Optical conductivity and damping of plasmons due to electron-electron interaction

We re-visit the issue of plasmon damping due to electron-electron interaction. The plasmon linewidth can related to the imaginary part of the charge susceptibility or, equivalently, to the real part of the optical conductivity, $\mathrm{Re}\sigma(q,\omega)$. Approaching the problem first via a standard semi-classical Boltzmann equation, we show that $\mathrm{Re}\sigma(q,\omega)$ of two-dimensional (2D) electron gas scales as $q^2T^2/\omega^4$ for $\omega\ll T$, which agrees with the results of Refs. [1] and [2] but disagrees with that of Ref. [3], according to which $\mathrm{Re}\sigma(q,\omega) \propto q^2T^2/\omega^2$. To resolve this disagreement, we re-derive $\mathrm{Re}\sigma(q,\omega)$ using the original method of Ref. {mishchenko:2004} for an arbitrary ratio $\omega/T$ and show that, while the last term is, indeed, present, it is subleading to the $q^2T^2/\omega^4$ term. We give a physical interpretation of both leading and subleading contributions in terms of the shear and bulk viscosities of an electron liquid, respectively. We also calculate $\mathrm{Re}\sigma(q,\omega)$ for a three-dimensional (3D) electron gas and doped monolayer graphene. We find that, with all other parameters being equal, finite temperature has the strongest effect on the plasmon linewidth in graphene, where it scales as $T^4\ln T$ for $\omega\ll T$.


I. INTRODUCTION
Collective modes of a Fermi liquid (FL) are the direct manifestation of its many-body nature.In a charged FL, the most well-studied mode is the plasmon.The plasmon dispersion and linewidth contain important information about the manybody dynamics in electron systems and is also a crucial parameter for plasmonic devices.Traditionally, plasmons have been observed by electron energy-loss spectroscopy [4].The interest to plasmon dynamics has recently intensified due to near-field optical spectroscopy of graphene-based devices [5][6][7] and momentum-resolved electron energy-loss spectroscopy of HTC cuprates and related compounds [8].On the theoretical side, within the random-phase approximation (RPA) and at  = 0 a plasmon has an infinitely long lifetime as along as it stays outside the particle-hole continuum and thus cannot decay via the Landau-damping mechanism.1Beyond RPA, plasmons do decay into multiple particle-hole pairs.In terms of Feynman diagrams, these processes are accounted for by dressing the free particle-hole bubbles with interaction lines.Damping of plasmons in three dimensions (3D) was studied by Dubois and M. G. Kivelson as early as in 1969 [9].Damping of plasmons in two dimensions (2D) was studied in Refs.[1][2][3][10][11][12][13] both in the collisionless and hydrodynamic regimes.However, the results of different papers for damping in the collisionless regime [1-3, 10, 13] do not always agree with each other, although all of them are obtained under the same assumptions, the most important of which is weak coupling.
The goal of this communication is to finalize the result for the lifetime of plasmons due to electron-electron interaction, at least at weak coupling.We will limit our attention to the collisionless regime, which occurs if the plasmon frequency is much higher than the rate of relaxation towards equilibrium, and consider the cases of 2D and 3D electron gases with parabolic dispersion, as well as of doped monolayer graphene.We will also consider only damping due to intraband excitations, although interband excitations need to be accounted for to explain real materials [14].
Formally, damping of plasmons is due to the fact that the imaginary part of the charge susceptibility,   (q, ) is finite outside the particle-hole continuum.Thanks to the Einstein relation, Re(, ) = 2   2 Im  (, ), the same condition can be reformulated in terms of the real part of the conductivity.To facilitate the comparison, we list below the results of different papers for Re(, ).2 Mishchenko, Reizer, and Glazman (MRG) [3] considered a Galilean-invariant 2D electron gas (2DEG), i.e., a 2D electron system with  k =  2 /2 dispersion.Using an original method to calculate the absorption rate of electromagnetic radiation by electrons, they obtained the following result for the conductivity at finite , , and : where   and  F are the Fermi momentum and velocity, respectively, and  = 2 2 is the inverse radius of the screened Coulomb interaction in 2D, defined by Equation ( 2) is derived under the following assumptions: max{, } ≪  F  ≲  F and  ≪ max{, }/ F .(We remind the reader that  F  ≪  F at weak coupling.)The results of the current paper will also be obtained under the same assumptions.Please note that henceforth  stands for the temperature of the electron system, which may be different from the lattice temperature; for example, Ref. [6] reports the electron temper-ature of graphene under near-field pumping to be as high as 3200 K.
The vanishing of Re(, ) in Eq. ( 2) at  = 0 reflects the fact that internal forces in a Galilean-invariant system do not affect the motion of the center-of-mass and, therefore, the dissipative part of the conductivity must vanish at  = 0 and finite .Such a constraint is no longer valid for graphene, which is not a Galilean-invariant system.The optical conductivity of graphene away from the charge neutrality was analyzed in Refs.[1] and [2], which identified the limiting form of the conductivity at  = 0.In addition, Refs.[1] and [2] presented an O ( 2 ) term, which is necessary for determining the plasmon linewidth.The complete result, as derived in Ref. [2], reads Re(, ) = Re 1 () + Re 2 (, ), (4a) where Re 1 () and Re 2 (, ) are the O ( 0 ) and O ( 2 ) contributions, respectively,  D is the Dirac velocity, which plays the role of  F for the Dirac spectrum, and  * =  F / D .3Note that although Re 1 is finite at  = 0, it is suppressed by a factor of (/ F ) 2 compared to a regular FL contribution [15].
It was further conjectured in Refs.[1] and [2] that the O ( 2 ) contribution should be the same for the Dirac and parabolic spectra, up to the re-definitions of the effective mass and inverse screening length, i.e., the optical conductivity of a 2DEG should read If this is the case (and we will show explicitly that it is), then there is a contradiction with the MRG result, Eq. ( 2).Indeed, Eqs. ( 2) and ( 5) differ by a factor of ( F /) 2 ∼  2 p ()/ 2 , where  p () is the plasmon dispersion.This implies that Eq. ( 5) is parametrically larger than Eq. ( 2) for  ≪  p ().
The difference between Eqs. ( 2) and ( 5) is not purely mathematical.In fact, the corresponding contributions arise from different physical processes and, on a general level, are related to the bulk and shear viscosities of the electron liquid, correspondingly.
The approaches employed in previous works [1][2][3] involve quite complicated computations.We find it instructive to start with a more straightforward approach, namely, with a semiclassical Boltzmann equation, which is valid for  ≪ .In this regime, Eqs.(2) and ( 5) reduce to and respectively, and it should be fairly easy to see which one is correct.This exercise is the subject of Sec.II.The rest of the paper is organized as follows.Section III A gives a brief review of the MRG method.In Sec.III B, we re-derive the result for the optical conductivity of a 2DEG using the MRG method and show that, in agreement with conjectures of Refs.[1] and [2], it is given by Eq. ( 5) rather than Eq. ( 2).We must emphasize that the error in Ref. [3] is purely computational and reflects neither on the MRG method itself nor on the results of this reference for plasmon damping due to electron-phonon interaction.Using the MRG method, we also calculate the optical conductivity of a 3D electron gas in Sec.III C and supply the details of the derivation of Eqs.(4a)-(4c) for graphene.In Sec.IV, we give a physical interpretation of our results in terms of the bulk and shear viscosities of an electron liquid.In Sec.V, we discuss the plasmon linewidth.Section VI presents our conclusions.

