Single-particle spectral function for the classical one-component plasma
aa r X i v : . [ phy s i c s . p l a s m - ph ] N ov Single-particle spectral function for the classical one-component plasma
C. Fortmann ∗ Institut f¨ur Physik, Universit¨at Rostock, 18051 Rostock, Germany
The spectral function for an electron one-component plasma is calculated self-consistently usingthe GW (0) approximation for the single-particle self-energy. In this way, correlation effects whichgo beyond the mean-field description of the plasma are contained, i.e. the collisional damping ofsingle-particle states, the dynamical screening of the interaction and the appearance of collectiveplasma modes. Secondly, a novel non-perturbative analytic solution for the on-shell GW (0) self-energy as a function of momentum is presented. It reproduces the numerical data for the spectralfunction with a relative error of less than 10% in the regime where the Debye screening parameteris smaller than the inverse Bohr radius, κ < a − . In the limit of low density, the non-perturbativeself-energy behaves as n / , whereas a perturbation expansion leads to the unphysical result of adensity independent self-energy [W. Fennel and H. P. Wilfer, Ann. Phys. Lpz. , 265 (1974)].The derived expression will greatly facilitate the calculation of observables in correlated plasmas(transport properties, equation of state) that need the spectral function as an input quantity. Thisis demonstrated for the shift of the chemical potential, which is computed from the analyticalformulae and compared to the GW (0) -result. At a plasma temperature of 100 eV and densitiesbelow 10 cm − , both approaches deviate less than 10% from each other. PACS numbers: 52.27.Aj, 52.65.Vv, 71.15.-mKeywords: Spectral function, Self-energy, GW approximation, One-component plasma
I. INTRODUCTION
The many-particle Green function approach [1] allows for a systematic study of macroscopic properties of correlatedsystems. Green functions know a long history of applications in solid state theory [2], nuclear [3], and hadron physics[4], and also in the theory of strongly coupled plasmas [5]. In the latter case, optical and dielectric properties [6, 7] havebeen studied using the Green function approach, as well as transport properties like conductivity [8] and stoppingpower [9, 10], and the equation of state [11]. Modifications of these quantities due to the interaction among theconstituents can be accessed, starting from a common starting point, namely the Hamiltonian of the system.The key quantity to electronic properties in a correlated many-body environment is the electron spectral function A ( p , ω ), i.e. the probability density to find an electron at energy (frequency) ω for a given momentum p . It is relatedto the retarded electron self-energy Σ( p , ω + iδ ) , δ > A ( p , ω ) = − p , ω + iδ )[ ω − ε p − Re Σ( p , ω )] + [Im Σ( p , ω + iδ )] . (1)Here, the single-particle energy ε p = p − µ e (2)has been introduced, µ e is the electron chemical potential. Note that here and throughout the paper, the Rydbergsystem of units is used, where ~ = 1, m e = 1 /
2, and e / πǫ = 2. Furthermore, the Boltzmann constant k B is setequal to 1, i.e. temperatures are measured in units of energy.The self-energy describes the influence of correlations on the behaviour of the electrons. A finite, frequency depen-dent self-energy leads to a finite life-time of single-particle states and a modification of the single-particle dispersionrelation. Hence, the calculation of the electron self-energy is the central task if one wants to determine electronicproperties, e.g. those mentioned above.The Hartree-Fock approximation [12] represents the lowest order in a perturbative expansion of the self-energy interms of the interaction potential [13]. Being a mean-field approximation, effects due to correlations in the systemcannot be described. Examples are the appearance of collective modes, the energy transfer during particle collisions,and the quasi-particle damping. The next order term is the Born approximation, where binary collision are taken ∗ Electronic address: [email protected]; URL: http://everest.mpg.uni-rostock.de/~carsten into account via a bare Coulomb potential. However, the Born approximation leads to a divergent integral, due tothe long-range Coulomb interaction. Therefore, the perturbation expansion of the self-energy has to be replaced bya non-perturbative approach, accounting for the dynamical screening of the interparticle potential.A non-perturbative approach to the many-particle problem is given by the theory of Dyson [14] and Schwinger[15, 16] generalized to finite temperature and finite density [17]. An excellent introduction to Dyson-Schwingerequations can also be found in [4]. The Dyson-Schwinger equation for the self-energy Σ contains the full Greenfunction G , the screened interaction W and the proper vertex Γ. Since each of these functions obeys a differentDyson-Schwinger equation itself, involving higher order correlation functions, the Dyson-Schwinger approach leads toa hierarchy of coupled integro-differential equations. In order to provide soluble equations, this hierarchy has to beclosed at some level, i.e. correlation functions of a certain order either have to be parametrized or neglected.One such closure of the Dyson-Schwinger hierarchy consists in neglecting the vertex, i.e. the three-point function,and considering only the particle propagators and their respective self-energies, i.e. two-point functions. One