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

Bashab Dey and Tarun Kanti Ghosh

Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India We investigate the nature of density response of linear triple component fermions by computing the Lindhard function, dielectric function, plasmon mode and long wavelength optical conductivity of the system and compare the results with those of Weyl fermions and three dimensional free electron gas. Linear triple component fermions are the low energy quasiparticles of linear triple component semimetals, consisting of linearly dispersive and dispersionless (flat band) excitations.

The presence of flat band brings about notable modifications in the response properties with respect to Weyl fermions such as induction of a new region in the particle-hole continuum, reduced plasmon energy gap, shift in absorption edge, enhanced rate of increase in energy absorption with frequency and forbidden intercone transitions in the long wavelength limit. The plasmon dispersion follows the usual ω ∼ ω0+ ω1q² 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^1–12 and Dirac^6,13–21 semimetals have become breeding grounds for plethora of intriguing physical phenomena such as Fermi arc surface states², chiral anomaly^22–24, anoma- lous Hall effect²⁵ etc. The quasiparticles close to the band-crossing nodes act as condensed-matter versions of Weyl²⁶ and massless Dirac²⁷ fermions theorized in high-energy physics. Recent studies have unveiled other classes of topological semimetals where more than two bands cross at a node and exhibit fermionic excitations with no counterpart in high energy physics^28–31. It is speculated that mirror and discrete rotational sym- metries in symmorphic crystals may lead to topologi- cally protected three-fold degenerate crossing points^32,33. First-principles calculations^34–37 have shown that ma- terials such as TaN, MoP, WC, RhSi, RhGe and ZrTe can host three-band crossings in the neighborhood of the Fermi level^35,38–42. In this paper, we deal with one such class of semimetals with three-band crossings, where quasiparticles around the nodes transform under pseudospin-1 representation. These are called triple- component semimetals⁴³ (TCSs) and their low energy excitations are called triple component fermions (TCFs).

The pseudospin degrees of freedom may emerge from spe- cific admixtures of orbital and spin projections^28,44,45.

The dynamics of the TCFs are governed by the Hamil- tonian H(k) = d(k) · S, where S = (Sx, Sy, Sz) de- note the usual spin-1 matrices and d(k) is a vector func- tion of k. The band structure consists of two dispersive bands and a flat band. The TCFs can be grouped into linear, quadratic and cubic, depending on the form of d(k). For linear TCFs, the energy scales linearly with all the three components of momentum. For quadratic and cubic TCFs, the energy scales linearly with kz, but as k_⊥² and k³_⊥ respectively in the k_x-k_y plane, where k_⊥=q

k²_x+ k_y². Time-reversal symmetric TCFs arise in materials with space group symmetry 199 and 214, e.g.

Pd3Bi2S2and Ag2Se2Au²⁸. Material realizations of time- reversal symmetry (TRS)-breaking TCFs is still absent but are predicted to be found in magnetically ordered systems. In this work, we analyse the density and cur- rent response functions for linear TCFs in non-interacting and interacting limits.

Linear Response: When a system is subjected to a time and space dependent electric field (or potential), the field couples with the charge degree of freedom of elec- trons and drives the system into a non-equilibrium phase.

The system responds by modifying its charge density and induction of charge currents. If the strength of the ex- ternal field is small enough to be treated as a perturba- tion, the response function is obtained from the Kubo formula⁴⁶. For stronger fields, the system shows nonlin- ear response⁴⁷ and may give rise to Floquet bands^48–50. The Kubo formula usually holds good for the typical am- plitudes of electric fields used in the experiments. It is used to obtain current and density responses of the sys- tem, which are given by its conductivity and polarizabil- ity respectively. The polarizability relates the induced density fluctuation of the electron gas to the external po- tential, while the conductivity relates the induced current to the external field. The conductivity can be calculated from the polarizability and vice-versa.

The polarizability function in momentum-frequency space is called the Lindhard function⁵¹. 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 and frequency, which is attributed to intraband or interband particle-hole excitations across the Fermi sea. This region in the momentum-frequency space is called the particle-hole continuum (PHC). At T → 0, energy is not absorbed from the field for fre- quencies and momenta outside the PHC. The shape of the PHC depends on the the chemical potential, band structure and overlap factor between the bands.

