Lindhard function, optical conductivity and plasmon mode of a linear triple component fermionic system


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1 Lindhard function, optical conductivity plasmon mode of a linear triple component fermionic system Bashab Dey Tarun Kanti Ghosh Department of Physics, Indian Institute of TechnoyKanpur, Kanpur8 6, India Pd Bi S Ag Se Au 8. Material realizations of timereversal symmetry (TRSbreaking TCFs is still absent but are predicted to be found in magnetically ordered systems. In this work, we analyse the density current response functions for linear TCFs in noninteracting interacting limits. Linear Response: When a system is subjected to a time space dependent electric field (or potential, the field couples with the charge degree of freedom of electrons drives the system into a nonequilibrium phase. The system responds by modifying its charge density induction of charge currents. If the strength of the external field is small enough to be treated as a perturbation, the response function is obtained from the Kubo formula 6. For stronger fields, the system shows nonlinear response 7 may give rise to Floquet bs 8 5. The Kubo formula usually holds good for the typical amplitudes of electric fields used in the experiments. It is used to obtain current density responses of the system, which are given by its conductivity polarizability respectively. The polarizability relates the induced density fluctuation of the electron gas to the external potential, while the conductivity relates the induced current to the external field. The conductivity can be calculated from the polarizability viceversa. The polarizability function in momentumfrequency space is called the Lindhard function 5. The imaginary part of Lindhard function is a measure of energy absorbed by the system. The electron gas allows absorption for a range of momentum frequency, which is attributed to intrab or interb particlehole excitations across the Fermi sea. This region in the momentumfrequency space is called the particlehole continuum (PHC. At T, energy is not absorbed from the field for frequencies momenta outside the PHC. The shape of the PHC depends on the the chemical potential, b structure overlap factor between the bs. On inclusion of Coulomb interaction between the electrons, the Lindhard function gets renormalized by the dielectric function within Rom Phase Approximation (RPA The renormalization gives rise to plasarxiv:9.595v condmat.meshall] Sep We investigate the nature of density response of linear triple component fermions by computing the Lindhard function, dielectric function, plasmon mode long wavelength optical conductivity of the system compare the results with those of Weyl fermions three dimensional free electron gas. Linear triple component fermions are the low energy quasiparticles of linear triple component semimetals, consisting of linearly dispersive dispersionless (flat b excitations. The presence of flat b brings about notable modifications in the response properties with respect to Weyl fermions such as induction of a new region in the particlehole continuum, reduced plasmon energy gap, shift in absorption edge, enhanced rate of increase in energy absorption with frequency forbidden intercone transitions in the long wavelength limit. The plasmon dispersion follows the usual ω ω ω q nature as observed in most of the three dimensional electronic systems. I. INTRODUCTION Three dimensional semimetals having linear energy spectra around the Fermi level viz. Weyl Dirac 6, semimetals have become breeding grounds for plethora of intriguing physical phenomena such as Fermi arc surface states, chiral anomaly, anomalous Hall effect 5 etc. The quasiparticles close to the bcrossing nodes act as condensedmatter versions of Weyl 6 massless Dirac 7 fermions theorized in highenergy physics. Recent studies have unveiled other classes of topoical semimetals where more than two bs cross at a node exhibit fermionic excitations with no counterpart in high energy physics 8. It is speculated that mirror discrete rotational symmetries in symmorphic crystals may lead to topoically protected threefold degenerate crossing points,. Firstprinciples calculations 7 have shown that materials such as TaN, MoP, WC, RhSi, RhGe ZrTe can host threeb crossings in the neighborhood of the Fermi level 5,8. In this paper, we deal with one such class of semimetals with threeb crossings, where quasiparticles around the nodes transform under pseudospin representation. These are called triplecomponent semimetals (TCSs their low energy excitations are called triple component fermions (TCFs. The pseudospin degrees of freedom may emerge from specific admixtures of orbital spin projections 8,,5. The dynamics of the TCFs are governed by the Hamiltonian H(k = d(k S, where S = (S x, S y, S z denote the usual spin matrices d(k is a vector function of k. The b structure consists of two dispersive bs a flat b. The TCFs can be grouped into linear, quadratic cubic, depending on the form of d(k. For linear TCFs, the energy scales linearly with all the three components of momentum. For quadratic cubic TCFs, the energy scales linearly with k z, but as k k respectively in the k xk y plane, where k = kx ky. Timereversal symmetric TCFs arise in materials with space group symmetry 99, e.g.