arrives atthe so-called GW -approximation, introduced in solid-state physics by Hedin [18]. Hedin was led by the idea, to includecorrelations in the self-energy by replacing the Coulomb potential in the Hartree-Fock self-energy by the dynamicallyscreened interaction W . In this way, one obtains a self-consistent, closed set of equations for the self-energy, thepolarization function Π, the Green function and the screened interaction.It can be shown [19], that the GW approximation belongs to the so-called Φ-derivable approximations [20, 21]. Assuch, it leads to energy, momentum, and particle number conserving expressions for higher order correlation functions.It was successfully applied in virtually all branches of solid state physics. An overview on theoretical foundations andapplications of the GW approximation can be found in the review articles [22, 23, 24].The drawback of the GW approximation is, that the Ward-Takahashi identities are violated. Ward-Takahashiidentities provide an exact relation between the vertex function Γ, i.e. the effective electron-photon coupling in themedium, and the self-energy and follow from the Dyson-Schwinger equations. They reflect the gauge invariance of thetheory. In GW , they are violated simply because corrections to the vertex beyond zero order are neglected altogether.This issue touches on a fundamental problem in many-body theory and field theory, namely the question how topreserve gauge invariance in an effective, i.e. approximative theory, without violating basic conservation laws. Adetailed analysis of this question with application to nuclear physics is presented in a series of papers by van Heesand Knoll [25, 26, 27].Approximations for the self-energy, that also contain the vertex are often referred to as GW Γ approximations.An example can be found in Ref. [28], where the spectral function of electrons in aluminum is calculated using aparametrized vertex function. An interesting result obtained in that work is that vertex corrections and self-energycorrections entering the polarization function, largely cancel. This can be understood as a consequence of Ward-Takahashi identities. Thus, and in order to reduce the numerical cost, it is a sensible choice to neglect vertexcorrections altogether, and to keep the polarization function on the lowest level, i.e. the random phase approximation(RPA) which is the convolution product of two non-interacting Green functions in frequency-momentum space. Thecorresponding self-energy is named the GW (0) self-energy and has been introduced by von Barth and Holm [29],who were also the first to study the fully self-consistent GW approximation [30]. Throughout this work, the GW (0) self-energy will be analyzed.Having been used in solid state physics traditionally, the GW (0) -method was also applied to study correlations inhot and dense plasmas, recently. The equation of state [31, 32], as well as optical properties of electron hole plasmasin highly excited semiconductors [33], and dense hydrogen plasmas [7] were investigated.In general, the calculation of such macroscopic observables of many-particle system involves numerical operationsthat need the spectral function as an input. Since the self-consistent calculation of the self-energy, even in GW (0) -approximation, is itself already a numerically demanding task, it is worth looking for an analytic solution of the GW (0) equations, which reproduces the numerical solution at least in a certain range of plasma parameters. Such an analyticexpression then also allows to study the self-energy in various limiting cases, such as the low density limit or the limitof high momenta, which are difficult to access in the numerical treatment. Furthermore, an analytic expression, beingalready close to the numerical solution permits the calculation of the full GW (0) self-energy using only few iterations.Analytical expressions for the single-particle self-energy have already been given by other authors, e.g. Fennel andWilfer [34] and Kraeft et al. [12]. They calculated the self-energy in first order of the perturbation expansion withrespect to the dynamically screened potential. Besides being far from the converged GW (0) self-energy, their resultis independent of density, i.e. the single-particle life-time is finite even in vacuum. As shown in [35], this unphysicalbehaviour is a direct consequence of the perturbative treatment. By using a non-perturbative ansatz, an expressionfor the self-consistent self-energy in a classical one-component plasma was presented, that reproduces the full GW (0) self-energy at small momenta, i.e. for slow particles. The behaviour of the quasi-particle damping at larger momentaremained open and will be investigated in the present work. Secondly, based on the information gathered about thelow and high momentum behaviour, an interpolation formula will be derived, that gives the quasi-particle dampingat arbitrary momenta.The work is organized in the following way: After a brief outline of the GW (0) -approximation in the next section,numerical results will be given in section III for the single-particle spectral function for various sets of parameterselectron density n e and electron temperature T . In section IV the analytic expression for the quasi-particle dampingwidth is presented and comparison to the numerical results are given. Section V deals with the application of thederived formulae to the calculation of the chemical potential as a function of density and temperature. An appendixcontains the detailed derivation of the analytic self-energy. As a model system, we focus on the electron one-componentplasma, ions are treated as a homogeneously distributed background of positive charges (jellium model). II. SPECTRAL FUNCTION AND SELF-ENERGY