On inclusion of Coulomb interaction between the elec- trons, the Lindhard function gets renormalized by the dielectric function within Random Phase Approxima- tion (RPA)^52–56. The renormalization gives rise to plas-

arXiv:2109.05195v1 [cond-mat.mes-hall] 11 Sep 2021

mon modes⁵³ which are perceived as collective oscilla- tions 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 and infinitesimal amounts of energy respec- tively to be excited at long wavelengths (q → 0). The curve in momentum-frequency space along which the di- electric function vanishes is called the plasmon disper- sion. Plasmons can be excited at wavelengths and fre- quencies given by the plasmon dispersion. They are probed using inelastic scattering experiments such as electron energy loss spectroscopy⁵⁹. The dimensional- ity and band structure of the system play crucial roles in determining the plasmon dispersion. It has been studied that 2D electron systems such as 2D free elec- tron gas (FEG)⁵⁸, graphene^60–66 and dice lattice⁶⁷(2D pseudospin-1 system) host a gapless plasmon mode with dispersion ∼ √

q at long wavelengths. In contrast, 3D FEG⁶⁸, 3D noncentrosymmetric metals⁶⁹ and doped Weyl^70,71and Dirac semimetals^72–74exhibit gapped plas- mon modes dispersing as ∼ ω₀+ ω₁q² in the long wave- length limit, where ω0 and ω1 are constants with ap- propriate dimensions. However, the study of response functions and plasmons in TCSs is still unexplored. We fill this gap in the research by making a comprehensive analysis of the Lindhard function, PHC, dielectric func- tion and optical conductivity for linear isotropic TCFs and compare the results with those of Weyl fermions and 3D FEG. Furthermore, we derive approximate analyti- cal expression of Lindhard function and plasmon disper- sion in the long wavelength limit. We investigate the interplay of three bands and the effect of flat band in particular in the response functions. For the rest of the paper, ‘TCF’ and ‘Weyl fermions/semimetals’ would re- fer to linear isotropic TCF and isotropic type-I Weyl fermions/semimetals repectively.

This paper is organized as follows. In Sec. (II), we review the low energy band structure and eigenstates of TCF. In Sec. (III A), we obtain the Lindhard function and PHC of doped linear TCS. The calculation of di- electric function and plasmon modes of the system are shown in Sec. (III B). A discussion on optical conductiv- ity is presented in Sec. (III C). Finally, the results are summarized in Sec. (IV).

II. MODEL HAMILTONIAN

The Hamiltonian of TCFs around a band touching node is given by

H(k) = ~v^FS · k. (1)

Here, vF is the Fermi velocity and S = (Sx, Sy, Sz) de- notes the usual spin-1 matrices. The band structure

comprises of three bands viz. Ek+ = ~v^Fk (conduction band), Ek− = −~v^Fk (valence band) and Ek0 = 0 (flat band). Denoting the pseudospin basis states {|si} as

| ↑i = (1 0 0)^T, |0i = (0 1 0)^T and | ↓i = (0 0 1)^T where T stands for transpose, the single-particle eigen- states {|λ(k)i} are given by

| + (k)i =





cos^{2 θ}₂

sin θ√ 2e^iφ sin^{2 θ}₂e^2iφ



, | − (k)i =





sin^{2 θ}₂

−^{sin θ√}

2e^iφ cos^{2 θ}₂e^2iφ



 (2) and

| 0 (k)i =







−^{sin θ√}

2

cos θe^iφ

sin θ√ 2e^2iφ





. (3)

III. RESPONSE FUNCTIONS OF TCFS A. LINDHARD FUNCTION

A brief review of the theory of linear density response for a multi-band system is presented in Appendix(A).

The dynamical polarization function or the Lindhard function (A16) of a non-interacting system of electrons is given by

χ(q, ω) = lim

η→0

g V

X

k,λ,λ⁰

Fλ,λ⁰(k, k + q)(fλ,k− fλ⁰,k+q)

~(ω + iη) + E^λ,k− Eλ⁰,k+q

, (4) where g is the degeneracy factor, F_λ,λ⁰(k, k + q) =

|hλ(k)|λ⁰(k + q)i|² is the overlap between the corre- sponding states and fλ,k = [e^{β(Eλ,k−EF)}+ 1]⁻¹ is the Fermi-Dirac distribution function.