II. OPTICAL CONDUCTIVITY VIA THE SEMI-CLASSICAL BOLTZMANN EQUATION
In this section, we calculate the longitudinal conductivity of a 2D Galilean-invariant electron system, using the semi-classical Boltzmann equation (BE).Assuming the electric field of the form E = E 0   (q•r−  ) , the BE for the Fourier transform of the non-equilibrium part of the distribution function   (q, ; k) ≡   k reads where is the electron-electron collision integral, and an infinitesimally small imaginary term 0 + was added to ensure the retarded nature of the response.With a definition where  k,p→k ′ p ′ is the scattering probability and ∫ k is a shorthand for ∫  2 /(2) 2 (and similarly for other momenta).The overall scale of the collision integral is given by the relaxation rate due to electron-electron interactions at finite , 1/ ee ().The temperature is assumed to be low enough so that the condition 1/ ee () ≪  is satisfied, yet  ≪ .In this case, Eq. ( 11) can be solved by subsequent iterations in the collision integral.To zeroth order, we neglect the collision integral and obtain At the next step, we substitute  k =  (0) k + (1)  k back into Eq.( 11) and neglect  (1)  k inside the collision integral, to obtain The conductivity is read off from the electrical current , as a coefficient of linear proportionality between j and E. As we are interested in the longitudinal part of the conductivity, we choose q||E.Then the conductivity can be found as where it is understood that   (q, ) denotes the conductivity calculated with both E and q being along the -axis.
The real part of the zeroth order conductivity, obtained from Eq. ( 12), is non-zero only within the particle-hole continuum, i.e., for || <  F , and is not relevant here, while the first-order correction in Eq. (13) yields To arrive at the last result, we used the symmetry properties of  k,p→k ′ p ′ [17,18].At q = 0, the square bracket in sec-ond line of Eq. ( 15) is reduced to which vanishes identically for the Galilean-invariant case, when v k = k/.A finite result for the conductivity is obtained by expanding Eq. ( 15) in .To order  2 , we obtain where q = q/.The general form of Re(, ) can be deduced already at this step.Indeed, Eq. ( 16) contains integrals over three independent energies (say,  k ,  p , and  k ′ ), each of them contributing a factor of  to the final result.Therefore, which is consistent with Eq. ( 9).The result (17) can be understood in the following way.A factor of  2 follows immediately from the facts that Re(, ) must vanish at  = 0 and be analytic in  (at finite  and ).The scaling 1/ 4 follows from the fact that we need to iterate the BE once and expand the result in  F /.Finally, the factor of  2 is the expected FL scaling of the scattering rate.