We start our discussion with the integral equation for the imaginary part of the single-particle self-energy in GW (0) approximation:Im Σ( p , ω + i δ ) = 1 n F ( ω ) X q Z ∞−∞ d ω ′ π V ( q ) A ( p − q , ω − ω ′ )Im ǫ − ( q , ω ′ ) n B ( ω ′ ) n F ( ω − ω ′ ) . (3) V ( q ) = 8 π/q Ω is the Fourier transform of the Coulomb potential with the normalization volume Ω . It is multipliedby the inverse dielectric function in RPA, ǫ RPA ( q, ω ) = 1 − V ( q ) X p n F ( ε p − q / ) − n F ( ε p + q / ) ω + ε p − q / − ε p + q / . (4)to account for dynamical screening of the interaction. Furthermore, the Fermi-Dirac and the Bose-Einstein distributionfunction, n F / B ( ε ) = [exp( ε/k B T ) ± − , were introduced. Note, that the dielectric function is only determined once,at the beginning of the calculation. In particular, the single-particle energies ε p = p − µ e entering equation (4) aredetermined from the non-interacting chemical potential, whereas during the course of the self-consistent calculationof the self-energy, the chemical potential is recalculated at each step via inversion of the density relation n e ( µ e , T ) = 2 X p Z dω π A ( p , ω ) n F ( ω ) . (5)Using the self-consistent chemical potential also in the RPA polarization function leads to violation of the f -sum rule,i.e. conservation of the number of particles.The real part of the self-energy is obtained via the Kramers-Kronig relation [2] All quantities (spectral function,self-energy, and chemical potential) have to be determined in a self-consistent way. This is usually achieved by solvingequations (1), (3), and (5) iteratively.The numerical algorithm is discussed in detail in Ref. [35]. III. NUMERICAL RESULTS
The spectral function was calculated for the case of a hot one-component electron plasma. Temperature and densitywere chosen such, that the plasma degeneracy parameter θ = TE F , (6)is always larger than 1, i.e. the plasma is non-degenerate. Furthermore, the temperature is fixed above the ionizationenergy of hydrogen, T ≫ T (cid:18) πn e (cid:19) / , (7)which gives the mean Coulomb interaction energy compared to the thermal energy, is smaller than 1 in all calculations,i.e. we are in the limit of weak coupling.In figure 1, we show contour plots of the spectral function in frequency and momentum space for two differentdensities ( n e = 7 × cm − , upper graph) and ( n e = 7 × cm − , lower graph). The temperature is set to T = 1000 eV in both calculations. The free particle dispersion ω = ε p is shown as solid black line.In the first case, the plasma is classical ( θ = 7 . × ) and weakly coupled (Γ = 4 . × − ). The spectral function issymmetrically broadened and the maximum is found at the free dispersion, i.e. there is no notable quasi-particle shiftat the present conditions. At increasing momentum, the width of the spectral function decreases, and the maximumvalue increases; the norm is preserved.The situation changes, when we go to higher densities, cf. the lower graph in figure 1. The chosen parameters aretypical solar core parameters [36]. The degeneracy parameter is now θ = 1 . . ω pl = 4 √ πn e which is about 23 Ry at the present density. In the former case of lower density no plasmarons appear,only a featureless, single resonance is obtained. At higher momenta, the plasmarons merge into the central peak. Asin the low-density case, the position of the maximum approaches the single-particle dispersion, due to the decreasingHartree-Fock shift at high p . Again, the width of the spectral function decreases with increasing momentum. ] -1B momentum p [a0 1 2 3 4 5 6 7 8 9 10 [ R y ] e µ + ω fr e qu e n c y ) [ / R y ] ω A ( p , -5 -4 -3 -2 -1 -1B momentum p [a0 1 2 3 4 5 6 7 8 9 10 [ R y ] e µ + ω fr e qu e n c y -40-20020406080100 ) [ / R y ] ω A ( p , -5 -4 -3 -2 -1 FIG. 1: (Color online) Contour plots of the spectral function as a function of momentum and frequency. The colour scale islogarithmic. Results are shown for two different densities ( n = 7 × cm − , upper graph) and ( n = 7 × cm − , lowergraph). For these parameters, the plasma coupling parameter is Γ = 4 . × − , and Γ = 9 . × − , respectively. Thedegeneracy parameter is θ = 7 . × and θ = 1 .
6, respectively. The black line indicates the free particle dispersion ω = ε p . This is visible more clearly in figures 2 - 7. Here, the solid curves represent the GW (0) spectral function as afunction of frequency. Results are shown for three different momenta, p = 0 a − (a), p = 50 a − (b), and p = 100 a − (c). Two different temperatures are considered, T = 100 eV (figures 2-4) and T = 1000 eV (figures 5-7) and for eachtemperature three different densities are studied. With increasing momentum p , the spectral function becomes moreand more narrow, converging eventually into a narrow on-shell resonance, located at the unperturbed single-particledispersion ω + µ e = p .As a general feature, one can observe an increase of the spectral function’s width with increasing density andwith increasing temperature. The increase with density is due to the increased coupling, while the increase withtemperature reflects the thermal broadening of the momentum distribution function n F ( ω ) that enters the self-energyand thereby also the spectral function. From these results, we see