For TCF, the interband and intraband overlaps be- tween the dispersive bands is given by

Fλ,λ⁰(k, k + q) = 1 4



1 + λλ⁰k · (k + q)

|k||k + q|

2

, λ, λ⁰= ±1 (5) and that between the flat and dispersive bands is

F0,λ(k, k + q) = Fλ,0(k, k + q) = 1 2

"

1 − k · (k + q)

|k||k + q|

2# . (6) At T → 0 K, the Lindhard function (4) takes the follow- ing form for E_F > 0 (i.e. doped TCS):

χ(q, ω) = χ⁽⁺⁾(q, ω) + χ⁽⁰⁾(q, ω) + χ⁽⁻⁾(q, ω), (7) where

χ⁽⁺⁾(q, ω) = lim

η→0

g V

X

k

 F+,+(k, k + q)(f+,k− f+,k+q)

~ω + iη + E+,k− E_+,k+q + F+,0(k, k + q)f+,k

~ω + iη + E+,k− E_0,k+q − F0,+(k, k + q)f+,k+q

~ω + iη + E0,k− E_+,k+q + F+,−(k, k + q)f+,k

~ω + iη + E+,k− E−,k+q

− F_−,+(k, k + q)f+,k+q

~ω + iη + E−,k− E+,k+q

 ,

(8)

χ⁽⁰⁾(q, ω) = lim

η→0

g V

X

k

 F0,+(k, k + q)f0,k

~ω + iη + E0,k− E+,k+q

− F+,0(k, k + q)f0,k+q

~ω + iη + E+,k− E0,k+q



(9)

and

χ⁽⁻⁾(q, ω) = lim

η→0

g V

X

k

 F_−,+(k, k + q)f_−,k

~ω + iη + E−,k− E+,k+q

− F_+,−(k, k + q)f_−,k+q

~ω + iη + E+,k− E_−,k+q



. (10)

Here, we have excluded the terms which represent the intraband transitions within flat and valence bands and the interband transitions between them. This is true only for E_F > 0.

On non-dimensionalizing the quantities as x = k/kF, Q = q/kF, Ω = limη→0~(ω + iη)/EF = limη→0(˜ω +

i~η/E^F), ˜ω = ~ω/E^F and ˜χ^(λ)(Q, Ω) = χ^(λ)(q, ω)/χF

(where EF = ~v^FkF and χF = gk_F²/(4π²~vF)) and con- verting the summation into continuous integrals, equa- tions (8),(9) and (10) simplify as

˜

χ⁽⁺⁾(Q, Ω) = Z 1

0

x(Ω + x)

4Q log Ω²+ 2Ωx − Q²+ 2xQ Ω²+ 2Ωx − Q²− 2xQ

 dx

+ Z 1

0

(Ω + x)h

−4Qx(Ω²+ 2Ωx + x²) − (Q²− x²)²log_(Q−x)2

(Q+x)²

+ (−Q²+ Ω²+ 2Ωx + 2x²)²log_Ω2+2Ωx−Q²+2xQ Ω²+2Ωx−Q²−2xQ

i

16Qx(Ω²+ 2Ωx + x²) dx

+ Z 1

0



−x − Q²− Ω²− 2Ωx − 2x²

4Q log Ω²+ 2Ωx − Q²+ 2xQ Ω²+ 2Ωx − Q²− 2xQ



dx

+ Z 1

0

Q² 2(Ω + x)

 Q²+ x²

2Q² +(Q²− x²)²

8Q³x log (Q − x)² (Q + x)²



dx + Ω ↔ −Ω
,

(11)

˜

χ⁽⁰⁾(Q, Ω) = Z Q

0

1 8xQ

 2

3(3Q²x + x³) + 2x(−2Q²+ Ω²− 2x²) + 2QxΩ −(Q − x)²(Q + x)²

Ω log Q + x Q − x



(Q − Ω − x)(Q + Ω − x)(Q − Ω + x)(Q + Ω + x)

Ω log x + Q − Ω

Q − x − Ω

  dx

+ Z Λ

Q

1 8xQ

 2

3(3x²Q + Q³) + 2Q(−2Q²+ Ω²− 2x²) + 2QxΩ −(Q − x)²(Q + x)²

Ω log x + Q x − Q



(Q − Ω − x)(Q + Ω − x)(Q − Ω + x)(Q + Ω + x)

Ω log x + Q − Ω

x − Q − Ω

 

dx + Ω ↔ −Ω

(12)

and

˜

χ⁽⁻⁾(Q, Ω) = Z Q

0

1 16xQ



2x(2Q²− Ω²+ 6Ωx − 11x²) − 2Qx(Ω − 5x) −2

3(3Q²x + x³) −(Q − x)²(Q + x)²

x − Ω log x + Q Q − x



−(Q²− (Ω − 2x)²)²

Ω − x log 2x + Q − Ω Q − x

  dx

+ Z Λ

Q

1 16xQ



2Q(2Q²− Ω²+ 6Ωx − 11x²) − 2Qx(Ω − 5x) −2

3(3x²Q + Q³) −(Q − x)²(Q + x)²

x − Ω log x + Q x − Q



−(Q²− (Ω − 2x)²)²

Ω − x log 2x + Q − Ω 2x − Q − Ω

 

dx + Ω ↔ −Ω
.

