of the polarization tensor derived in the literature in the framework of both ordinary and thermal QFT.
The paper is organized as follows.In Section II, we consider general expression for the polarization tensor of graphene at nonzero temperature in the space-time of D dimensions.Section III is devoted to the zerotemperature part of the polarization tensor and its analytic properties.In Section IV, the convergence properties of the polarization tensor in both three-and fourdimensional space-time are considered.In Section V, the reader will find our conclusions and a discussion.
Recall that use the system of units where = c = 1.

II. REPRESENTATION OF THE POLARIZATION TENSOR AT NONZERO TEMPERATURE IN D DIMENSIONS
In the framework of QFT the one-loop polarization tensor of graphene was considered in many papers (see, e.g., Refs.[9, 16-18, 30, 31, 35, 76]).It is represented by a simple diagram of Fig. 1 where the solid lines depict the propagators of fermionic quasiparticles which move with the Fermi velocity v F and satisfy the Dirac equation in 2+1 dimensions Here, γ ν are the standard Dirac matrices, γ1, is the vector potential of the electromagnetic field, and m is the mass of quasiparticles bearing the electric charge e.
The important feature of Eq. ( 1) is that the interaction of charged quasiparticles with the electromagnetic field is introduced by the standard substitution where, for a graphene sheet in the plane x 3 = 0, it holds x ν = (t, x 1 , x 2 , 0), ν = 0, 1, 2, 3. Note that Eq. ( 2) contains the speed of light in the factor e/c (we recall that here c = 1).This reflects the fact that the electromagnetic field, although it interacts with the quasiparticles confined in a graphene plane, exists in the 3+1 dimensional bulk.As a consequence, in the Dirac model of graphene, the electric charge in the system of units with = c = 1 is not dimensional (as it holds in the strictly 2+1 dimensional electrodynamics [1]) but dimensionless and results in the standard fine structure constant α = e 2 ≈ 1/137.Calculation of the diagram shown in Fig. 1 includes an integration over the internal momentum q = (q 0 , q) and taking the trace of γ-matrices (see Refs. [17,30] for details).Keeping in mind that we are looking for the polarization tensor of graphene at any temperature T , within the Matsubara formalism, an integration over q 0 should be replaced with a summation over the pure imaginary fermionic Matsubara frequencies where n = 0, ±1, ±2, . .., and k B is the Boltzmann constant.In so doing, the zero component of the external, photon, wave vector k = (k 0 , k) is equal to the pure imaginary bosonic Matsubara frequencies Although here and below we deal with graphene, which is the two-dimensional sheet of carbon atoms, in the following we use the D-dimensional vectors (q 0 , q) = (q 0 , q 1 , . . ., q D−1 ) and (k 0 , k) = (k 0 , k 1 , . . ., k D−1 ), where the dimension of the spatial part is D − 1, and respective integration measures.The metric tensor is defined as g µν = diag(1, −1, −1, . . ., −1) and the product of two vectors is qk = q ν k ν = q 0 k 0 − qk.The trace of the metric tensor is g ν ν = D.The point is that, in general, the polarization tensor is ultraviolet divergent like most radiative corrections in QFT.For instance, simple power counting shows a divergence also in (2+1)-dimensions.For this reason, a regularization is necessary.By introducing the D-dimensional space-time, we take the dimensional regularization which amounts in formally taking a complex dimension D (see, e.g., Sec.11.2 in the textbook [77]).This allows to find the analytic properties of the polarization tensor as the function of D.