that the spectral function has a quite simple formin the limit of low coupling, i.e. at low densities and high temperatures.The numerical results are compared to a Gaussian ansatz for the spectral function, shown as the dashed curve infigures 2 - 7. The explicit form of the Gaussian is given as equation (11), below. It’s sole free parameter is the width,denoted by σ p . An analytic expression for σ p will be derived in section IV. The coincidence is in general good at highmomenta, whereas at low momenta, the spectral function deviates from the Gaussian. In particular, the steep wingsand the smoother plateau that form at low momenta are not reproduced by the Gaussian. Also, the plasmaron peaksappearing in the spectral function at high density (see figure 7, cannot be described by the single Gaussian.Determining σ p via least-squares fitting of the Gaussian ansatz to the numerical data at each p leads to the solidcurve in figure 8), obtained in the case of n = 7 × cm − and T = 100 eV. Starting at some finite value σ at p = 0, the width falls off slowly towards larger p . The dashed curve shows σ p as obtained from the analytic formulathat will be derived in the now following section. IV. ANALYTICAL EXPRESSION FOR THE QUASI-PARTICLE SELF-ENERGY
The solution of the GW (0) -equation (3) requires a considerable numerical effort. So far (see e.g. the work byFennel and Wilfer in [34]), attempts to solve the integral (3) analytically were led by the idea to replace the spectralfunction on the r.h.s. by its non-interacting counterpart, A (0) ( p , ω ) = 2 πδ ( ε p − ω ), i.e. to go back to the perturbationexpansion of the self-energy and neglect the implied self-consistency. At the same time, the inverse dielectric functionis usually replaced by a simplified expression, e.g. the Born approximation or the plasmon-pole approximation [12].Whereas the second simplification is indispensable due to the complicated structure of the inverse dielectric function,the first one, i.e. the recursion to the quasi-particle picture, is not necessary, as was shown by the author in Ref. [35].In fact, the result that one obtains in the quasi-particle approximation is far from the converged result, at least inthe high temperature case. Secondly, using the quasi-particle approximation, the imaginary part of the self-energy isnot density dependent, i.e. a finite life-time of the particle states is obtained even in vacuum. This unphysical resultcan only be overcome if one sticks to the self-consistency of the self-energy, i.e. if one leaves the imaginary part ofthe self-energy entering the r.h.s. of equation (3) finite.Using the statically screened Born approximation, which describes the binary collision among electrons via astatically screened potential, a scaling law Im Σ( p , ω QP ( p )) ∝ Γ / was found [35]. Hence, the spectral functionswidth vanishes when the plasma coupling parameter Γ (see equation (7)) tends to 0. An expression for the self-energywas found, that reproduces the converged GW (0) calculations at small momenta, p ≪ κ . At higher momenta, thederived expression ceases to be valid.In this work, a different approximation to the dielectric function is studied, namely the plasmon-pole approximation[12]. This means, that the inverse dielectric function is replaced by a sum of two delta-functions that describe thelocation of the plasmon resonances,Im ǫ − ( q , ω ′ ) −→ Im ǫ − ( q , ω ′ ) = − π ω ω q [ δ ( ω − ω q ) + δ ( ω + ω q )] . (8)For classical plasmas, the plasmon dispersion ω q can be approximated by the Bohm-Gross dispersion relation [38], ω q = ω (cid:18) q κ (cid:19) + q . (9)Many-particle and quantum effects on the plasmon dispersion have recently been studied in [39].The plasmon-pole approximation allows to perform the frequency integration in equation (3), resulting in theexpressionIm Σ( p , ω + iδ ) = ω X q V ( q ) 1 ω q (cid:20) A ( p − q , ω − ω q ) n B ( ω q ) exp( ω q /T ) − A ( p − q , ω + ω q ) n B ( − ω q ) exp( − ω q /T ) (cid:21) . (10)We will first study the case of high momenta, i.e. momenta that are large against any other momentum scale orinverse length scale, such as the mean momentum with respect to the Boltzmann distribution, ¯ p = p T / κ = p πn/T .As discussed in the previous section, the numerical results for the spectral function at high momenta can well bereproduced by a Gaussian. Thus, we make the following ansatz for the spectral function: A Gauss ( p , ω ) = − √ πσ p exp (cid:18) − ( ω − ε p − Σ HF ( p )) σ p (cid:19) . (11)Note, that only the Hartree-Fock contribution to the real part of the self-energy appears, the frequency dependentpart is usually small near the quasi-particle dispersion, ω QP ( p ) = ε p + Re Σ( p , ω ) | ω = ω QP ( p ) , (12)which therefore can be approximated as ω QP ( p ) = ε p + Re Σ HF ( p ). In the following, we make use of the knowledgeabout the width parameter σ p that we gathered already through simple least-squares fitting of the Gaussian ansatzto the spectral function in order to solve the integrals in equation (10).First, we replace the spectral function on the r.h.s by the Gaussian ansatz (11) and evaluate the emerging equationat the single-particle dispersion ω QP ( p ). By claiming that the Gaussian and the spectral function have the samevalue at the quasi-particle energy, we identify σ p = p π/ p , ω QP ( p ))). Figure 8 shows, that the quasi-particledamping σ p is a smooth function of p , that varies only little on the scale of the screening parameter κ . Since thelatter defines