(13)

3

2

1

0 1 2 3

FIG. 1: Different regions of PHC for TCF. The dotted, vi- olet and red regions indicates flat to conduction, valence to conduction and intra-conduction-band transitions respec- tively for EF > 0.

We restrict the limits of integration in Eqs. (12) and (13) to an ultraviolet cutoff Λ = k_c/k_F  1. Thus, the dimensionless form of Lindhard function is

˜

χ(Q, Ω) = ˜χ⁽⁺⁾(Q, Ω) + ˜χ⁽⁰⁾(Q, Ω) + ˜χ⁽⁻⁾(Q, Ω). (14) A diagram of the PHC for doped TCS (EF > 0) is shown in Fig.[(1)]. Like Weyl semimetals, the PHC for intraband transitions within the conduction band is bounded by ˜ω = Q, ˜ω = 0 and ˜ω = Q − 2 lines, while the interband transitions between valence and conduc- tion bands occur in the region bounded by ˜ω = Q and

˜

ω = −Q + 2 lines. The flat band introduces a new region of PHC which is absent in Weyl semimetals. The PHC for interband transitions between the flat and conduction bands is above ˜ω = 1 line. So, the flat-to-conduction PHC overlaps those of intercone and intracone ones.

These features were observed in dice lattice also⁶⁷. The

FIG. 2: Density plot of the natural logarithm of Im [ ˜χ(Q, Ω)]

as functions of Q and Ω for TCF.

numerical plot of natural logarithm of Im [ ˜χ(Q, Ω)] as functions of Q and ˜ω (shown in Fig.[2]) reveals the char- acteristics of the PHC reasonably well albeit the sharp demarcations of different regions of absorption.

The static Lindhard function Re [ ˜χ(Q, 0)] as a function of Q is plotted in Fig.(3) for TCF, Weyl semimetals and 3D FEG. The function rises monotonically with Q for both TCF and 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 = 2.

B. DIELECTRIC FUNCTION AND PLASMONS For TCF, the dielectric function (A18) can be written as

ε(Q, Ω) = 1 − C

Q²χ(Q, Ω),˜ (15)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0

2 4 6 8 10 12

14 Weyl

TCF FEG

Re

FIG. 3: Plots of Re[ ˜χ(Q, 0)] vs Q for TCF, weyl semimetal and free electron gas(FEG). The Re[ ˜χ(Q, 0)] increases mono- tonically with Q for TCF and Weyl semimetals, but a de- creasing function of Q for free electron gas. Also, magnitude of Re[ ˜χ(Q, 0)] for TCF is greater than that of Weyl semimetal for the same set of parameters.

FIG. 4: Density plot of the natural logarithm of loss function (27) as a function of q/kF and ~ω/EF. 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 particle-hole excitations.

where C = e²g/(4εrε0π²~vF). For vF = 4 × 10⁵ m/s, εr = 10 and g = 2, we get C ≈ 0.347. The undamped plasmon modes ˜ωp for TCF can be obtained by solving the following equation for Ω and Q:

1 − C

Q²Re [ ˜χ(Q, Ωp)] = 0. (16)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

FIG. 5: Comparison of analytical solution of plasmon mode (dotted curve) for long wavelength (Q  1) regime given by Eq.(25) and numerically obtained plasmon mode in the loss function plot. The agreement is good for low Q as expected.

FIG. 6: Density plot of the natural logarithm of loss function as functions of εr and Ω for TCF for very small wavelengths Q  1 . The plasmon mode (bright yellow curve) remains undamped and and its frequency decreases with r.