III. OPTICAL CONDUCTIVITY VIA THE MISHCHENKO-REIZER-GLAZMAN (MRG) METHOD
In this section, we resolve the disagreement between the results of Refs.[3], and [1] and [2], and finalize the correct expression for the optical conductivity of a 2DEG.Using the MRG method, we will show that, in addition to the contribution found by MRG [Eq.( 2)], there is also another contribution given by Eq. (5).For completeness, we will also derive the expressions for the optical conductivities of a 3D electron gas and graphene in Secs.III C and III D, respectively.

A. MRG method
In the MRG method, one calculates the rate at which electromagnetic radiation is absorbed by a system of interacting electrons.The differential probability of an electron-electron collision in the presence of a photon is written via the Fermi golden rule as where p, k (p ′ , k ′ ) are the initial (final) momenta of electrons,  is the total spin of two electrons in the initial state,   is the spin projection on the quantization axis, and L ,  is the matrix element which depends on  and, in general, on   .
To first order in the screened Coulomb interaction, the matrix elements for the singlet and triplet states are given by and respectively, where  0 is the in-plane component of the electric field of an electromagnetic wave.Next, one derives the total probability of absorption using Eq. ( 19) with matrix elements from Eq. (20), which is then used to calculate the dissipation rate.The latter is then related to the real part of the longitudinal conductivity, which is given by [3] The matrix elements can be written as L 0,0 =  0 (A + A ex )/ 2 , and L 1,0 =  0 (A − A ex )/ 2 , where and A ex is the exchange term obtained by interchanging p ′ ↔ k ′ in Eq. ( 23).5 From now on, we will neglect the exchange term A ex , which contains the interaction potential at large momenta transfers and is, therefore, small for a weakly-screened Coulomb interaction.It is also convenient to introduce the momentum and energy transfers via Interestingly, the last formula can be expressed as a convolution of two free-electron response functions.For example, as shown in Appendix D, the dominant contribution to the conductivity of a Galilean-invariant system can be cast into the following form where  B () is the Bose function,   = 8/15 for  = 3 and   = 1/2 for  = 2, and where we have introduced the imaginary parts of the density-density response function [22] Im  (, ) ≡ −2 and of the transverse current-current response function (The factors of two in the equations above account for spin degeneracy.)In the zero-temperature limit the Ω-integral is restricted to the range 0 < Ω <  (for positive ).Furthermore, because both response functions vanish linearly at low frequency, we see that the integral goes as  3 , and the final result for the conductivity is proportional to  2 / 2 , as it should.Similar formulas for the subdominant contributions to the conductivity of Galilean-invariant systems are provided in Appendix D. Equation (25) helps to elucidate the nature of excitations responsible for plasmon damping, as will be discussed in detail in Section IV.We now proceed with applying Eq. ( 24) to specific cases.