the scale on which contributions to the q -integral are most important, we can neglect the momentumshift in the self-energy on the r.h.s., i.e. we can replace the spectral function on the r.h.s of equation (10) as A ( p − q , ε p + Σ HF ( p ) ± ω q ) −→ − √ πσ p exp (cid:18) − ( ε p ± ω q − ε p − q ) σ p (cid:19) , (13)and can now perform the integral over the angle θ between the momenta p and q , √ πσ p Z − d cos θ exp (cid:18) − ( ε p ± ω q − p − q + 2 pq cos θ + µ e ) σ p (cid:19) = π pq Erf " ( p + q ) − p ∓ ω q √ σ p − Erf " ( p − q ) − p ± ω q √ σ p . (14)The remaining integration over the modulus of the transfer momentum q can be performed after some furtherapproximations, explained in detail in appendix A. For large p , one finally obtains the transcendent equation σ p = − . r π ω pl p [ n B ( ω pl ) exp( ω pl /T ) − n B ( − ω pl ) exp( − ω pl /T )] − r π T p ln( κ p /σ p ) (15)The solution of this equation can be expanded for large arguments of the logarithm, yielding σ p = − r π Tp ϕ ( p ) , ϕ ( p ) = " ξ ( p ) − ln ξ ( p ) + ln ξ ( p ) ξ ( p ) − ln ξ ( p ) ξ ( p ) + ln ξ ( p ) ξ ( p ) − ξ ( p )2 ξ ( p ) + ln ξ ( p )2 ξ ( p ) + ln ξ ( p )3 ξ ( p ) + O ( p ) − ξ ( p ) = ln( r π κp exp( A/T ) /T ) , A = − . ω pl n B ( ω pl ) exp( ω pl /T ) − n B ( − ω pl ) exp( − ω pl /T )] . (16)Equation (16) is a solution of equation (15) provided the argument of the inner logarithm is larger than Euler’sconstant e , i.e. q π κp exp( A/T ) /T > e , i.e. at large p . The case of small p , where the previous inequality does nothold, has to be treated separately, see appendix B.Together with an expression for the quasi-particle damping at vanishing momentum taken from [35] and scaled suchthat the maximum of the spectral function at p = 0 is reproduced, σ = − π √ κT /
2, an interpolation formula (Pad´eformula) was derived that covers the complete p -range: σ Pad´e p = a + a p b p + b p ˜ ϕ ( p ) , ˜ ϕ ( p ) = " ˜ ξ ( p ) − ln ˜ ξ ( p ) + ln ˜ ξ ( p )˜ ξ ( p ) − ln ˜ ξ ( p )˜ ξ ( p ) + ln ˜ ξ ( p )˜ ξ ( p ) − ˜ ξ ( p )2 ˜ ξ ( p ) + ln ˜ ξ ( p )2 ˜ ξ ( p ) + ln ˜ ξ ( p )3 ˜ ξ ( p ) ˜ ξ ( p ) = ln( e + r π κp exp( A/T ) /T ) ,a = − π √ κT , a = − κ (cid:16) π (cid:17) / , b = r πκ T , b = πκ T . (17)The function ˜ ξ ( p ) in the last equation differs from ξ ( p ) in equation (16) in that Euler’s constant e ≃ . ϕ ( p ) is regularized at small p and tends to 1 at p = 0, i.e. the quasi-particle damping goes to the correct low- p limit. At large p , this modification is insignificant,since the original argument rises as p . For the detailed derivation, see appendix B.Expression (17), used in the Gaussian ansatz (11), leads to a spectral function that well reproduces the numericaldata from full GW (0) -calculations: Figure 8 (dashed curve) shows the effective quasi-particle damping width σ p as afunction of momentum p for the case n = 7 × cm − and T = 100 eV. The solid curve gives the best-fit value for σ p obtained via least-square fitting of the full GW (0) -calculations assuming the Gaussian form (11), see section III.Both curves coincide to a large extent. The largest deviations are observed in the range of p ≃ a − . At this point,the validity of expression (16) as the solution of equation (15) ceases, since the argument of the logarithm becomessmaller than e . As already mentioned, we circumvented this problem by regularizing the logarithms, adding e to itsargument. The deviation at p ≃ a − of up to 15% is a residue of this procedure. At higher momenta, the deviationis generally smaller than 10% and the analytic formula evolves parallel to the fit parameters. s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=0 a B-1 GW (0) Gauss fit (a) p = 0 s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=50 a B-1 GW (0) Gauss fit (b) p = 50 a − s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=100 a B-1 GW (0) Gauss fit (c) p = 100 a − FIG. 2: Spectral function in GW (0) -approximation (solid lines) and Gaussian ansatz (dashed lines) with quasi-particle dampingwidth σ p taken from equation (17) for three different momenta. Plasma parameters: n = 7 × cm − , T = 100 eV. Theplasma coupling parameter is Γ = 1 . × − , the degeneracy parameter is θ = 1 . × , the Debye screening parameter is κ = 6 . × − a − . At smaller densities, the correspondence is even better as can be seen by comparing the spectral functions shownin figures 2 - 7. The dashed curves give the Gaussian ansatz for the spectral function with the quasi-particle widthtaken from the interpolation formula (17). As a general result, the analytic expression for the quasi-particle damping σ p leads to a spectral function that nicely fits the numerical solution for the spectral function at least at finite p . Atvery small values of p , the overall correspondence is still fair, i.e. the position of the maximum and the overall widthmatch, but the detailed behaviour does not coincide. In particular, the steep wings and the central plateau, thatforms in the GW (0) -calculation, is not reproduced by the one-parameter Gaussian. For this situation, the analyticformula for self-energy given in [35] should be used instead.By comparing the numerical data for the spectral function to the Gaussian ansatz at different densities, it is found,that the Gaussian spectral function is a good approximation as long as the Debye screening parameter κ is smallerthan the inverse Bohr radius, κ < a − . This becomes immanent by comparing figures 6 and 7. In the first case( n e = 7 × cm − , T = 1000 eV), we have κ = 0 .