Since the exact solution of the Eq.(16) cannot be ob- tained analytically, we deduce an approximate expression of long wavelength (Q  1) and low frequency (˜ω  1) plasmon mode of this system using the expansion of (14) in orders of Q. The Lindhard function for small Q can

(eV)

(eV)

FIG. 7: Density plot of the natural logarithm of loss function as functions of EF and Ω for TCF for very small wavelengths Q  1 . The plasmon mode (bright yellow curve) remains undamped and and its frequency increases with EF.

be written as

˜

χ(Q, Ω) = ˜χcc(Q, Ω) + ˜χf c(Q, Ω) + ˜χvc(Q, Ω), (17) where ˜χ_cc(Q, Ω), ˜χ_{f c}(Q, Ω) and ˜χ_vc(Q, Ω) are intra- conduction band, flat-to-conduction and valence-to- conduction (intercone) contributions respectively, given by

˜

χ_cc(Q, Ω) =

 2

3Ω²Q²+ 2

5Ω⁴Q⁴+ O(Q⁶)



, (18)

˜

χf c(Q, Ω) = Z 1

0

 −4x

3(Ω²− x²)Q²+ 4

15x(Ω²− x²)Q⁴

 dx

+ Z Λ

0

 (4x)

3(Ω²− x²)Q²+4x²(Ω⁴− 5Ω²x²) 15(−Ω²x + x³)³ Q⁴

 dx,

(19) and

χ˜_vc(Q, Ω) =

 Z 1 0

4x²

−15Ω²x³+ 60x⁵+ Z Λ

0

4x² 15Ω²x³− 60x⁵

 dx Q⁴.

(20)

Firstly, we obtain the plasmon energy gap ˜ω⁽⁰⁾p =

˜

ωp(Q → 0) by substituting the real part of Eq.(17) upto order of Q² in Eq.(16). The simplified form of Eq.(17) containing only the term proportional to Q²can be writ- ten as

˜

χ(Q², Ω) = 2 3

 1 Ω² + log

 1 − Ω² Λ²− Ω²



Q². (21)

0.0 0.5 1.0 1.5 2.0

4 3 2 1 0 1 2 3 4

TCF Weyl FEG

FIG. 8: Plots of real part of dielectric function Re [ε(0, Ω)]]

vs ˜ω for TCF, Weyl semimetal and FEG. The Re [ε(0, Ω)]]

vanishes at plasmon frequencies Ω⁽⁰⁾p (marked by small circles) of the respective systems. They are peaked at ˜ω = 1 and

˜

ω = 2 for TCF and Weyl semimetals respectively.

Substituting the real part of the above expression in Eq.(16) gives

1 −2 3C

 1

(˜ωp⁽⁰⁾)² + log

1 Λ²

+



−1 + 1 Λ²



(˜ω⁽⁰⁾_p )²+ O((˜ω⁽⁰⁾_p )³)



= 0.

(22)

Considering ˜ω⁽⁰⁾p  1 i.e ~ωp⁽⁰⁾  E_F, we neglect the terms of the order of (˜ωp⁽⁰⁾)² and higher in the above equation to get the plasmon gap as

˜ ω_p⁽⁰⁾=

s ₂

3C

1 + ²₃C log Λ². (23) The plasmon gap depends on the cut-off Λ. For Λ = 10, Ω⁽⁰⁾p ≈ 0.33. In terms of EF, we have

ω⁽⁰⁾_p =EF

~

s ₂

3C

1 +²₃C log Λ². (24) So, plasmon gap is linearly proportional to EF for large values of E_F. The variation of Re [ε(Q → 0, Ω)] with ˜ω is shown in Fig.(8) for TCF, Weyl semimetals and FEG.

The points marked by small circles are the plasmon en- ergy gaps for the respective systems. The gaps show the following trend : (˜ω⁽⁰⁾p )TCF < (˜ωp⁽⁰⁾)Weyl < (˜ω⁽⁰⁾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 wave- length regime can be obtained by taking into account higher order terms of Eq.(17). The plasmon dispersion

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0

1 2 3 4 5 6

TCF Weyl FEG

FIG. 9: Plots of Re [˜σ(Q → 0, Ω)] vs ˜ω for TCF, Weyl semimetals and FEG. The divergence at Ω → 0 refers to the Drude weights of the respective systems. The optical absorp- tion for TCF and Weyl semimetals begin at ˜ω = 1 and ˜ω = 2 respectively.

upto the order of Q²is

˜

ωp = ˜ω⁽⁰⁾_p 1 + ξ(˜ωp⁽⁰⁾)C

2 (1 + (2C/3) log Λ²)Q²

!

, (25)

where

ξ(˜ω⁽⁰⁾_p ) = 4 15

3

2(˜ωp⁽⁰⁾)⁴ − 1 2(˜ω⁽⁰⁾p )²

+3 8

!