B. Two-dimensional electron gas
First, we consider a 2D electron gas with a parabolic dispersion  k =  2 /2.As are we interested in the limit of  F / ≪ 1, we expand A in Eq. ( 23) in 1/ as The A 1 term in the equation above is the one that was found in Ref. [3].However, as will be shown below, one also needs to keep the A 2 term, despite the fact that it appears to be next order in 1/.[Note that we have already neglected O ( 4 ) terms in A 2 .]Expanding the interaction potential as  Q+q =  Q +q • ∇ Q  Q and retaining only up to O ( 2 ) terms, we re-write A 1 in Eq. (28b) as The MRG result, Eq. ( 2), is reproduced by keeping the first term in the equation above.Indeed, each of the three energy integrations in Eq. ( 24) (over  k ,  p , and Ω) contribute a factor of , thereby canceling out a factor of 1/ 3 .Next, each of the two delta-functions contributes a factor of 1/ which, in 2D, leads to a logarithmic divergence in the integral over  at the lower limit.Cutting off this divergence at  ∼ / F , we reproduce the structure of Eq. ( 2).Since the second term in Eq. ( 29) contains an extra factor of , the resultant integration does not lead to a logarithmic divergence, and is thus subleading in the leading logarithmic sense.Now, we turn to the "new" (compared to MRG) term, A 2 in Eq. (28c).Expanding this term to order  2 , we obtain As to be expected (and indeed shown to be the case in Appendix C 1), a typical value of |k − p| ∼   ≫  ≳ .Then A 2 can be estimated as |A 2 | ∼  2     / 2 .The resultant integral over  is convergent at  = 0 but needs to be cut off at  ∼   at the upper limit, upon which one reproduces the structure of Eq. ( 5).(The cross-term, A 1 A 2 , vanishes to leading order upon angular integration.) We pause here to emphasize a non-trivial structure of the expansion in 1/.Indeed, the expansion of the individual components in Eq. ( 23) starts with the 1/ terms, which cancel each other, followed by the O (v k,p • q/ 2 ) terms, which are supposed to be the leading ones.However, the latter also almost cancel each other, and one needs to keep two O ( 2 ) terms: the  2 /2 terms in Eq. (28b) and the entire A 2 term in Eq. (28c).For  ≪  p (), the "new" term (A 2 ) exceeds the "old" one.Deferring all further details to Appendix C 1, we present here the final result for the optical conductivity of a 2DEG with parabolic dispersion: The second ("new") term coincides with Eq. ( 5), which confirms the conjecture made in Refs.[1] and [2], while the second one is subleading to the first one for  ≪  p ().

C. Three-dimensional electron gas
For a 3D electron gas with parabolic dispersion, the form of A as is the same in Eqs.(28a)-( 30), but the integrals are different due to a change of the phase space.Deferring the details to Appendix C 2, we present here only the final result for the optical conductivity in the limit of  ≪  F : where the inverse screening length is given by  2 = 8 2   , and   =   /2 2 is the density of states per spin.

D. Doped monolayer graphene
The optical conductivity of graphene was calculated in Refs.[1] and [2], but the most complete result was given in Ref. [2] without a derivation.Here, we re-derive this result using the MRG method.
Without loss of generality, we assume that the Fermi energy is located in the upper Dirac cone.In the low-energy limit, i.e., for max{, } ≪  F , one can neglect the presence of the lower Dirac cone.6 Also, for a long-range Coulomb interaction, one can neglect processes that lead to swapping of electrons between the  and  ′ valleys, as well as the exchange parts of both intra-and intervalley scattering amplitudes.For the same reason, the phase factors in the matrix elements between spinor states can be replaced by unities.With all these simplification, electrons in graphene can be described by the following lowenergy Hamiltonian MRG's notations [3].6 As shown in Ref. [23], the interaction between electrons in the upper and lower Dirac cones gives a subleading (in the leading-logarithm sense) con-tribution to the optical conductivity.
where  = ±1 labels the  and  ′ valleys, k is the electron momentum measured from the center of the corresponding valley,  = ±1 is the spin projection, and  (0) Q = 2 2 / is the bare Coulomb potential.Within this approximation, the valley index plays the role of a conserved isospin.Therefore, the optical conductivity can be calculated by applying the MRG method to then case of spin-1/2 fermions occupying a single valley, i.e., using Eq. ( 24) and then multiplying the result by 2 2 = 4.
In contrast to 2D and 3D electron gases graphene is a non-Galilean-invariant system.Therefore, its optical conductivity is finite even at  = 0.However, we will see that in order to determine the plasmon linewidth accurately, one needs to retain both O (1) and O ( 2 ) terms in the optical conductivity.Recalling the factor of 1/ 2 in Eq. ( 24), we see that A should have terms of O () and O ( 2 ), which would give the O (1) and O ( 2 ) contributions to the conductivity, respectively.As shown in Appendix C 3, the result for A to required accuracy is given by where p is obtained by interchanging k with p in  k , and  nn ′ denotes the angle between the vectors n and n ′ .The -independent part of Re is exactly the same as calculated in Ref. [2], and we will give only a brief summary of the computational steps.Once A 2  1 is substituted into Eq.( 24), the differences of electronic dispersions in Eq. (34c) can be expressed via the frequencies  and Ω, using the constraints imposed by the delta-functions in Eq. (24).Suppose that  = 0.Then, given that Ω ∼ , we have A 2  1 ∝  2 .The three energy integrations (over  k ,  p , and Ω) are effectively constrained to the interval (0, ) by the Fermi functions and, therefore, contribute a factor of  each.Altogether, it follows that Re 1 () ∝  2 ×  3 / 3 =  2 .This is the leading contribution to the conductivity for  ≪  F .Therefore, one can neglect the frequencies in the delta-functions, upon which they impose purely geometric constraints on the angular variables; for  ≪  F , we simply have cos  pQ = cos  kQ = 0.The -integration is same the as for a 2DEG (see Appendix C 1): given that each of the two delta-function gives a factor of 1/, the integral over  diverges logarithmically at the lower limit.Cutting this divergence off at  = ||/ D , we reproduce the  = 0 limit of Eq. (4b).The reasoning for the opposite limit of  ≪  is essentially the same, except for now the dispersions in A 2  1 give a factor of  2 , another  3 comes from the energy integrations, and the factor of 1 −  − / in Eq. ( 24) is reduced to /.Cutting off the -integration at  = / D , we obtain Re 1 () ∝ (/) ×  5 × ln/ 3 =  4 ln/ 2 , which is the  ≪  limit of Eq. (4b).Details of calculating the O ( 2 ) can be found in Appendix C 3, and the final result is given by Eqs.(4a)-(4c).