19, while in the second case ( n e = 7 × cm − , T = 1000 eV), κ = 1 . s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=0 a B-1 GW (0) Gauss fit (a) p = 0 s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=50 a B-1 GW (0) Gauss fit (b) p = 50 a − s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=100 a B-1 GW (0) Gauss fit (c) p = 100 a − FIG. 3: Spectral function in GW (0) -approximation (solid lines) and Gaussian ansatz (dashed lines) with quasi-particle dampingwidth σ p taken from equation (17) for three different momenta. Plasma parameters: n = 7 × cm − , T = 100 eV. Theplasma coupling parameter is Γ = 2 . × − , the degeneracy parameter is θ = 3 . × , the Debye screening parameter is κ = 1 . × − a − . s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=0 a B-1 GW (0) Gauss fit (a) p = 0 s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=50 a B-1 GW (0) Gauss fit (b) p = 50 a − s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=100 a B-1 GW (0) Gauss fit (c) p = 100 a − FIG. 4: Spectral function in GW (0) -approximation (solid lines) and Gaussian ansatz (dashed lines) with quasi-particle dampingwidth σ p taken from equation (17) for three different momenta. Plasma parameters: n = 7 × cm − , T = 100 eV. Theplasma coupling parameter is Γ = 4 . × − , the degeneracy parameter is θ = 7 . × , the Debye screening parameter is κ = 6 . × − a − . spectral function at increased densities. Since the position of the plasmaron peak is given approximatively by theplasma frequency ω pl , whereas the width of the central peak at small p is just the quasi-particle width σ , we canidentify the ratio of these two quantities, − ω pl /σ ∝ √ κ as the parameter which tells us if plasmaron peaks appearseparately ( ω pl > − σ ) or not ( ω pl < − σ ). Since the plasma frequency increases as a function of n / , whereas thequasi-particle width scales as n / (c.f. equation (17)), the transition from the single peak behaviour to the morecomplex behaviour including plasmaron resonances, appears at increased density. Neglecting numerical constants oforder 1 in the ratio of plasma frequency to damping width, we see that − ω pl /σ < κ <
1, whichwas our observation from the numerical results. Therefore, we can identify the range of validity of the presentedexpressions for the spectral function and the quasi-particle damping. It is valid for those plasmas, where we havedensities and temperatures, such that κ < κ < < T − / . Since we restrict ourselves to plasma temperatures, where bound states can be excluded,i.e.
T > < GW (0) calculations and the parametrized spectral function atsmall momenta is not as good as in the case of large momenta, the parametrized spectral function can be applied inthe regime of validity to the calculation of plasma observables without introducing too large errors. As an example,this will be shown for the case of the chemical potential µ in the next section. s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=0 a B-1 GW (0) Gauss fit (a) p = 0 s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=50 a B-1 GW (0) Gauss fit (b) p = 50 a − s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=100 a B-1 GW (0) Gauss fit (c) p = 100 a − FIG. 5: Spectral function in GW (0) -approximation (solid lines) and Gaussian ansatz (dashed lines) with quasi-particle dampingwidth σ p taken from equation (17) for three different momenta. Plasma parameters: n = 7 × cm − , T = 1000 eV. Theplasma coupling parameter is Γ = 4 . × − , the degeneracy parameter is θ = 7 . × , the Debye screening parameter is κ = 1 . × − a − . s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=0 a B-1 GW (0) Gauss fit (a) p = 0 s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=50 a B-1 GW (0) Gauss fit (b) p = 50 a − s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=100 a B-1 GW (0) Gauss fit (c) p = 100 a − FIG. 6: Spectral function in GW (0) -approximation (solid lines) and Gaussian ansatz (dashed lines) with quasi-particle dampingwidth σ p taken from equation (17) for three different momenta. Plasma parameters: n = 7 × cm − , T = 1000 eV. Theplasma coupling parameter is Γ = 2 . × − , the degeneracy parameter is θ = 3 . × , the Debye screening parameter is κ = 1 . × − a − . V. APPLICATION: SHIFT OF THE CHEMICAL POTENTIAL
To demonstrate the applicability of the presented formulae for quick and reliable calculations of plasma properties,we calculate the shift of the electron’s chemical potential ∆ µ = µ − µ free , i.e. the deviation of the chemical potential ofthe interacting plasma µ from the value of the non-interacting system µ free . The chemical potential of the interactingsystem µ is obtained by inversion of the density as a function of T and µ , equation (5). The free chemical potential µ free is obtained in a similar way by inversion of the free density n free ( T, µ free ) = 2 X p n F ( ε p − µ free ) . (18)Figure 9 shows the shift of the chemical potential as a function of the plasma density n for a fixed plasma temperature T = 100 eV. Results obtained by inversion of equation (5) using the parametrized spectral function (11) with thequasi-particle damping width taken from equation (17) (solid curve) are compared to those results taking the numerical GW (0) spectral function (dashed curve).The GW (0) result gives slightly smaller shifts than the parametrized spectral function, i.e. the usage of the analyticaldamping width leads to an overestimation of the the shift of the chemical potential. However, the deviation remainssmaller than 20% over the range of densities considered here, i.e. for κ <
1. At small densities, i.e. for n ≤ cm − ,the parametrized spectral function yields the same result as the full GW (0) calculation.The deviation at increased density can be reduced by improving the parametrization of the spectral function atsmall momenta. To this end, the behaviour of the quasi-particle damping width at high momenta, equation (16)should be combined with the frequency dependent solution for σ p at vanishing momentum, as presented in Ref. [35].0 s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=0 a B-1 GW (0) Gauss fit (a) p = 0 s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=50 a B-1 GW (0) Gauss fit (b) p = 50 a − s p ec t r a l f un c ti on A ( p , ω ) [ / R y ] frequency ω+µ e [Ry]p=100 a B-1 GW (0) Gauss fit (c) p = 100 a − FIG. 7: Spectral function in GW (0) -approximation (solid lines) and Gaussian ansatz (dashed lines) with quasi-particle dampingwidth σ p taken from equation (17) for three different momenta. Plasma parameters: n = 7 × cm − , T = 1000 eV. Theplasma coupling parameter is Γ = 9 . × − , the degeneracy parameter is θ = 1 .