2 mon modes 5 which are perceived as collective oscillations of the interacting electrons in the uniform positively charged background of the system (jellium model. They correspond to the poles of the renormalized Lindhard function or the zeroes of the dielectric function. They may be grouped into gapped or gapless modes, which require finite infinitesimal amounts of energy respectively to be excited at long wavelengths (q. The curve in momentumfrequency space along which the dielectric function vanishes is called the plasmon dispersion. Plasmons can be excited at wavelengths frequencies given by the plasmon dispersion. They are probed using inelastic scattering experiments such as electron energy loss spectroscopy 59. The dimensionality b structure of the system play crucial roles in determining the plasmon dispersion. It has been studied that D electron systems such as D free electron gas (FEG 58, graphene 6 66 dice lattice 67 (D pseudospin system host a gapless plasmon mode with dispersion q at long wavelengths. In contrast, D FEG 68, D noncentrosymmetric metals 69 doped Weyl 7,7 Dirac semimetals 7 7 exhibit gapped plasmon modes dispersing as ω ω q in the long wavelength limit, where ω ω are constants with appropriate dimensions. However, the study of response functions plasmons in TCSs is still unexplored. We fill this gap in the research by making a comprehensive analysis of the Lindhard function, PHC, dielectric function optical conductivity for linear isotropic TCFs compare the results with those of Weyl fermions D FEG. Furthermore, we derive approximate analytical expression of Lindhard function plasmon dispersion in the long wavelength limit. We investigate the interplay of three bs the effect of flat b in particular in the response functions. For the rest of the paper, TCF Weyl fermions/semimetals would refer to linear isotropic TCF isotropic typei Weyl fermions/semimetals repectively. This paper is organized as follows. In Sec. (II, we review the low energy b structure eigenstates of TCF. In Sec. (III A, we obtain the Lindhard function PHC of doped linear TCS. The calculation of dielectric function plasmon modes of the system are shown in Sec. (III B. A discussion on optical conductivity is presented in Sec. (III C. Finally, the results are summarized in Sec. (IV. II. MODEL HAMILTONIAN The Hamiltonian of TCFs around a b touching node is given by H(k = v F S k. ( Here, v F is the Fermi velocity S = (S x, S y, S z denotes the usual spin matrices. The b structure comprises of three bs viz. E k = v F k (conduction b, E k = v F k (valence b E k = (flat b. Denoting the pseudospin basis states { s } as = ( T, = ( T = ( T where T sts for transpose, the singleparticle eigenstates { λ(k } are given by (k = III. cos θ sin θ e iφ sin θ eiφ (k = sin θ, (k = sin θ e iφ cos θ eiφ ( sin θ cos θe iφ. ( sin θ e iφ RESPONSE FUNCTIONS OF TCFS A. LINDHARD FUNCTION A brief review of the theory of linear density response for a multib system is presented in Appendix(A. The dynamical polarization function or the Lindhard function (A6 of a noninteracting system of electrons is given by g χ(q, ω = lim η V F λ,λ (k, k q(f λ,k f λ,kq (ω iη E λ,k E λ k,λ,λ,kq ( where g is the degeneracy factor, F λ,λ (k, k q = λ(k λ (k q is the overlap between the corresponding states f λ,k = e β(e λ,k E F ] is the FermiDirac distribution function. For TCF, the interb intrab overlaps between the dispersive bs is given by F λ,λ (k, k q = ] k (k q λλ, λ, λ = ± k k q (5 that between the flat dispersive bs is F,λ (k, k q = F λ, (k, k q = ( ] k (k q. k k q (6 At T K, the Lindhard function ( takes the following form for E F > (i.e. doped TCS: where χ(q, ω = χ ( (q, ω χ ( (q, ω χ ( (q, ω, (7,
