The GW Plus Cumulant Method and Plasmonic Polarons: Application To

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GW pcrlpoete fteHGaedsusdi Sec. in discussed are HEG the of properties spectral rsn acltoso h osfnto fteHGi the in HEG the of The function RPA. loss the of calculations present h auciti raie sflos nSec. In follows. as organized is manuscript The nti ok epeetasuyo h pcrlfunc- spectral the of study a present we work, this In ytepamnenergy. plasmon the by lscmln ( cumulant plus etrsadrnraie hi energy their renormalizes and features l GW 25,26 17,18 xodO13H ntdKingdom United 3PH, OX1 Oxford le pcrlfnto ae nthe on based function spectral olved GW lsoi oao ad,ta is, that bands, polaron plasmonic f ge 1.5 rgies prxmto oilsrt h mrec of emergence the illustrate to approximation ∼ suigmn-oyperturbation many-body using as h uuatapoc rvdueu also useful proved approach cumulant The 17,19,20 n ple odsrb h pcrlsig- spectral the describe to applied and ω prxmto n t plcto othe to application its and approximation 31 pl 28–30 ω eo h uspril nry con- energy, quasiparticle the below pl GW semiconductors, eo h quasiparticle the below C prah efis review first We approach. +C) 8,16 GW eietefis cumulant first the Beside 8 prxmto n the and approximation int the to tion h uuatexpan- cumulant The 27 15 14,21–24 n lrfs quasi- ultrafast and ,the s, GW hsmdli ex- is model This 23,24,31 Capproach +C GW n models and n con- and Cap- +C II we , III , 2

Figure 1. Schematic representation of the parabolic band dispersion of the HEG. The red circles (‘1’, ‘2’, and ‘3’) indicate some representative ﬁnal states for electron-hole excitations starting from the initial state ‘0’ at the Fermi energy µ. (b)-(c) RPA loss function of the HEG evaluated in the GW approximation at the densities rs = 3 (b) and rs = 5 (c). Here we considered frequencies with a small imaginary component (η ∼ 0.004×µ) to clearly visualize the plasmon peak. The case η = 0 is shown in (d) for rs = 5. Energy and momentum are expressed in units of the plasmon energy ωpl and Fermi momentum kF, respectively. The circles (0, 1, 2, and 3) in panel (b) are placed at points where the intensity of the loss function arises from the transitions illustrated in panel (a). whereas in Sec. IV we present calculations of the HEG as:7 spectral function based on the GW +C approach. Finally, ′ dk dω ′ ′ our conclusions are presented in Sec. V. χ0(q,ω)= −i G0(q + k,ω + ω )G0(k,ω ). Z (2π)4 (3)

II. SIGNATURES OF PLASMONS IN THE We introduced here the non-interacting Green’s function, DIELECTRIC FUNCTION defined by: 1 Electronic excitations of the HEG can be characterized G0(k,ω)= , (4) ω + µ − ǫk + isign(µ − ǫk) through the computation of the loss function:32 2 where µ is the Fermi energy, ǫk = k /2, and η a positive −1 L(q,ω)=Im ǫ (q,ω), (1) infinitesimal. The convolution in Eq. (3) may be carried out analytically,33 yielding an explicit expression for the where ǫ is the dielectric function. Since the dielectric dielectric function of the HEG:9,15 function vanishes at the frequencies resonant with the αrs excitations of plasmons,33 the loss function exhibits pro- ǫ(q,ω)=1+ [H(q + u/q) − H(q − u/q)], (5) 8πq3 nounced singularities at the plasmon energies ωpl(q). 34 The condition ǫ[q,ωpl(q)] = 0, which defines the plas- where H(q) = 2q + (1 − q2)ln [(q + 1)/(q − 1)], α = mon energy, provides a rational to distinguish between (4/9π)1/3. Here we followed the notation of Ref. 9 where spectral signatures of plasmon and electron-hole pairs in q denotes momenta in units of 2kF and u are energies in the loss function. In particular, plasmons are expected to units of 4µ. rs denotes the Wigner-Seitz radius. The cal- induce Dirac-delta-like features in Eq. (1), well separated culation of the loss function is reduced to the evaluation from the continuum of electron-hole pair excitations. of the Eq. (5) for several frequencies and momenta. In a Green’s function formalism, the dielectric function In Fig. 1 we report the loss function of the HEG for may be expressed as: rs = 3 (b) and rs = 5 (c). A detailed discussion of the loss function may be found in many textbooks.15,35 ǫ(q,ω)=1 − v(q)χ0(q,ω), (2) Briefly, the broad band of width 2kF is the continuum of electron-hole excitation. To exemplify the origin of where we introduced the irreducible polarizability χ0 and these features we report in Fig. 1 (a) a schematic repre- the bare Coulomb interaction v(q)=4π/|q|2. Here and sentation of the free electron energy band of the HEG. in the following we adopted Hartree atomic units, unless The labels ‘1’, ‘2’, and ‘3’ denote possible final states otherwise stated. In the RPA, whereby electron-hole in- for the excitation of an electron at the Fermi energy teractions are neglected, χ0 may be expressed explicitly (red dot labelled ‘0’). The transition to 1 involves a 3