6, the Debye screening parameter is κ = 1 . a − .Here, the Gaussian fit is no longer sufficient due to the appearance of plasmaron resonances in the spectral function (shouldersat ω ≃ −
30 Ry and ω ≃
20 Ry). p [a B-1 ]-0.5-0.4-0.3-0.2 qu a s i - p a r ti c l e d a m p i ng σ p [ R y ] GW (0) -fitInterpolation formula FIG. 8: Effective quasi-particle damping σ p as a function of momentum p for plasma density n = 7 × cm − and temperature T = 100 eV. The fit-parameters for the Gaussian fit to the full GW (0) -calculations are given as dots, the solid line denotes theanalytic interpolation formula (17). However, this task goes beyond the scope of this paper, where we wish to present comparatively simple analyticexpressions for the damping width that yield the correct low density behaviour of plasma properties.
VI. CONCLUSIONS AND OUTLOOK
In this paper, the GW (0) -approximation for the single-particle self-energy was evaluated for the case of a classicalone-component electron plasma, with ions treated as a homogeneous charge background. A systematic behaviour ofthe spectral function was found, i.e. a symmetrically broadened structure at low momenta and convergence to a sharpquasi-particle resonance at high p . At increased densities, plasmaron satellites show up in the spectral function assatellites besides the main peak.In the second part, an analytic formula for the imaginary part of the self-energy at the quasi-particle dispersion ω QP ( p ) = ε p +Σ HF ( p ) was derived as a two-point Pad´e formula that interpolates between the exactly known behaviourat p = 0 and p → ∞ . The former case was studied in [35], while an expression for the asymptotic case p → ∞ wasderived here. The result is summarized in equation (17). In contrast to previously known expressions for the quasi-particle damping, based on a perturbative approach to the self-energy [34], the result presented here shows a physicallyintuitive behaviour in the limit of low densities, i.e. it vanishes when the system becomes dilute. Using the Gaussian1 n [cm -3 ]1e-021e-01 - ∆ µ [ R y ] Gaussian spectral function GW (0) T = 100 eV
FIG. 9: Shift of the chemical potential as a function of the plasma density for a plasma temperature T = 100 eV. Resultsfor ∆ µ using the parametrized spectral function (solid line) are compared to full numerical calculations, using the GW (0) -approximation. ansatz (11) for the spectral function in combination with the quasi-particle width leads to a very good agreementwith the numerical data for the spectral function in the range of plasma parameters, where κ < a − ; the relativedeviation is smaller than 10% under this constraint.Thus, a simple expression for the damping width of electrons in a classical plasma has been found, that can beused to approximate the full spectral function to high accuracy. This achievement greatly facilitates the calculationof observables that take the spectral function or the self-energy as an input, such as optical properties (inversebremsstrahlung absorption), conductivity, or the stopping power.Furthermore, it was demonstrated that the derived expressions allow for quick and reliable calculations of plasmaproperties without having to resort to the full self-consistent solution of the GW (0) -approximation. As an example,the shift of the chemical potential was calculated using the parametrized spectral function and compared to GW (0) results. For densities of n < cm − , both approaches coincide with a relative deviation of less than 10%, goingeventually up to 20% as the density approaches 10 cm − . At low densities both approaches give identical results.This shows the extreme usefulness of the presented approach for the calculation of observables via the parametrizedspectral function.As a furhter important application of the results presented in this paper, we would like to mention the calculation ofradiative energy loss of particles traversing a dense medium, i.e. bremsstrahlung. A many-body theoretical approachto this scenario is given by Knoll and Voskresensky [40], using non-equilibrium Green functions. They showed, that afinite spectral width of the emitting particles leads to a decrease in the bremsstrahlung emission. This effect is knownas the Landau-Pomeranchuk-Migdal effect [41, 42]. It has been experimentally confirmed in relativistic electronscattering