3 χ ( g (q, ω = lim η V k F, (k, k q(f,k f,kq ω iη E,k E,kq F, (k, k qf,k ω iη E,k E,kq F,(k, k qf,kq ω iη E,k E,kq F, (k, k qf,k ω iη E,k E,kq ], F,(k, k qf,kq ω iη E,k E,kq (8 χ ( g (q, ω = lim η V k F, (k, k qf,k ω iη E,k E,kq F ],(k, k qf,kq ω iη E,k E,kq (9 χ ( g (q, ω = lim η V k F, (k, k qf,k ω iη E,k E,kq ] F, (k, k qf,kq ω iη E,k E,kq. ( Here, we have excluded the terms which represent the intrab transitions within flat valence bs the interb transitions between them. This is true only for E F >. On nondimensionalizing the quantities as x = k/k F, Q = q/k F, Ω = lim η (ω iη/e F = lim η ( ω i η/e F, ω = ω/e F χ (λ (Q, Ω = χ (λ (q, ω/χ F (where E F = v F k F χ F = gkf /(π v F converting the summation into continuous integrals, equations (8,(9 ( simplify as χ ( (Q, Ω = x(ω x Q ( Ω Ωx Q xq Ω Ωx Q xq Qx(Ω Ωx x (Q x dx ( (Q x (Ω x (Qx ( Q Ω Ωx x 6Qx(Ω Ωx x x Q Ω Ωx x ( Ω Ωx Q ] xq Q Ω Ωx Q dx xq Q Q x (Ω x Q (Q x ( ] (Q x 8Q x (Q x dx ( Ω Ω, ( ] Ω Ωx Q xq Ω Ωx Q xq ( dx Q χ ( (Q, Ω = 8xQ (Q Ω x(q Ω x(q Ω x(q Ω x Λ Q 8xQ (Q x x x( Q Ω x QxΩ (Q x (Q x Ω ( ] x Q Ω dx Ω Q x Ω (x Q Q Q( Q Ω x QxΩ (Q x (Q x Ω ( ] x Q Ω dx ( Ω Ω x Q Ω (Q Ω x(q Ω x(q Ω x(q Ω x Ω ( x Q x Q ( Q x Q x (
4 Q χ ( (Q, Ω = x(q Ω 6Ωx x Qx(Ω 5x 6xQ (Q x x (Q x (Q x x Ω (Q (Ω x ( ] x Q Ω dx Ω x Q x Λ Q(Q Ω 6Ωx x Qx(Ω 5x Q 6xQ (x Q Q (Q x (Q x x Ω (Q (Ω x ( ] x Q Ω dx ( Ω Ω. Ω x x Q Ω ( x Q x Q ( x Q Q x ( FIG. : Different regions of PHC for TCF. The dotted, violet red regions indicates flat to conduction, valence to conduction intraconductionb transitions respectively for E F >. We restrict the limits of integration in Eqs. ( ( to an ultraviolet cutoff Λ = k c /k F. Thus, the dimensionless form of Lindhard function is χ(q, Ω = χ ( (Q, Ω χ ( (Q, Ω χ ( (Q, Ω. ( A diagram of the PHC for doped TCS (E F > is shown in Fig.(]. Like Weyl semimetals, the PHC for intrab transitions within the conduction b is bounded by ω = Q, ω = ω = Q lines, while the interb transitions between valence conduction bs occur in the region bounded by ω = Q ω = Q lines. The flat b introduces a new region of PHC which is absent in Weyl semimetals. The PHC for interb transitions between the flat conduction bs is above ω = line. So, the flattoconduction PHC overlaps those of intercone intracone ones. These features were observed in dice lattice also 67. The FIG. : Density plot of the natural arithm of Im χ(q, Ω] as functions of Q Ω for TCF. numerical plot of natural arithm of Im χ(q, Ω] as functions of Q ω (shown in Fig.] reveals the characteristics of the PHC reasonably well albeit the sharp demarcations of different regions of absorption. The static Lindhard function Re χ(q, ] as a function of Q is plotted in Fig.( for TCF, Weyl semimetals D FEG. The function rises monotonically with Q for both TCF Weyl fermions with the slope being higher in the former. This nature is contrary to that of FEG where the function decreases monotonically with Q with a slope discontinuity at ω =. B. DIELECTRIC FUNCTION AND PLASMONS For TCF, the dielectric function (A8 can be written as ε(q, Ω = C χ(q, Ω, (5 Q