. (26)

The plasmon mode can be traced numerically from the loss function which is defined as

−Im

 1

ε(q, ω)



= V (q)Im [χ]

(1 − V (q)Re [χ])²+ (V (q)Im [χ])² . (27) Figure (4) shows the density plot of loss function for TCF. The plasmon mode can be spotted as the bright curve originating outside the PHC and finally merging into it. The part of the plasmon mode outside the PHC is undamped while that inside the PHC gets damped into particle-hole 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.(25) (labelled by dotted line) is plotted alongside the numeri- cally obtained mode for comparison. The agreement be- tween the two solutions holds good for low Q as expected.

C. OPTICAL CONDUCTIVITY

The optical conductivities in the non-interacting and interacting limits are related to the respective Lindhard

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Ω

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Re[σ(0,)]Ω

TCF Weyl FEG

FIG. 10: Plots of Re [˜σⁱ(Q → 0, Ω)] vs ˜ω for TCF, Weyl semimetals and FEG. Electron-electron interaction induces sharp peaks in the optical conductivities, which correspond to the plasmon modes. The optical absorption edges for TCF and Weyl semimetals begin at ˜ω = 1 and ˜ω = 2 which is sim- ilar to the non-interacting case but with reduced intensities.

functions as⁵⁵

σ(q, ω) = iωe²

q² χ(q, ω) (28)

and

σⁱ(q, ω) = iωe²

q² χⁱ(q, ω) (29) respectively. The real part of optical conductivity corresponds to dissipation/absorption of energy in the medium. Using Eq.(21) in Eqs.(28) and (29), we get

Re[˜σ(Q → 0, Ω)] = − ω˜

Q²Im [ ˜χ(Q², Ω)] (30) and

Re [˜σⁱ(Q → 0, Ω)] =

−˜ω Im [ ˜χ(Q², Ω)]/Q²

(1 − C Re [ ˜χ(Q², Ω)]/Q²)²+ (C Im [ ˜χ(Q², Ω)]/Q²)², (31) where

Im [ ˜χ(Q², Ω)] = −2 3

 2π

ω˜ δ(˜ω) + πΘ(˜ω²− 1)



Q² (32) and

Re [ ˜χ(Q², Ω)] = 2 3

 1

˜

ω² − π²δ²(˜ω) − log    

Λ²− ˜ω² 1 − ˜ω²

 Q². (33) Here, we have defined Re [˜σ(Q → 0, ω)] = Re [σ(q → 0, ω)]/σF with σF = e²gkF/(4π²~). The variation of

Re [˜σ(Q → 0, Ω)] and Re [˜σⁱ(Q → 0, Ω)] with ˜ω for TCFs, Weyl semimetals and 3D FEG are plotted in Figs.(9) and (10) respectively. The zero frequency peak accounts for the intraband absorption and is evident in all the three systems. The interband absorption edges of TCF and Weyl semimetals commence at ~ω = E^F and ~ω = 2EF respectively and the absorption grows linearly with frequency. For TCF, the absorption edge corresponds to the onset of flat-to-conduction absorption whereas for Weyl semimetals, it indicates valence-to-conduction (or intercone) absorption. The intercone absorption of TCF vanishes in the Q → 0 limit since ˜χvc(Q, Ω) is of the order of Q⁴for small Q [see Eq.(20)], which makes ˜σvc(Q, Ω) ∼ O⁽Q²).

In the interacting limit, the zero frequency peak van- ishes and new peaks emerge at frequencies correspond- ing to the plasmon gaps. The magnitudes of interband absorption gets suppressed for both Weyl fermions and TCF but the location of the absorption edges remain un- altered.

IV. CONCLUSION

We have explored the Lindhard function, PHC, loss function, plasmon mode and optical conductivity of TCF and compared the results with those of Weyl fermions and 3D free electron gas. The flat band endows the re- sponse functions with several new features which were absent in Weyl semimetals. The PHC gets extended due to transitions between flat and conduction bands which occur for frequencies above EF/~. An approxi- mate 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 Q²which represent intra-conduction band and flat-to-conduction transitions, while valence-to-conduction transitions are of the order of Q⁴. The plasmon frequency shows the usual dependence ω ∼ ω0+ ω1q² as observed in most of the 3D electronic systems. The plasmon energy gap is proportional to EF

for E_F  ~ω and is a decreasing function of background dielectric constant. The plasmon energy gap is reduced as compared to Weyl semimetals for the same set of pa- rameters and no plasmon mode occurs as EF → 0. We obtain the analytical expression of real part of optical conductivity in the Q → 0 limit for both nonteracting and interacting cases. Unlike Weyl semimetals, the inter- band optical absorption for TCF begins at ~ω = E^F and the optical transitions between valence and conduction bands 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 electron-electron interactions, the energy absorption gets reduced in both the systems and plasmon peaks show up at the plasmon energy gaps.