Note that for graphene the Dirac cones are located at the two points at the corners of the Brillouin zone [13].Then, after taking the trace over the gamma matrices, the resulting polarization tensor in the momentum rep-resentation is given by [30] Π where and where the infinitely small additions i0 originate from the fermion propagators, the scaled momentum is k = (k 0 , v F k), q 0 = q 0n = iq Dn and k0 = k 0l = ik Dl in accordance with Eqs. ( 3) and (4).For instance, Z00 (ik Dl , k; iq Dn , q) = −q Dn (q Dn − k Dl ) etc., and It is common knowledge that electrodynamics is the gauge invariant theory.This means that the Fourier transformed vacuum current should be invariant under the gauge transformation where χ(k) is an arbitrary function [78].As a consequence, Thus, for the polarization tensor, the gauge invariance is realized in the form of the transversality condition It is easily seen that the polarization tensor of graphene (5) satisfies this condition like that in full QED.Really, using Eqs.( 6) and (7), by a simple rewriting, one obtains where kν = η ν β k β .Then, from Eqs. ( 5) and ( 15) we find Note that q 0 = q 0n = iq Dn given by Eq. ( 3).The integral in Eq.( 16) converges under the condition ReD < 2. Using this condition, the seemingly divergent integral/sum is regularized allowing the shift of variables q → Q + k where and q Dn , k Dl are defined in Eqs. ( 3) and ( 4).As a result, the integrand itself vanishes, i.e., the polarization tensor (5) satisfies the transversality condition ( 14) even before carrying out the momentum integration.Now we represent the polarization tensor (5) as the part, which is independent on temperature, and the thermal correction to it.For this purpose, the right-hand of Eq. ( 5) is rewritten as where (below we omit the already specified repeated arguments).
Using the Cauchy residual theorem, the sum ( 18) can be represented in the form where the integration contour in the complex q D -plane shown in Fig. 2 consists of the paths γ 1 and γ 2 .The validity of Eq. ( 20) becomes evident when taking into account that the poles of the expression under the integral are at the points q Dn = 2πk B T (n + 1/2) shown as dots in Fig. 2 and calculating the sum of the residues at these poles.

iΓ −iΓ
The complex qD-plane containing the integration paths γ1 and γ2.The dots indicate the poles at the fermionic Matsubara frequencies.The four additional poles are shown as crosses (see the text for further discussion).
Substituting Eq. ( 20) in Eq. ( 5) and interchanging the order of integrations, one obtains Here, the integrand in the second term is decreasing in the lower half-plane.To get the integrand in the first term decreasing in the upper half-plane, in the integral along γ 1 , we use the identity Substituting it to Eq. ( 21), we bring the polarization tensor to the form and Note that the sign minus in front of ( 24) appeared because the direction of the path γ 1 is against the real axis in the complex plane q D .
The first term on the right-hand side of Eq. ( 23) given by Eq. ( 24) has the meaning of the polarization tensor at zero temperature (till the moment it is calculated at the bosonic Matsubara frequencies).As to the second term given by Eq. ( 25), it explicitly depends on T and has the meaning of the thermal correction.
We begin from calculation of the thermal correction.This can be done by closing the integration paths γ 1 and γ 2 with the help of semicircles of the infinitely large radii in the upper and lower half planes, respectively, and applying again the Cauchy residue theorem.In the upper half-plane, there are two poles of the function Z µν /R at the roots of R.These are q D = iΓ(q) and q D = i Γ(q) + k Dl where Γ and Γ are defined in Eq. (10).In the lower half-plane, the poles of the function Z µν /R are at q D = −iΓ(q) and q D = −i Γ(q) + k Dl .All these poles are shown in Fig. 2 as crosses.
Calculating the residues at all the four poles and taking into account that the integrals along both semicircles vanish, we rewrite the thermal correction (25) as When obtaining this equation, it was used that exp[−iλk Dl /(k B T )] = 1 due to Eq. ( 4).Equation ( 26) can be further simplified because the integrand is symmetric under the substitution q → k − q.Making this substitution and the replacement λ → −λ in the second term of this equation, one obtains Taking into account that Γ ∼ |q| when |q| → ∞, it is seen that the integral in Eq. ( 27) converges exponentially fast for any D. Note that Eq. ( 27) is easily generalized for the case of graphene with a nonzero chemical potential µ.This is done by the replacement [81] 1 Thus, the problem of convergence of the polarization tensor reduces to the question of whether its zerotemperature part (24) converges.Note that the thermal correction in the form of Eq. ( 27) admits an immediate analytic continuation to the real frequency axis by putting ik Dl = k 0 = ω (compare with similar results obtained for the temperature Green functions in Refs.[79,80] and with Ref. [9]).In a similar way, the polarization tensor at zero temperature along the real frequency axis is obtained from Eq. ( 24) by putting ik Dl = k 0 = ω and q D = −iq 0 .With this substitution, it takes the form where k = (k 0 , k), q = (q 0 , q) and d D q = dq 0 dq.