5 5. Weyl TCF FEG Re FIG. : Plots of Re χ(q, ] vs Q for TCF, weyl semimetal free electron gas(feg. The Re χ(q, ] increases monotonically with Q for TCF Weyl semimetals, but a decreasing function of Q for free electron gas. Also, magnitude of Re χ(q, ] for TCF is greater than that of Weyl semimetal for the same set of parameters FIG. 5: Comparison of analytical solution of plasmon mode (dotted curve for long wavelength (Q regime given by Eq.(5 numerically obtained plasmon mode in the loss function plot. The agreement is good for low Q as expected. FIG. : Density plot of the natural arithm of loss function (7 as a function of q/k F ω/e F. The plasmon mode appears as bright curve in the region where Im ( χ vanishes. Hence, the mode is undamped. It continues to extend into the PHC where it gets damped into particlehole excitations. where C = e g/(ε r ε π v F. For v F = 5 m/s, ε r = g =, we get C.7. The undamped plasmon modes ω p for TCF can be obtained by solving the following equation for Ω Q: C Q Re χ(q, Ω p] =. (6 FIG. 6: Density plot of the natural arithm of loss function as functions of ε r Ω for TCF for very small wavelengths Q. The plasmon mode (bright yellow curve remains undamped its frequency decreases with ɛ r. Since the exact solution of the Eq.(6 cannot be obtained analytically, we deduce an approximate expression of long wavelength (Q low frequency ( ω plasmon mode of this system using the expansion of ( in orders of Q. The Lindhard function for small Q can
6 6 TCF Weyl FEG (ev (ev FIG. 7: Density plot of the natural arithm of loss function as functions of E F Ω for TCF for very small wavelengths Q. The plasmon mode (bright yellow curve remains undamped its frequency increases with E F. be written as χ(q, Ω = χ cc (Q, Ω χ fc (Q, Ω χ vc (Q, Ω, (7 where χ cc (Q, Ω, χ fc (Q, Ω χ vc (Q, Ω are intraconduction b, flattoconduction valencetoconduction (intercone contributions respectively, given by ( χ cc (Q, Ω = Ω Q 5Ω Q O(Q 6, (8 χ fc (Q, Ω = Λ ] x (Ω x Q 5x(Ω x Q dx (x (Ω x Q x (Ω 5Ω x 5( Ω x x Q x χ vc (Q, Ω = 5Ω x 6x 5 Λ x ] 5Ω x 6x 5 dx Q. ] dx, (9 ( Firstly, we obtain the plasmon energy gap ω p ( = ω p (Q by substituting the real part of Eq.(7 upto order of Q in Eq.(6. The simplified form of Eq.(7 containing only the term proportional to Q can be written as χ(q, Ω = ( ] Ω Ω Λ Ω Q. ( FIG. 8: Plots of real part of dielectric function Re ε(, Ω]] vs ω for TCF, Weyl semimetal FEG. The Re ε(, Ω]] vanishes at plasmon frequencies Ω ( p (marked by small circles of the respective systems. They are peaked at ω = ω = for TCF Weyl semimetals respectively. Substituting the real part of the above expression in Eq.(6 gives C ( ω p ( Λ ( Λ ] ( ( ω p ( O(( ω p ( =. Considering ω p ( i.e ω p ( E F, we neglect the terms of the order of ( ω p ( higher in the above equation to get the plasmon gap as ω ( p = C. ( C Λ The plasmon gap depends on the cutoff Λ. For Λ =,.. In terms of E F, we have Ω ( p ω ( p = E F C. ( C Λ So, plasmon gap is linearly proportional to E F for large values of E F. The variation of Re ε(q, Ω] with ω is shown in Fig.(8 for TCF, Weyl semimetals FEG. The points marked by small circles are the plasmon energy gaps for the respective systems. The gaps show the following trend : ( ω p ( TCF < ( ω p ( Weyl < ( ω p ( FEG. Hence, for the same set of parameters, the plasmon gap of TCFs is smaller than that of (doped Weyl semimetal. The approximate plasmon dispersion in the long wavelength regime can be obtained by taking into account higher order terms of Eq.(7. The plasmon dispersion