ACKNOWLEDGEMENTS

We would like to thank Sonu Verma for useful discus- sions.

Appendix A: Theory of linear density response The Hamiltonian operator of an electron gas in low energy continuum model of a lattice (excluding electron- electron interactions) is given by

H =ˆ X

k,λ

Eλkc^†_λkcλk, (A1)

where c^†_λk and c_λk are creation and annihilation opera- tors of the single-particle states |ψλki ≡ |λ(k)i|ki with energies Eλkand λ is the band index. The density oper- ator ˆρ(r) is given by

ˆ

ρ(r) = ˆΨ^†(r) ˆΨ(r). (A2) The field operators Ψˆ^†(r) and Ψ(r) are generallyˆ expressed in terms of operators corresponding to momentum-spin basis {|ψs,ki} (i.e. {|si|ki}), which gives

ˆ ρ(r) = 1

V X

q

e^iq·r

 X

k,s

c^†_skc_sk+q



. (A3)

For a three-band system, the Hamiltonian is diagonal in {|ψλki} basis and hence it is convenient to expand ˆ

ρ(r) in operators corresponding to this basis. The basis transformation equations are given by

csk=X

λ

hs|λ(k)icλk, c^†_sk=X

λ

hs|λ(k)i^∗c^†_λk, (A4)

where λ is summed over (−1, 0, 1). Using Eqs.(A3) and (A4), we get

ˆ ρ(r) = 1

V X

q

e^iq·r

 X

k,λ₁,λ₂

hλ1(k)|λ2(k + q)ic^†_λ

1kcλ₂k+q

 . (A5) When the system is in thermodynamic equilibrium with a reservoir at temperature T , the equilibrium electron density ρ(r) given by

ρ(r) ≡ h ˆρ(r)i0= 1 Z0

X

{N }

hN |ˆρ(r)e^{−β ˆH}|N i, (A6)

where Z0=P

{N }hN |e^{−β ˆH}|N i is the canonical partition function, β = (kBT )⁻¹ and the summation runs over all the N -particle fermionic eigenstates of ˆH. When the system is subjected to an external electric field E_ext(r, t), a perturbation of the form

V (t) =ˆ Z

ˆ

ρ(r⁰)φext(r⁰, t)dr⁰Θ(t − t0) (A7)

ext

−eRr⁰

Eext(r, t)·r dr is the electric potential and t0is the time when the field is switched on. The new Hamiltonian is

Hˆ⁰(t) = ˆH + ˆV (t). (A8) The time evolution of the states are now governed by Hˆ⁰(t), which drives the system out of equilibrium and the electron density becomes a function of both space and time in general. Considering magnitude of the perturba- tion very small compared to h ˆHi0, the nonequilibrium expectation value of density upto linear order in φ_ext is given by the Kubo formula as⁵⁵

hˆρ(r)i = h ˆρ(r)i₀+ Z

dr⁰ Z ∞

t₀

dt⁰χ(r, r⁰, t, t⁰)φ_ext(r⁰, t⁰) (A9) or,

ρ_ind(r, t) = Z

dr⁰ Z ∞

t₀

dt⁰χ(r, r⁰, t, t⁰)φ_ext(r⁰, t⁰), (A10)

where ρ_ind(r, t) ≡ h ˆρ(r)i − h ˆρ(r)i₀ is the induced density and χ(r, r⁰, t − t⁰) is the retarded density-density correla- tion function or polarizability given by

χ(r, r⁰, t, t⁰) = −iΘ(t − t⁰)h[ ˆρ_I(r, t), ˆρ_I(r⁰, t⁰)]i₀/~. (A11) Here, h...i0 denotes the expection value taken with re- spect to the equilibrium state and ˆρI(r, t) is the density operator in the interaction picture, which is defined as