IV. PHYSICAL INTERPRETATION
To clarify the physical meaning of the results obtained in the previous sections, we start from the relation between the real part of the conductivity and the viscosities of the electron liquid [24].This relation can be inferred from the equation of motion for the current density and has the form where   = / and   = / are the bulk and shear viscosities, respectively, (also known as longitudinal and transverse kinematic viscosities respectively) of the electron gas with number density , evaluated in the collisionless regime.
(Not to be confused with the hydrodynamic viscosities, which are non-perturbative in the Coulomb interaction).Interestingly, Eq. ( 35) can be viewed as an extension of the Einstein relation for the conductivity.The standard Einstein relation (for electrons in the presence of impurities) reads where  (0) = lim →0   (, 0) is the density of states at the Fermi level,   (, ) is the charge susceptibility, and D is the diffusion coefficient.Equation (35) can be obtained by replacing  (0) = lim →0   (, 0) by lim →0   (, ) = − 2 / 2 at finite frequency (notice, however, the change of sign in front of   ) and D by   + (2 − 2/)   , which is the diffusion coefficient of the momentum density.
The key question now is: what is the low-frequency behavior of the transport coefficients   and   ?The answer is that   tends to a finite value at  → 0 (plus corrections of order  2 ), while   vanishes at  → 0 as  2 .Thus we see that the terms proportional to  −2 arise from the shear viscosity, while the -independent terms arise from the  2 contributions to the shear viscosity as well as the bulk viscosity.(We stress that this is valid in dimensions  > 1, where a transverse viscosity can be defined.) What is the physical reason for the difference?The two viscosity coefficients can be expressed in terms of the stressstress response function as follows: where [•, •] is the commutator of two operators, and ⟨•⟩ denotes the statistical average over the equilibrium density matrix.The time evolution of Â is generated by the non-interacting Hamiltonian.The spectral density of an observable Ô is obtained from Eq. ( 39) by calculating Im⟨⟨ Ô; Ô⟩⟩  .Furthermore, the stress-tensor operator is given by [25] P = T + Ŵ , where is the bare Coulomb potential, and nQ = p â † p−Q âp is the number density operator.
Let us consider the bulk viscosity first.The trace of the stress tensor is easily seen to be given by P = T + Ĥ, where T is the kinetic energy operator and Ĥ is the total Hamiltonian of the electron system, i.e., the sum of the kinetic and interaction parts.Ĥ is a constant of the motion and thus does not contribute to the response function.
Then we are left with which can be rewritten in terms of the time derivatives of T as where we used that The second line of this equation follows the from timetranslational invariance of the response function.Equation (44) is obtained by noting that the commutator of two Hermitian operators is anti-Hermitian, and therefore all its eigenvalues are imaginary.Thus, the term ⟨[ T, T]⟩, obtained by applying Eq. (45) to ⟨⟨ T; T⟩⟩  twice, is purely real and can only contribute to the real part of the response function.
Due to the Coulomb interaction, the kinetic energy operator depends on time, and its time derivative is given by (Notice that T is proportional to the scalar product of the Coulomb force density FQ = −Q (0)  nQ and the longitudinal current density ĵ−Q .) In the limit of a large number of fermion flavors [26] the spectral function Im⟨⟨ T; T⟩⟩  is the convolution of two electron-hole spectral functions, associated with density fluctuations longitudinal current density fluctuations The spectral function of density fluctuations vanishes as  at  → 0, while that of longitudinal current-density fluctuations vanishes as  3 , as one can see from the well-known relation [22].Therefore, at zero temperature we have Substituting the last result into Eq.( 44) gives   ∝  2 as announced.The essential reason for this result is the scarcity of longitudinal electron-hole pair excitations at low frequency: their spectral density vanishes as  3 .
Let us now consider the shear viscosity, Eq. ( 38).Without loss of generality we can orient the  asis along Q and the  axis perpendicular to Q.It can be easily shown that the averages ⟨⟨ Ŵ ; Ŵ ⟩⟩  and ⟨⟨ T ; Ŵ ⟩⟩  involve only longitudinal current density fluctuations and, therefore, vanish as  2 as before.However the average ⟨⟨ T ; T ⟩⟩  now involves transverse current density fluctuations.As before, we rewrite   () as where ĵ,−Q is the component of the current density in the  direction, i.e., perpendicular to Q and . . .stand for subleading terms that contain the longitudinal component of the current-density operator.Now we see that the spectral function Im⟨⟨ T ; T ⟩⟩  is the convolution of two electron-hole spectral functions, one associated with density fluctuations and another one associated with transverse current density fluctuations.The spectral density of transverse current density fluctuations vanishes as  as opposed to  3 .Therefore we now have Substituting the last result into Eq.( 48) gives a finite value of   in the  → 0 limit.The reason for the different behavior of the longitudinal and transverse current spectral functions is illustrated in Fig. 1.
The horizontal arrows represent the momenta of lowenergy electron-hole pair excitations, Q and −Q.Such excitations contribute to plasmon damping.Due to kinematic constraints, the initial momenta of the excited electrons are confined to the crescentshaped regions (external to the dashed lines), within the Fermi surface (solid circle).The short red segments denote the widths of allowed regions of the initial momenta for excitations of a given (small) energy.Notice that these excitations are essentially transverse, meaning that the average momentum of the initial and final state of the excited electron (black arrow) is almost orthogonal to the momentum of the excitation, Q.The size of the integration region (red segments) vanishes linearly as the energy of the excitation tends to zero, consistent with the fact that the spectral function of density fluctuations vanishes linearly at low frequency.By contrast, longitudinal current excitations with the same wave vector Q (indicated by the red horizontal lines) have much higher energies: their contribution to the low-frequency spectrum vanishes as the third power of the excitation energy.