7 TCF Weyl FEG TCF Weyl FEG Reσ(, Ω ] Ω FIG. 9: Plots of Re σ(q, Ω] vs ω for TCF, Weyl semimetals FEG. The divergence at Ω refers to the Drude weights of the respective systems. The optical absorption for TCF Weyl semimetals begin at ω = ω = respectively. upto the order of Q is ( where ω p = ω ( p ξ( ω ( p = 5 ξ( ω ( p C ( (C/ Λ Q, (5 ( ( ω p ( ( ω p (. (6 8 The plasmon mode can be traced numerically from the loss function which is defined as ] V (qim χ] Im = ε(q, ω ( V (qre χ] (V (qim χ]. (7 Figure ( shows the density plot of loss function for TCF. The plasmon mode can be spotted as the bright curve originating outside the PHC finally merging into it. The part of the plasmon mode outside the PHC is undamped while that inside the PHC gets damped into particlehole excitations, acquiring a finite lifetime. The zoomed version of the above plot is shown in Fig.(5, where the analytically obtained plasmon mode in Eq.(5 (labelled by dotted line is plotted alongside the numerically obtained mode for comparison. The agreement between the two solutions holds good for low Q as expected. C. OPTICAL CONDUCTIVITY The optical conductivities in the noninteracting interacting limits are related to the respective Lindhard FIG. : Plots of Re σ i (Q, Ω] vs ω for TCF, Weyl semimetals FEG. Electronelectron interaction induces sharp peaks in the optical conductivities, which correspond to the plasmon modes. The optical absorption edges for TCF Weyl semimetals begin at ω = ω = which is similar to the noninteracting case but with reduced intensities. functions as 55 σ(q, ω = iωe χ(q, ω (8 q σ i (q, ω = iωe q χi (q, ω (9 respectively. The real part of optical conductivity corresponds to dissipation/absorption of energy in the medium. Using Eq.( in Eqs.(8 (9, we get Re σ(q, Ω] = ω Q Im χ(q, Ω] ( Re σ i (Q, Ω] = ω Im χ(q, Ω]/Q ( C Re χ(q, Ω]/Q (C Im χ(q, Ω]/Q, where Im χ(q, Ω] = ( ] π ω δ( ω πθ( ω Q ( Re χ(q, Ω] = ( ω π δ ( ω Λ ω ω Q. ( Here, we have defined Re σ(q, ω] = Re σ(q, ω]/σ F with σ F = e gk F /(π. The variation of
8 8 Re σ(q, Ω] Re σ i (Q, Ω] with ω for TCFs, Weyl semimetals D FEG are plotted in Figs.(9 ( respectively. The zero frequency peak accounts for the intrab absorption is evident in all the three systems. The interb absorption edges of TCF Weyl semimetals commence at ω = E F ω = E F respectively the absorption grows linearly with frequency. For TCF, the absorption edge corresponds to the onset of flattoconduction absorption whereas for Weyl semimetals, it indicates valencetoconduction (or intercone absorption. The intercone absorption of TCF vanishes in the Q limit since χ vc (Q, Ω is of the order of Q for small Q see Eq.(], which makes σ vc (Q, Ω O ( Q. In the interacting limit, the zero frequency peak vanishes new peaks emerge at frequencies corresponding to the plasmon gaps. The magnitudes of interb absorption gets suppressed for both Weyl fermions TCF but the location of the absorption edges remain unaltered. IV. CONCLUSION We have explored the Lindhard function, PHC, loss function, plasmon mode optical conductivity of TCF compared the results with those of Weyl fermions D free electron gas. The flat b endows the response functions with several new features which were absent in Weyl semimetals. The PHC gets extended due to transitions between flat conduction bs which occur for frequencies above E F /. An approximate expression for low energy plasmon