Figure 2. GW self-energy of the HEG for rs =5 at k = 0.6kF (a), k = kF (b), and k = 1.4kF (c). The energy is in units of the plasmon energy, ωpl = 4.2 eV. very small change of energy and a momentum transfer In principle, the evaluation of the self-energy in of ∼ 2kF . The contribution of these transition to the Eq. (6) should employ a Green’s function obtained self- loss function is infinitesimal due to the small phase space consistently from the solution of the Dyson’s equation: available for the transitions. By considering finite en- −1 −1 [G(k,ω)] = [G (k,ω)] − Σ(k,ω). (8) ergy changes, electron-hole excitations must necessarily 0 involve a change of momentum. The maximum (mini- Self-consistent GW denotes the procedure in which mum) momentum transfer would correspond to transi- Eqs. (2), (3), (6), (7), and (8) are iterated until con- tion of the type 2 (3). The corresponding signatures of vergence is reached. For atoms and molecules it is well these transitions in the loss function are indicated by 1, 2, established that self-consistent GW improves the de- and 3 in Fig. 1 (b). According to the previous discussion, scription of quasiparticle energies36–41 as compared to the high-intensity feature at small momentum transfer non-self-consistent calculations. Similar conclusions have may not be attributed to electron-hole excitations. This been obtained for the total energies of atoms,37,42,43 feature is the plasmon peak and stems from the zeros of molecules,44–46 and the homogeneous electron gas.27,47,48 dielectric function. In particular, for k = 0, the plasmon For what concerns plasmon satellites in the spectral func- peak occurs exactly at the plasma energy ωpl. At energies tion, however, Holm and Von Barth have shown that self- and momenta at which the plasmons and the electron- consistent GW deteriorates the spectral function due to hole excitations coexist, the plasmon peak is broadened a spurious renormalization of the satellite intensity.47 In out and its spectral features are not distinguishable from the following we will limit the discussion to ‘one-shot’ the electron-hole continuum. GW (or G0W0), in which Eq. (6) is evaluated at the first- To visualize the plasmon peak of the loss function we iteration of the self-consistent procedure. The GW self- considered frequencies with a small imaginary part (η = energy [Eq. (6)] has been obtained from the numerical 0.004×µ). If purely real frequencies were considered, the integration Eq. (89)-(91) of Ref. 9. In Fig. 2 we illustrate plasmon peak would not be visible in the loss function the real and imaginary part of the self-energy (in units owing to the finite momentum resolution in the figure of the plasmon energy ωpl) for rs = 5. These results, [Fig. 1 (d), for rs = 5]. We now move on to discuss the based on the calculation of the RPA dielectric function, spectral function of the HEG in the GW approximation are in excellent agreement with the results reported by and the spectral signatures of plasmons. Lundqvist based on the plasmon-pole approximation.6 The GW self-energy exhibits a sharp pole at the energy ǫk ± ωpl, where the +/− signs hold for empty/occupied III. GW SELF-ENERGY AND SPECTRAL states. It is evident from Eqs. (6) and (7) that this feature FUNCTION OF THE HEG stems primarily from the plasmon peak in the dielectric function, which introduces a singularity in the screened In the GW approximation, the electron self-energy for Coulomb interaction W owing to the vanishing ǫ. the HEG takes the form: Having reviewed the self-energy in the GW approximation, we move now to discuss the signatures of plas- i ′ Σ(k,ω)= dωdqG(k + q,ω + ω )W (q,ω). (6) mon excitations in the spectral function of the HEG. The 2π Z spectral function is given by: ′′ The screened Coulomb interaction W can be expressed 1 |Σk(ω)| as: A(k,ω)= ′ 2 ′′ 2 , (9) π [ω − ǫk − Σk(ω)] + [Σk(ω)] v(q) ′ ′′ W (q,ω)= . (7) where Σ and Σ denote the real and imaginary part of ǫ(q,ω) the GW self-energy, respectively. In independent-particle 4