A. Definitions
We now derive the expressions for the plasmon linewidth, using the results Re(, ) obtained in previous sections.The linewidth, Γ(), is calculated at the plasmon pole and is given by in 2D and in 3D.It is also customary to define a dimensionless inverse quality factor (IQF) B. Electron gas with parabolic dispersion

2D electron gas
In a 2DEG, the plasmon frequency is  p () =  F √︁ /2 with  = 2 2 .Substituting the leading term in the optical conductivity [Eq.( 5)] into Eq.( 51), we obtain the plasmon linewidth as Defining  p ≡  F  ∼  p (), we see that Γ() behaves as  2 for  ≪ (/ p ) 2 and as  2 for  ≫ (/ p ) 2 .The corresponding IQF can be written in a dimensionless form as 3  e ln 1 where  e =  2 / F is dimensionless coupling constant of the Coulomb interaction, q = /, and T = / p .

3D electron gas
Neglecting plasmon dispersion, the plasmon frequency of a For a 3D electron gas is given by  p = √︁ 4 2 /, where  =  3  /3 2 is the number density.Substituting Re from Eq. (32) into Eq.( 52), we obtain where  2 = 8 2   with   =   /2 2 , and where the dimensionless parameters are the same as defined after Eq. ( 55).The asymptotic forms of Γ() and () are the same as for a 2DEG.

C. Doped monolayer graphene
So far, we discussed graphene in the isotropic approximation, which neglects trigonal warping of the Fermi surfaces around the  and  ′ points.The corresponding optical conductivity is given by the sum of Re 1 and Re 2 from Eqs. (4b) and (4c), respectively.Trigonal warping breaks the degeneracy of the  and  ′ valleys, and intervalley scattering is now allowed to contribute to the conductivity, even if there is no swapping of electrons between the valleys.The corresponding contribution to the conductivity was found in Ref. [2]: where  is the distance between two nearest carbon atoms.This contribution is of a regular FL type [15], i.e., it is finite at  = 0 and, in contrast to Re 1 , is not suppressed due to partial Galilean invariance.However, being a lattice effect, it becomes significant only at sufficiently high filling, when the product  F  is not too small.Accordingly, the plasmon linewidth is written as the sum of three parts: where  p () =  D √  is the plasmon frequency and  = 4 2  F / D .For  ≪  √︁ / p , the linewidth is determined primarily by Γ 1 () in Eq. (59b), which is independent of  and scales as  4 ln in this limit.In the opposite limit of  ≫  √︁ / p , the contributions from Γ 1 and Γ 2 are of the same order-of-magnitude (although Γ 2 is numerically smaller) and both scale as  2 .On the other hand, the contribution from Γ 3 is much smaller than that from Γ 1 for any reasonable value of  F .
In terms of the dimensionless variables introduced after Eq. ( 55), the IQF for graphene is written as where  g =  2 / D .