According to Eqs. ( 6) and ( 7), Z µν ∼ q 2 and R ∼ q 4 in the limit q 2 → ∞.These simple power-counting arguments show that the integral (29) may contain the ultraviolet divergences of the order of q D−2 , i.e., diverge linearly and quadratically in three-and four-dimensional space-time, respectively.Below we show how these expectations are modified by the gauge invariance of the polarization tensor.

III. ZERO-TEMPERATURE PART AND ITS ANALYTIC EXPRESSION IN D DIMENSIONS
In this section, we calculate the zero-temperature polarization tensor (29) in the case of D-dimensional spacetime.For this purpose, we use the following representation for the propagators entering Eq. ( 29) [75]: For the momenta q ν ′ entering Z µ ′ ν ′ in Eq. ( 7), we use This substitution is made for all q entering the function Z µ ′ ν ′ , i.e., Substituting Eqs. ( 6) and (30) in Eq. ( 29) with account of the definition of R in Eq. ( 7) and using Eq. ( 32), the polarization tensor at zero temperature is presented as where the quantity M is defined as This expression for M can be identically rewritten in the form It is seen that only the first term in the expression (35) for M depends on q.Then, the integration with respect to q in Eq. ( 33) can be easily performed.For this purpose, we use the well known formulas [75] where s + t > 0 and j = 1, 2, . . ., D − 1.
Combining the necessary number of expressions in Eq. ( 37), for the D-dimensional space-time one obtains By applying Eq. ( 38) with a necessary shift of the integration variable q in Eqs.(33) and (35), one obtains where (40) The functional form of the quantity Z µ ′ ν ′ is presented in the first line of Eq. (7).It is seen that, in order to calculate the quantity (40), one should find how the operators obtained from q µ ′ , q µ ′ q ν ′ , and q 2 by the replacement of q µ ′ with −i∂/∂ξ µ ′ act on the exponent in Eq. (40).As an example, By putting here ξ = 0, one finds In a similar way, calculating the remaining derivatives and putting ξ = 0 in the obtained results, we arrive at where we accounted for g µν g µν = D. Using Eqs. ( 7), (42), and (43), we bring Eq. ( 40) to the form It is convenient to rewrite the polarization tensor (39) in terms of new integration variables ρ and λ defined as so that where ρ is the so-called Feynman parameter (frequently denoted by x).
It is easily seen that where the factor λ in Eq. ( 47) comes from the Jacobian.In terms of the variables (45), the quantities Z µ ′ ν ′ 1 from Eq. ( 44) and H from Eq. ( 36) take the form Then, the polarization tensor ( 39) is given by where and H 1 are defined in Eq. ( 48).Note that the limit of large momenta corresponds to small λ.
The integral over λ in Eq. ( 49) can be calculated using the formula [82] ∞ where Γ(w) is the gamma function.Note that the integral on the left-hand side of Eq. ( 50) is equal to the gamma function only under the conditions Re(−iH 1 ) > 0 and Re w > 0. The first of them is satisfied due to the presence of i0 in Eq. ( 48).Below we apply Eq. ( 50) for the space-time with ReD < 2 where Re w > 0. The results for the cases ReD > 2 are obtained by the standard analytic continuation (see the next section for the differences between the cases D = 3 or D = 4).Using Eq. ( 50) in Eq. ( 49), one finds Using the property the integral (51) can be rewritten in a simpler form Inserting Eq. ( 53) in Eq. ( 49), we arrive at From the tensor structure of Eq. ( 54) it becomes evident that for Π µν 0 the transversality condition ( 14) is satisfied as it must be for both the zero-temperature part of the polarization tensor and for the thermal correction to it.