Figure 3. Spectral function of the HEG evaluated in the GW approximation for (a) rs = 3, (b) rs = 4, and (c) rs = 5. Energy and momentum are expressed in units of the plasmon energy ωpl and Fermi momentum kF, respectively. To facilitate a comparison on the same scale, the intensity of the spectral function has been multiplied by ωpl. The red dotted lines indicate the Fermi energy. (d) Quasiparticle energy versus momentum dispersion relations within the Hartree-Fock (HF) approximation, and for non-interacting electrons (free). approximations, such as the Hartree-Fock approximation or Kohn-Sham density functional theory, the self-energy is static (independent of frequency) and real. The spectral function reduces to a Dirac delta function:

A(k,ω)= δ(ω − ǫk − Σk). (10) In the case of the Hartree-Fock approximation, the self- energy may be obtained analytically,49 and the evaluation of the spectral function is straightforward. The Hartree-Fock spectral function is reported in Fig. 3(d), alongside with the free-electron spectral function. At each k point, the spectral function of the HEG exhibits Dirac-delta-like structures at the energy of quasiparticle excitations. However, there are no structures that may be attributed to collective excitations induced by electronic correlation. In the GW approximation, on the other hand, the self-energy is characterized by a com- Figure 4. Spectral function of the HEG evaluated from GW plex frequency dependence which introduces several ad- (a) and GW +C (b) for k =0at rs = 5. (c) Ratio between the ditional signatures of electron correlation in the spectral intensity of the plasmon satellite and the quasiparticle peak function of the HEG. (as obtained form GW +C) as a function of rs. The GW spectral function is illustrated in Fig. 3 for the HEG at three different electron densities (rs = 3, 4, and 5). The quasiparticle band is the bright band that the following, the cumulant expansion approach provides appears at ω ≃ −ωpl for k = 0 and increases quadrati- an ideal way to address this problem, as it improves cally with momentum. At variance with the independent the plasmon-induced spectral features of the HEG at the particle approximation, the quasiparticle peaks acquire a same computational cost of a GW calculation. broadening (vanishing at the Fermi energy) which stems from electronic correlation and is related to the finite lifetime of electronic excitations. IV. SPECTRAL FUNCTION FROM THE GW Beside the quadratic quasiparticle band, the spectral PLUS CUMULANT APPROACH function presents pronounced spectral features at energies 1.5 ωpl below the quasiparticle energy. These fea- For the computation of spectral properties, it is com- tures are additional solutions of the quasiparticle equa- mon practice to combine the cumulant expansion with tion, that is, they arise from the zeros of ω − ǫk − ′ the GW approximation. In the resulting GW +C ap- Σk(ω) in Eq. (9). These spectral features, first reported proach, the spectral function can be expressed as:17 by Lundqvist, have been originally attributed to plasmarons, a new type of quasiparticle stemming from the A(k,ω) = [AQP(k,ω)+ AQP(k,ω) ∗ AC(k,ω)]. (11) strong coupling between holes and plasmons.11,12 How- ever, subsequent work have shown that plasmarons are This expression corresponds to the first-order cumulant an artifact of the GW approximation.8,14 As shown in expansion, and it ignores processes in which multiple 5

Figure 5. Spectral function of the HEG evaluated from the GW +C approach for (a) rs = 3, (b) rs = 4, and (c) rs = 5. Energy and momentum are expressed in units of the plasmon energy ωpl and Fermi momentum kF, respectively. To facilitate a comparison on the same scale, the spectral function intensity has been multiplied by ωpl. The dashed lines indicate the Fermi energy. plasmons are excited. Multi-plasmon processes may closely the momentum dependence of the quasiparticle be accounted for by including higher-order cumulant bands. This indicates that also the HEG is character- terms.14,20,25 The first term in Eq. (11) is the ordinary ized by the formation of a well-defined plasmonic po- quasiparticle spectral function, defined as: laron band. Plasmonic polaron bands are a manifesta-