D. Discussion
The IQFs for 2D and 3D electrons gases are plotted as a function of q = / in Fig. 2. At other parameters being equal, damping is stronger in 2D (solid curve) than in 3D (dashed curve over a wide range of .This is primarily due to the fact that finite temperature has little effect on damping is 3D.Indeed, the temperature dependence of () in Eq. ( 57) is weak as long as T ≪ 1 (or  ≪  p ), and () scales as q2 for any q.On the contrary, () in 2D [Eq.(55)] decreases with q slower than in 3D, i.e., as q1/2 T2 for q ≪ T2 (or  ≪  √︁ / p ).As q approaches 1 from below, damping in 2D and 3D becomes comparable.
The partial components of () as well as total () in graphene are plotted in Fig. 3. First, we note that  1 () dom-inates over the other two components for almost entire range of .Next, we see that the effect of finite temperature on damping is graphene is even stronger than for a 2DEG: for q ≪ 1, () ≈  1 () scales as T4 / √ q, i.e., the (relative) plasmon linewidth decreases with increasing q.In this regime, the usual numerical prefactor of 2, amplifying the effect of finite temperature on collective modes, plays a very important role, as it leads to an enhancement of the linewidth by a factor of 2(2) 4 ≈ 3120.An apparent divergence of () at  → 0 signals a breakdown of the collisionless approximation and a crossover to the hydrodynamic regime.As  increases, () goes through a shallow minimum at q = 4 2 ( √ 13 − 2)/9 T2 ≈ 7.0 T2 and then increases as q3/2 .Comparing the vertical scales of Figs. 2 and 3, we see that, at other parameters being equal, damping is, in general, stronger in graphene than in an electron gas with parabolic dispersion, even though the IQF for the latter is higher order in the coupling constant (third vs second).This is, again, due to a higher  sensitiviy of the linewidth of graphene to finite .For q ≪ T2 , the ratio ()| 2DEG /()| graphene ∼ q/ T2 ≪ 1. Damping of plasmons in graphene at finite  was discussed by Lucas and Das Sarma [13], who conjectured that Γ() ∝  2 in the collisionless regime.This result is consistent with Eq. (54) for a 2DEG with parabolic dispersion and with the subleading, Γ 2 () contribution to the linewidth in graphene [Eq.(59c)].However, the leading contribution to the linewidth in graphene is given by Γ 1 () [Eq.(59b)], which is independent of  and scales as  4 ln for small  ≪ (/ p ) 2 ).The discrepancy between our result and that of Ref. [13] is due to the assumption of Ref. [13] that the relevant relaxation rate in graphene scales in a canonical FL way, i.e., as  2 .However, whereas the single-particle relaxation rate in graphene indeed scales as  2 (modulo a factor of ln), the current relaxation rate, defined as Re =  2 / *  2  j (,) with  being the carrier number density, scales as 1/ j ∝ max{ 4 ln ||, 4 ln }; see, e.g., Ref. [21] and references therein.This property is not unique to graphene but common for any 2D and 3D FL with an isotropic but non-parabolic spectrum (except for there is no ln factor in 3D) [2,23].