The analytic continuations of Eq. ( 54) to the cases of D = 4 and D = 3 are considered in the next section.

IV. THREE-AND FOUR-DIMENSIONAL SPACE-TIMES
begin with the case of four-dimensional space-time D = 4. Keeping in mind the necessity of regularization, let us put D = 4 − 2ε, where ε vanishes when D goes to 4. In this case Eq. ( 54) takes the form In fact the gamma function on the right-hand side of Eq. ( 55) can be analytically continued to the entire plane of complex ε with exception of the poles at ε = 0, −1, −2, . . . .This allows to perform the dimensional regularization of the polarization tensor (55) and subsequent renormalization by subtracting the pole contribution in the form of 1/ε.do so, we expand the gamma function according to [82] where γ is the Euler constant.The factor H −ε 1 is repre-sented as where C is an arbitrary constant with the dimension of H 1 .
Substituting Eqs. ( 56) and (57) in Eq. ( 55), one obtains where It is convenient to rewrite this result in the form where and, in accordance with Eq. ( 48), The renormalization in quantum electrodynamics with D = 4 consists in discarding the pole term in Eq. ( 60) which corresponds to the logarithmic ultraviolet divergence.This divergence is by two powers less than it follows from a simple power counting for D = 4 discussed in the end of Sec.II.The decrease in the divergence power is the result of transversality (gauge invariance) of the polarization tensor ensured by the tensor structure of Eq. (59).By imposing the normalization condition Π ren 4 (k 2 = 0) = 0 (which is justified by the general theory of renormalization in QED), one can fix the arbitrary constant C ′ = −m 2 and arrive at (62) This is the well known result of the standard QED [75] if we put v F = 1 and consider one Dirac point in place of two as for graphene.Now we pass to the case D = 3, i.e., to the polarization tensor of graphene at zero temperature.In this case Eq. ( 54) takes the form This equation, similar to Eq. ( 55), is obtained by the analytic continuation of Eq. (54).However, as opposed to Eq. ( 55), it is finite and does not contain the pole terms.Thus, no subtraction of infinities is needed for obtaining the final physical result, i.e., the polarization tensor of graphene behaves like that in the truly three-dimensional QED which is the super-renormalizable theory (as mentioned especially in [2]), unlike the standard theory in four dimensions which is "only" renormalizable.
Using the same representation as in Eq. ( 59) one obtains from Eq. ( 63) The last integral is easily calculated [82].Thus, Using the most convenient expressions for this integral in different regions of parameters, for k2 < 0 we obtain Under the conditions k2 > 0, 2m > k2 we have Finally, under the conditions k2 > 0, 2m < k2 one obtains Note that there is a threshold at k2 = 2m.The two convenient independent quantities characterizing the polarization tensor are Π 00 and trΠ µν = g µν Π µν .Using Eq. ( 64), these are given by where Π 3 (k 2 ) is defined in Eqs. ( 67)-( 69) for different regions of the involved parameters.From Eq. ( 64) it is seen that if the mass shell equation k 2 0 − k 2 = 0 is satisfied, it holds Π µν (k 0 = 0) = 0.
Equations ( 64) and ( 67)-( 70) coincide with the results of Refs.[16,30,35] for the polarization tensor of graphene at zero temperature.It should be added also that the equivalent results [26] were found in the literature by the method of correlation functions in the random-phase approximation [83][84][85][86].The obtained results are unique and neither Π 00 0 nor trΠ µν 0 can be modified in any way.As to the thermal correction to the polarization tensor ∆ T Π µν , in Sec.II it was shown that it is finite for any D and defined uniquely.Because of this, it is not the subject of regularization which refers to only the zerotemperature case.
We underline that Eqs. ( 64) and ( 67)-( 69) for the polarization tensor of graphene at zero temperature, where the Fermi velocity v F is put equal to unity, are in agreement with the well known results of Refs.[1,2] obtained long ago in the framework of the standard (2+1)dimensional QED (the extra factor 2 is explained by the presence of two Dirac points for graphene).