′′ qp tion of the simultaneous excitation of a hole (for instance QP 1 |Σk(ǫk )| via the absorption of a photon) and the excitation of A (k,ω)= ′ qp 2 ′′ qp 2 , (12) π [ω − ǫk − Σk(ǫk )] + [Σk(ǫk )] a plasmon, and they manifest themselves as broadened

qp ′ qp band-structure replica, shifted by the plasmon energy ǫk = ǫk +Σk(ǫk ) is the quasiparticle energy. The term with respect to the ordinary quasiparticle bands. These C A is deﬁned as: spectral features have recently been predicted in the context of sp-bonded semiconductors23,24 and conﬁrmed ∂βk βk(ω) −βk(ǫk) −(ω − ǫk) by angle-resolved photoemission spectroscopy measure- ∂ω 31 C ǫk ments of silicon. In the case of silicon, plasmonic po- A (k,ω)= , (13) (ω − ǫk)2 laron bands replicate the entire set of valence bands. The

−1 HEG, on the other hand, is characterized by a single where βk(ω)= π ImΣk(ǫk − ω)θ(µ − ω). Equation (12) band. Correspondingly, a single plasmonic polaron band accounts for the contribution of quasiparticle excitations can be observed in Fig. 5. to the spectral functions in absence of plasmons. The Our calculations show that the intensity of the satellite second term in Eq. (11) accounts for processes in which features in the GW +C spectral function decreases with an electron is emitted and a plasmon is excited. increasing density (that is with decreasing rs), as shown We evaluated the GW +C angle-resolved spectral func- in Fig. 4(c). This behaviour may be attributed to the dif- tion of the HEG by combining the GW self-energy de- ferent scaling of the Coulomb interaction and the kinetic fined in Eq. (6) with the cumulant expansion defined by energy with the changes of the electron density:1 at large Eqs. (11)-(13). In Fig. 4 we compare the spectral function densities, the kinetic energy increases more rapidly than at rs = 5 and k = 0 obtained from GW (a) and GW +C the Coulomb interaction and, correspondingly, the effect (b). The GW and the GW +C approaches provide a sim- of electron correlation becomes less important as com- ilar description of the quasiparticle peak. In the GW +C pared to the kinetic term. In the limit of infinite elec- approach, these features stems from Eq. (12) which co- tron density, the HEG can be approximately described incides with the GW spectral function [Eq. (9)] at the qp by a non-interacting HEG, as the Coulomb interaction quasiparticle energy ǫk . The changes introduced by the becomes negligible, and the satellite is expected to dis- cumulant approach affect primarily the low-energy part appear completely. Conversely, the Coulomb interaction of the spectral function. At variance with the GW spec- dominates at low densities (large rs) and one may expect tral function, whereby the plasmon peak is red-shifted by a more pronounced effect of electron correlation on the approximately 1.5 ωpl with respect to the quasiparticle spectral properties. band, the GW +C yields a satellite structure separated by ∼ ωpl from the quasiparticle energy. As compared to the GW spectral function, these spectral features are more broad and less intense. V. SUMMARY AND CONCLUSIONS Inspecting the angle-resolved spectral function, shown in Fig. 5 for (a) rs = 3, (b) rs = 4, and (c) rs = 5, In summary, we have presented a study of spectral we note that the dispersion of GW +C satellite follows function of the homogeneous electron gas, with an em- 6 phasis on the signatures of electron-plasmon interactions. energy difference to the quasiparticle band to the plasma In particular, we reviewed the analysis of the loss func- energy ωpl. Consistently with previous work on semi- tion of the HEG in the random phase approximation and conductors, the present study reveals that also the HEG computed the spectral function of the HEG from the GW is characterized by the emergence of plasmonic polaron approximation and the GW +C approach. bands, that is, plasmon-induced band structure replica At variance with calculations in the independent- red-shifted by the plasmon energy. particle approximation, the explicit treatment of electron-electron interaction within the GW approximation introduces a non-trivial frequency dependence ACKNOWLEDGMENTS which, in turn, leads to the emergence of additional low- energy features in the spectral function. At the GW This work was supported by the Leverhulme Trust level, for k < kF the spectral function exhibits the spuri- (Grant No. RL-2012-001) and the European Research ous plasmaron peak at an energy of approximately 1.5 ωpl Council (EU FP7/ERC Grant No. 239578 and EU below the quasiparticle energy. A more advanced descrip- FP7/Grant No. 604391 Graphene Flagship). Calcu- tion of electron-plasmon coupling within the GW +C ap- lations were performed at the Oxford Supercomputing proach, however, reduces significantly the intensity of the Centre50 and at the Oxford Materials Modelling Labora- plasmon-induced spectral features and renormalizes their tory.

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