ˆ

ρI(r, t) = e^{i ˆHt/~}ρ(r)eˆ ^{−i ˆHt/~}. (A12) It can be seen that the polarizability is non-local in space and retarded in time, i.e. the response at a particu- lar point in space at a given instant of time is corre- lated to the value of external field at some other point in space at any previous instant of time. Moreover, h[ ˆρ_I(r, t), ˆρ_I(r⁰, t⁰)]i₀ is always a function of (t − t⁰) and for translationally invariant systems, it is a function of r − r⁰. For such systems, χ(r, r⁰, t, t⁰) ≡ χ(r − r⁰, t − t⁰) and hence ρind(r, t) becomes the convolution of χ and φ_ext in both time and space coordinates. By convolution theorem, we get

ρind(q, ω) = χ(q, ω)φext(q, ω), (A13) where

χ(q, ω) = Z

d(r − r⁰) Z ∞

−∞

d(t − t⁰)χ(r − r⁰, t − t⁰)×

e^{−i[q·(r−r0)−ω(t−t0)]}

(A14) and

φext(q, ω) = Z

dr⁰ Z ∞

−∞

dt⁰φext(r⁰, t⁰)e^{−i(q·r0−ωt0)} (A15)

reduces to χ(q, ω) = lim

η→0

g V

X

k,λ,λ⁰

F_λ,λ⁰(k, k + q)(f_λ,k− f_λ⁰_,k+q)

~(ω + iη) + Eλ,k− Eλ⁰,k+q

. (A16) This is called the Lindhard function. In (A16), g is the degeneracy factor, Fλ,λ⁰(k, k + q) = |hλ(k)|λ⁰(k + q)i|² is the overlap between the corresponding states and f_λ,k = [e^{β(Eλ,k−EF)}+ 1]⁻¹ is the Fermi-Dirac distribu- tion function.

On incorporating electron-electron interactions, the Lindhard function obtained within Random Phase Ap- proximation (RPA) is given by

χⁱ(q, ω) = χ(q, ω)

ε(q, ω), (A17)

where superscript i stands for ‘interactions’, χ(q, ω) is the non-interacting Lindhard function given by Eq.(A16), and ε(q, ω) is the dielectric function which has the fol- lowing form :

ε(q, ω) = 1 − V (q)χ(q, ω). (A18) Here V (q) = e²/(εrε₀q²) is the Fourier transform of Coulomb potential energy between electrons in SI units in a medium of background dielectric constant εr. The real space-time dielectric function ε(r, t) is the inverse Fourier transform of Eq.(A18) and acts as a response function between φext and φtotal:

φ_ext(r, t) = Z

dr⁰ Z ∞

t₀

dt⁰ε(r − r⁰, t − t⁰)φ_total(r⁰, t⁰).

(A19) The poles of the interacting Lindhard function in Eq.(A17) or the zeroes of the dielectric function in Eq.(A18) correspond to the collective modes of electron oscillations and are known as plasmon modes. They can be damped or undamped depending on the values of Q and Ω of the external perturbation. The undamped plasmon modes Ωp 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 → 0), Re [σ_xx(ω)] (exclud- ing the zero frequency peak) can be analytically derived from Kubo formula as

Re [σxx(ω)] = πge² (2π)^dω

X

λ,λ⁰

Z

d^dk(f_λ⁰(k) − f_λ(k))|v_x^λ0λ|²δ(∆E_λλ⁰− ~ω), (B1) where ∆Eλλ⁰ = Eλ(k) − Eλ⁰(k), d is the dimensionality, g is the degeneracy and v^λ_x^0λ = hψλ⁰(k)|ˆvx|ψλ(k)i with

ˆ

vx= ∂k_xH/~ being the x-component of velocity operator.

For TCF, the above expression reduces to Re [σ_xx(ω)] =

ge² 8π²ω

Z d³k



(f−(k) − f+(k))|v_x⁻⁺|²δ(2~v^Fk − ~ω) + (f0(k) − f+(k))|v_x⁰⁺|²δ(~vFk − ~ω)

 .

(B2) For TCF, ˆvx = vFSx, v_x⁻⁺ = 0 and |v_x⁰⁺|² = v_F²(3 − cos 2φ + 2 cos²φ cos 2θ)/8. Using these results, Eq.(B2)

gives

Re [σxx(ω)] = ge²ω

6π~v^FΘ(ω − vFkF), (B3)

where Θ(x) is the usual step function. Unlike Weyl semimetals, the absorption between the linearly disper- sive bands is absent in TCF. This feature is also seen in dice lattice, where it was attributed to zero (modulo 2π) Berry phase of the charge carriers⁷⁵.

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