dispersion has been derived within RPA using small Q expansion of the Lindhard function. The dominant contributions to the Lindhard function are of the order of Q which represent intraconduction b flattoconduction transitions, while valencetoconduction transitions are of the order of Q. The plasmon frequency shows the usual dependence ω ω ω q as observed in most of the D electronic systems. The plasmon energy gap is proportional to E F for E F ω is a decreasing function of background dielectric constant. The plasmon energy gap is reduced as compared to Weyl semimetals for the same set of parameters no plasmon mode occurs as E F. We obtain the analytical expression of real part of optical conductivity in the Q limit for both nonteracting interacting cases. Unlike Weyl semimetals, the interb optical absorption for TCF begins at ω = E F the optical transitions between valence conduction bs are forbidden in the long wavelength limit. The rate of increase in optical absorption with frequency is higher in TCFs than Weyl semimetals. On incorporating electronelectron interactions, the energy absorption gets reduced in both the systems plasmon peaks show up at the plasmon energy gaps. ACKNOWLEDGEMENTS We would like to thank Sonu Verma for useful discussions. Appendix A: Theory of linear density response The Hamiltonian operator of an electron gas in low energy continuum model of a lattice (excluding electronelectron interactions is given by Ĥ = k,λ E λk c λk c λk, (A where c λk c λk are creation annihilation operators of the singleparticle states ψ λk λ(k k with energies E λk λ is the b index. The density operator ˆρ(r is given by ˆρ(r = ˆΨ (r ˆΨ(r. (A The field operators ˆΨ (r ˆΨ(r are generally expressed in terms of operators corresponding to momentumspin basis { ψ s,k } (i.e. { s k }, which gives ˆρ(r = ( e iq r c sk V c skq. (A q k,s For a threeb system, the Hamiltonian is diagonal in { ψ λk } basis hence it is convenient to exp ˆρ(r in operators corresponding to this basis. The basis transformation equations are given by c sk = λ s λ(k c λk, c sk = λ s λ(k c λk, (A where λ is summed over (,,. Using Eqs.(A (A, we get ˆρ(r = ( e iq r λ (k λ (k q c λ V c k λ kq. q k,λ,λ (A5 When the system is in thermodynamic equilibrium with a reservoir at temperature T, the equilibrium electron density ρ(r given by ρ(r ˆρ(r = N ˆρ(re βĥ N, Z {N} (A6 where Z = {N} N e βĥ N is the canonical partition function, β = (k B T the summation runs over all the Nparticle fermionic eigenstates of Ĥ. When the system is subjected to an external electric field E ext (r, t, a perturbation of the form ˆV (t = ˆρ(r φ ext (r, tdr Θ(t t (A7
9 9 gets added to the Hamiltonian Ĥ, where φ ext(r, t = e r E ext (r, t r dr is the electric potential t is the time when the field is switched on. The new Hamiltonian is Ĥ (t = Ĥ ˆV (t. (A8 The time evolution of the states are now governed by Ĥ (t, which drives the system out of equilibrium the electron density becomes a function of both space time in general. Considering magnitude of the perturbation very small compared to Ĥ, the nonequilibrium expectation value of density upto linear order in φ ext is given by the Kubo formula as 55 ˆρ(r = ˆρ(r dr dt χ(r, r, t, t φ ext (r, t t (A9 or, ρ ind (r, t = dr dt χ(r, r, t, t φ ext (r, t, (A t where ρ ind (r, t ˆρ(r ˆρ(r is the induced density χ(r, r, t t is the retarded densitydensity correlation function or polarizability given by χ(r, r, t, t = iθ(t t ˆρ I (r, t, ˆρ I (r, t ] /. (A Here,... denotes the expection value taken with respect to the equilibrium state ˆρ I (r, t is the density operator in the interaction picture, which is defined as ˆρ I (r, t = e iĥt/ ˆρ(re iĥt/. (A It can be seen that the polarizability is nonlocal in space retarded in time, i.e. the response at a particular point in space at a given instant of time is correlated to the value of external field at some other point in space at any previous instant of time. Moreover, ˆρ I (r, t, ˆρ I (r, t ] is always a function of (t t for translationally invariant systems, it is a function of r r. For such systems, χ(r, r, t, t χ(r r, t t hence ρ ind (r, t becomes the convolution of χ φ ext in both time space coordinates. By convolution theorem, we get where χ(q, ω = ρ ind (q, ω = χ(q, ωφ ext (q, ω, (A d(r r d(t t χ(r r, t t e iq (r r ω(t t ] (A φ ext (q, ω = dr dt φ ext (r, t e i(q r ωt (A5 are the Fourier transforms. On simplication, Eq.(A reduces to g χ(q, ω = lim η V F λ,λ (k, k q(f λ,k f λ,kq (ω iη E λ,k E k,λ,λ λ,kq (A6 This is called the Lindhard function. In (A6, g is the degeneracy factor, F λ,λ (k, k q = λ(k λ (k q is the overlap between the corresponding states f λ,k = e β(e λ,k E F ] is the FermiDirac distribution function. On incorporating electronelectron interactions, the Lindhard function obtained within Rom Phase Approximation (RPA is given by χ i (q, ω = χ(q, ω ε(q, ω,. (A7 where superscript i sts for interactions, χ(q, ω is the noninteracting Lindhard function given by Eq.(A6, ε(q, ω is the dielectric function which has the following form : ε(q, ω = V (qχ(q, ω. (A8 Here V (q = e /(ε r ε q is the Fourier transform of Coulomb potential energy between electrons in SI units in a medium of background dielectric constant ε r. The real spacetime dielectric function ε(r, t is the inverse Fourier transform of Eq.(A8 acts as a response function between φ ext φ total : φ ext (r, t = dr dt ε(r r, t t φ total (r, t. t (A9 The poles of the interacting Lindhard function in Eq.(A7 or the zeroes of the dielectric function in Eq.(A8 correspond to the collective modes of electron oscillations are known as plasmon modes. They can be damped or undamped depending on the values of Q Ω of the external perturbation. The undamped plasmon modes Ω p are obtained from the zeroes of Re ε(q, ω] in the region where Im χ(q, ω] vanishes. Appendix B: Alternative derivation of real part of optical conductivity In long wavelength limit (q, Re σ xx (ω] (excluding the zero frequency peak can be analytically derived from Kubo formula as Re σ xx (ω] = πge (π d ω λ,λ d d k(f λ (k f λ (k v λ λ x δ( E λλ ω, (B where E λλ = E λ (k E λ (k, d is the dimensionality, g is the degeneracy v λ λ x = ψ λ (k ˆv x ψ λ (k with
10 ˆv x = kx H/ being the xcomponent of velocity operator. For TCF, the above expression reduces to Re σ xx (ω] = ge 8π d k (f (k f (k vx δ( v F k ω ω ] (f (k f (k vx δ( v F k ω. (B For TCF, ˆv x = v F S x, vx = vx = vf ( cos φ cos φ cos θ/8. Using these results, Eq.(B gives Re σ xx (ω] = ge ω 6π v F Θ(ω v F k F, (B where Θ(x is the usual step function. Unlike Weyl semimetals, the absorption between the linearly dispersive bs is absent in TCF. This feature is also seen in dice lattice, where it was attributed to zero (modulo π Berry phase of the charge carriers 75. C. Herring, Phys. Rev. 5, 65 (97. X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov, Phys. Rev. B 8, 5 (. G. Xu, H. Weng, Z. Wang, X. Dai, Z. Fang, Phys. Rev. Lett. 7, 8686 (. A. Burkov, L. Balents, Phys. Rev. Lett. 7, 75 (. 