VI. CONCLUSIONS
Prompted by some discrepancies between results of different theoretical papers, we re-visited the optical conductivity, Re(, ), of an electron system due to electron-electron interaction and a related issue of plasmon damping, focusing on the collisionless (as opposed to hydrodynamic) regime.Setting aside more sophisticated methods, we first showed that a semi-classical Boltzmann equation for a two-dimensional electron gas (2DEG) with parabolic dispersion yields the optical conductivity which behaves as  2  2 / 4 for  F  ≪  ≪ .This behavior is consistent with the results of Refs.[1,2] but not with those of Ref. [3].Next, we re-derived the full expression for Re(, ) of a 2DEG [Eq.( 5)], using the original and elegant method of Ref. [3], and identified the reason for discrepancy between the results of Ref. [3] and Refs.[1,2].We showed the results of Ref. [3] and Refs.[1,2] correspond to physically distinct contributions, arising from the bulk and shear viscosities of an electron liquid.While the bulk viscosity vanishes at  → 0, the shear one approaches a finite value.This explains why at sufficiently low frequency, the conductivity found in Refs.[1,2] is larger than one found in Ref. [3].For completeness, we also calculated Re(, ) of a 3D electron gas and doped graphene, using the same method.The optical conductivity of a 3D electron gas [Eq.(32)] is found to be similar to the 2D case: in both cases, Re(, ) vanishes at  → 0 due to Galilean invariance.However, the case of doped graphene is different due to broken Galilean invariance.As a result, Re(, ) is finite at  = 0 and scales as  2 ln || at  = 0.This result was derived in our previous work [2], using the equations-of-motion method.Here, we also re-derived the O ( 2 ) term in the conductivity and found that it is exactly the same as in a 2D electron gas (up to a re-definition of an effective electron mass), in agreement with a conjecture of Refs.[1,2].
Using our results for the optical conductivity, we analyzed the behavior of the plasmon linewidth, Γ(), in various systems.Given that the electron temperature in an experiment may be significantly higher than the lattice one, we paid special attention to the -dependence of Γ().We found that, with all parameters being equal, the effect of finite  on plasmon damping is the strongest in graphene: Γ() is independent of  and scales as  4 ln for  ≪  2 / 2 D , where  D is the Dirac velocity and  is the inverse screening length.This scaling reflects the fact that the current relaxation rate in a non-Galilean-invariant but isotropic system behaves as max{ 4 , 4 } (with an extra ln max{||, } factor in 2D) [2,23].Plasmon linewidth in graphene can be measured via near-field spectroscopy [5,6].Also, a recent experiment has demonstrated that plasmons in a system with strong spin-orbit coupling can be observed directly by Raman spectroscopy [27].We hope that the results of our paper will be useful for the interpretation of these and future experiments.
is the polarization bubble (Lindhard function).The plasmon linewidth is determined by the imaginary part of Π(q, ).For small , In two dimensions ( = 2) and for a parabolic dispersion with mass , Changing the integration variable from  to  =  2 /2, expressing  in units of  F as  =  F , and defining  ≡ / F , we obtain We are interested in the plasmon range, when  ≫ 1, such that  ≥  ≫ 1.In the degenerate case, we also have  F ≫ , thus the argument of cosh is always large, and cosh  ≈   /2.The integral can be then simplified as (This result was presented in Ref. [3] without a factor of   F / .) The plasmon linewidth is given by where  p () =  F √︁ /2 is the plasmon frequency,  = 2 2 is the inverse Thomas-Fermi screening radius, and  (0) q = 2 2 /.According to Eq. (53), the inverse quality factor is given by which is exponentially small for  ≪  and  ≪  F .

and introducing the energy transfer via
where   = /2 is the density of states, and  ab is the angle between vectors a and b.Expanding the factor in the square brackets in Eq. (B1) in /  to order  2 and projecting the electron momenta onto the Fermi surface, we obtain Writing  qp =  qQ +  Qp ,  qk =  qQ +  Qk and integrating over  qQ , we get To integrate over  kQ and  pQ , we use the constraints imposed by energy conservation via the delta-functions in Eq. (B1) With Ω = 0 and O ( 2 ) terms discarded, these constraints are  kQ = ±/2 and  pQ = ±/2.The last term in Eq. (B3) ensures that only the combinations of  kQ = − pQ = ±/2 give non-zero contributions.Summing up these contributions, we obtain Next, the energy integration gives Substituting Eqs.(B4) and (B5) back into Eq.(B1), and calculating the -integral to leading logarithmic accuracy with an upper limit cutoff at  ∼   , we arrive at the final result which is Eq. ( 18) of the main text.
Appendix C: Optical conductivity via MRG method

2D electron gas
In this Appendix, we derive Eq. (31) of the main text.We start with a 2D version of Eq. ( 24) with A given by the sum of the first term in Eq. (29) and the entire Eq.(30): Replacing the integration over p(k) integral by that over  p ( k ) and angles  pQ ( kQ ), and confining the energy integrations to the narrow vicinity of the Fermi energy, we obtain where   = /2 is the density of states per spin orientation.Writing  qp =  qQ +  Qp and  qk =  qQ +  Qk , and folding the angular integrations down to the (0, ) intervals, we get where is the projection of A onto the Fermi surface.
To perform the angular integrations in Eq. (C3), we use the constraints imposed by the delta functions in Eq. (C4), i.e., Substituting Eq. (C6) back into Eq.(C4), we get We anticipate (and will prove it later) typical values of the energy and momentum transfers to be such that Ω ∼  ≪  F  and  ≪  F .If so, then the factor in square brackets in Eq. ( C7a) is small as and similarly for other terms in Eq. (23).Substituting these expansions into Eq.( 23), we obtain The momentum transfer  is defined by p − p ′ = Q, and from momentum conservation we have k ′ − k = Q + q.As for the parabolic case, the difference between  k−k ′ =  Q+q and  p−p ′ =  Q can be neglected.Next, we split A into two parts as A = B + C, where Expanding B to O ( 2 ), we obtain As the remaining part of Eq. (C22), C, is already proportional to  2 , we can neglect  everywhere else in that part.Then,

Figure 2 .
Figure 2. Inverse qualify factor, () [Eq.(53)], for 2D (dashed) and 3D (solid) electron gases as a function of /, where  is the inverse screening radius.The electron temperature  = 0.03  F , where  F is the Fermi velocity, and the coupling constant of the Coulomb interaction  =  2 / F = 0.8.