V. CONCLUSIONS AND DISCUSSION
In the foregoing, we have analyzed the problem of convergence of the polarization tensor of graphene in the framework of the Dirac model.This is an interesting example regarding application of methods of the lowdimensional thermal QFT to a material of big practical importance.Although in the framework of QFT the polarization tensor of graphene is described by a simple oneloop diagram, which was calculated long ago, there are contradictory statements in the literature mentioned in Sec.I concerning its convergence, the necessity of its regularization and validity of the obtained results.Taking into account that the quantum field theoretical approach to the polarization tensor of graphene suggests the most direct and fundamental way for investigating the electrical conductivity and reflectance of graphene, as well as the Casimir effect in graphene systems, it seems necessary to clarify all the raised points.
For this purpose, we have performed a detailed calculation of the polarization tensor of graphene and analyzed its analytic properties as a function of the number of space-time dimensions.It is underlined that this tensor consists of the zero-temperature part plus the thermal correction.In so doing, the thermal correction is represented as an integral which converges in the space-time of any dimensionality.Thus, the question of regularization is irrelevant to the thermal correction and may be raised only with respect to the zero-temperature part of the polarization tensor.
For experts in QFT, calculation of the polarization tensor in the framework of (2+1)-dimensional QED is a rather simple exercise.Because of this, in the classical papers [1,2] the results of this calculation were presented without derivation.In Refs.[16,30,35], again with no detailed derivation, these results were modified for the case of graphene by taking into account the presence of two fundamental velocities.
As discussed in Sec.I, some of the theoretical predictions made using the quantum field theoretical polarization tensor (and especially its trace) are in disagreement with those found with the polarization tensor derived by the Kubo formula.To bring both tensors in agreement, an alternative regularization procedure was suggested [73] by imposing an artificial additional condition irrelevant to the rigorous formalism of quantum field theory.
Our detailed analysis of the convergence of the polarization tensor in D = (2 + 1)-dimensional space-time shows that, although it is formally represented by a divergent integral, its finite value is obtained by the analytic continuation.In so doing, one need not to discard any pole terms which do not appear in the case D = 3, i.e., the renormalization is not needed.Just this was meant in Refs.[27,28,30] stating that for D = 3 the ultraviolet divergences do not appear.After putting the Fermi velocity equal to the speed of light, our results for the zero-temperature polarization tensor are found in agreement with the well known results of Refs.[1,2].If the two fundamental velocities are present, our results coincide with those given for graphene in Refs.[16,30,35].
We remind that the situation is different in the case of the standard QED with D = 3 + 1.In this case, the zero-temperature polarization tensor is also obtained by the analytic continuation.However, for obtaining the finite result, it is necessary to discard the pole term which arises for D = 4.This pole corresponds to the ultraviolet divergence deleted by means of the renormalization procedure, which must be performed after a regularization.Therefore, there is a principal difference between the character of divergences of the polarization tensor for the three-and four-dimensional space-times.In both cases, however, the final results, obtained by the analytic continuation from the case of lower dimensionality and (for D = 4 only) by discarding the pole term and using the normalization condition, are unique and not a subject to any modification.
It is also necessary to stress also that the presence of a double pole at zero frequency in the transverse dielectric permittivity of graphene proven by using the polarization tensor [41] plays a decisive role in reaching an agreement between theory and measurements of the Casimir force in graphene systems [49][50][51][52].It is well known that for metallic test bodies the theoretical predictions are in agreement with the results of numerous precise experiments on measuring the Casimir force only if the response of metals to the low-frequency electromagnetic field is described by the dissipationless plasma model possessing a double pole at zero frequency [87,88].This problem was considered as a failure of the dissipative Drude model, possessing the single pole at zero frequency, in the re-gion of transverse electric evanescent waves [89].Thus, a prediction of the double pole in the transverse dielectric permittivity of graphene in the framework of quantum field theory, as opposed to the Kubo formula, is in favor of the former.
To conclude, the analysis performed in this paper opens opportunities for a wider use of quantum field theoretical methods for investigation of the properties of graphene and other novel materials.