5 D. Bulmash, C.X. Liu, X.L. Qi, Phys. Rev. B 89, 86 (. 6 S. Murakami, New J. Phys. 9, 56 (7. 7 G. B. Halász, L. Balents, Phys. Rev. B 85, 5 (. 8 S.M. Huang, et al., Nat. Commun. 6, 77 (5. 9 B. Lv, et al., Phys. Rev. X 5, (5. B. Lv, et al., Nat. Phys., 7 (5. H. Weng, C. Fang, Z. Fang, B. A. Bernevig, X. Dai, Phys. Rev. X 5, 9 (5. S.Y. Xu, et al., Science 9, 6 (5. A. Abrikosov, S. Beneslavskii, Sov. Phys. JETP, 699 (97. Z. Wang, Y. Sun, X.Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, Z. Fang, Phys. Rev. B 85, 95 (. 5 S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, A. M. Rappe, Phys. Rev. Lett. 8, 5 (. 6 S. Murakami, S. Iso, Y. Avishai, M. Onoda, N. Nagaosa, Phys. Rev. B 76, 5 (7. 7 Steinberg, J. A., S. M. Young, S. Zaheer, C. L. Kane, E. J. Mele, A. M. Rappe, Phys. Rev. Lett., 6 (. 8 S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Buechner, R. J. Cava, Phys. Rev. Lett., 76 (. 9 Z. Liu, et al., Nat. Mater., 677 (. Z. Liu, et al., Science, 86 (. M. Neupane, et al., Nat. Commun. 5, 786 (. S. L. Adler, Phys. Rev. 77, 6 (969. J. S. Bell, R. W. Jackiw, Nuovo Cimento 6, 7 (969. H. B. Nielsen, M. Ninomiya, Phys. Lett. B, 89 (98. 5 K.Y. Yang, Y.M. Lu, Y. Ran, Phys. Rev. B 8, 759 (. 6 H. Weyl, Proc. Natl. Acad. Sci. U.S.A. 5, (99. 7 P. A. M. Dirac, Proc. R. Soc. A 7, 6 (98. 8 B. Bradlyn, J. Cano, Z. Wang, M. Vergniory, C. Felser, R. Cava, B. A. Bernevig, Science 5, aaf57 (6. 9 A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, B. A. Bernevig, Nature (London 57, 95 (5. Y. Xu, F. Zhang, C. Zhang, Phys. Rev. Lett. 5, 65 (5. B. J. Wieder, Y. Kim, A. M. Rappe, C. L. Kane, Phys. Rev. Lett. 6, 86 (6. G. Chang, S.Y. Xu, S.M. Huang, D. S. Sanchez, C.H. Hsu, G. Bian, Z.M. Yu, I. Belopolski, N. Alidoust, H. Zheng, T.R. Chang, H.T. Jeng, S. A. Yang, T. Neupert, H. Lin, M. Z. Hasan, Scientific Reports 7, 688 (7. I. C. Fulga A. Stern, Phys. Rev. B 95, 6 (7. H. Weng, C. Fang, Z. Fang, X. Dai, Phys. Rev. B 9, 65 (6. 5 H. Weng, C. Fang, Z. Fang, X. Dai, Phys. Rev. B 9, (6. 6 C.H. Cheung, R. C. Xiao,M.C. Hsu, H.R. Fuh, Y.C. Lin, C.R. Chang, arxiv: J. Li, Q. Xie, S. Ullah, R. Li, H. Ma, D. Li, Y. Li, X.Q. Chen, Phys. Rev. B 97, 55 (8. 8 B. Q. Lv, Z.L. Feng, Q.N. Xu, X. Gao, J.Z. Ma, L.Y. Kong, P. Richard, Y.B. Huang, V. N. Strocov, C. Fang, H.M. Weng, Y.G. Shi, T. Qian, H. Ding, Nature (London 56, 67 (7. 9 J. B. He, D. Chen, W. L. Zhu, S. Zhang, L. X. Zhao, Z. A. Ren, G. F. Chen, Phys. Rev. B 95, 9565 (7. G. Chang, S.Y. Xu, B. J. Wieder, D. S. Sanchez, S.M. Huang, I. Belopolski, T.R. Chang, S. Zhang, A. Bansil, H. Lin, M. Z. Hasan, Phys. Rev. Lett. 9, 6 (7. P. Tang, Q. Zhou, S.C. Zhang, Phys. Rev. Lett. 9, 6 (7. Z. Zhu, G. W. Winkler, Q. S. Wu, J. Li, A. A. Soluyanov, Phys. Rev. X 6, (6. S. Ny, S. Manna, D. Calugaru, B. Roy, Phys. Rev. B, 5 (9. B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, B. A. Bernevig, Topoical quantum chemistry, Nature (London 57, 98 (7. 5 G. Chang, B. J. Wieder, F. Schindler, D. S. Sanchez, I. Belopolski, S.M. Huang, B. Singh, D. Wu, T.R. Chang, T. Neupert, S.Y. Xu, H. Lin, M. Z. Hasan, Nat. Mater. 7, 978 (8. 6 R. Kubo, Journal of the Physical Society of Japan,, 57 ( M. G. Papadopoulos, A. J. Sadlej, J Leszczynski, Nonlinear optical properties of matter, Springer (6.
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