experiments using dense targets, e.g. lead [43, 44]. In [45], it is shown that also thermal bremsstrahlungfrom a plasma is reduced due to the finite spectral width of the electrons in the plasma. In the cited papers, the quasi-particle damping width was either set as a momentum- and energy independant parameter (in [40]), or calculatedself-consistently using simplified approximations of the GW (0) theory (in [45]), which itself is a very time-consumingtask and prohibited investigations over a broad range of plasmas parameters. Now, based on this work’s results,calculations on the level of full GW (0) become feasible, since analytic formulae have been found that reproduce the GW (0) self-energy. Effects of dynamical correlations on the bremsstrahlung spectrum can be studied starting from aconsistent single-particle description via the GW (0) self-energy. Acknowledgments
The author acknowledges many helpful advice from Gerd R¨opke and fruitful discussion with W.-D. Kraeft as well aswith C. D. Roberts. Financial support was obtained from the German Research Society (DFG) via the CollaborativeResearch Center “Strong Correlations and Collective Effects in Radiation Fields: Coulomb Systems, Clusters, andParticles” (SFB 652).2
APPENDIX A: ANALYTIC SOLUTION FOR THE GW (0) SELF-ENERGY USING THE PLASMON POLEAPPROXIMATION
After the angular integration which was performed in equation (14), the imaginary part of the self-energy at thequasi-particle dispersion ω = ε p readsIm Σ( p , p ) = ω p Z ∞ dqq ω q ( " Erf q + 2 pq + ω q √ σ p ! − Erf q − pq + ω q √ σ p ! n B ( ω q ) exp( ω q /T ) − " Erf q + 2 pq − ω q √ σ p ! − Erf q − pq − ω q √ σ p ! n B ( − ω q ) exp( − ω q /T ) ) . (A1)This equation represents a self-consistent equation for Im Σ( p , ω = p ) = p /πσ p .Our aim is to derive an analytic expression, that approximates the numerical solution of equation (A1) for arbitrary p . To this end, we first look at the case of large momenta, p ≫ κ , and later combine that result with known expressionsfor the limit of vanishing momentum p →
0, to produce an interpolation (“Pad´e”) formula that covers the complete p -range.We perform a sequence of approximations to the integral in (A1). First, we observe, that at large p , the term 2 pq dominates in the argument of the error function. We rewrite equation (A1) as σ p = r π p , p ) = r π ω p Z ∞ dqq ω q ( " Erf pq √ σ p ! − Erf − pq √ σ p ! n B ( ω q ) exp( ω q /T ) (A2) − " Erf pq √ σ p ! − Erf − pq √ σ p ! n B ( − ω q ) exp( − ω q /T ) ) (A3)= r π ω p Z ∞ dqq ω q pq √ σ p ! [ n B ( ω q ) exp( ω q /T ) − n B ( − ω q ) exp( − ω q /T )] , (A4)The integrand in equation (A4) contains a steeply rising part at q < − σ p /p and a smoothly decaying part for atlarge q , i.e. when q ≫ − σ p /p . Therefore, we separate the integral in equation into two parts, one going from q = 0to q = ¯ q = − σ p /p and the other from ¯ q to infinity. In the first part of the integral, the values for q are so small, thatwe can replace the plasmon dispersion by the plasma frequency ω pl . In the second term, the argument of the errorfunction is large and the error function can be replaced by its asymptotic value at infinity, lim x →∞ Erf( x ) = 1. Thisleads to σ p = r π ω p ( Z ¯ q dqq ω pl pq √ σ p ! [ n B ( ω pl ) exp( ω pl /T ) − n B ( − ω pl ) exp( − ω pl /T )]+ 2 Z ∞ ¯ q dqq ω q [ n B ( ω q ) exp( ω q /T ) − n B ( − ω q ) exp( − ω q /T )] ) , (A5)Finally, we expand last term in powers of ω q /T , which is justified at low densities ( ω q ∝ ω pl ), and keep only the firstorder, n B ( ω q ) exp( ω q /T ) − n B ( − ω q ) exp( − ω q /T ) = 2 Tω q + O ( ω q ) − . (A6)We obtain σ p = r π ω p ( Z ¯ q dqq ω pl pq √ σ p ! [ n B ( ω pl ) exp( ω pl /T ) − n B ( − ω pl ) exp( − ω pl /T )] + 4 T Z ∞ ¯ q dqq ω q ) . (A7)3Both integrals can be performed analytically, Z ¯ q dqq Erf pq √ σ p ! = − r π p ¯ qσ p F (1 / , /
2; 3 / , / − p ¯ q /σ p )= − r π F (1 / , /
2; 3 / , / −
2) = − . , Z ∞ ¯ q dqq ω (1 + q /κ ) = 12 ln(1 + κ / ¯ q ) = 12 ln(1 + κ p /σ p ) , (A8)where ¯ q = − σ p /p was used. Note that in the second integral, the q term in the plasmon dispersion (9) is omitted. F ( a , a ; b , b ; z ) is the generalized hypergeometric function [46].We arrive at the equation σ p = − . r π ω p [ n B ( ω pl ) exp( ω pl /T ) − n B ( − ω pl ) exp( − ω pl /T )] − r π T p ln(1 + κ p /σ p ) . (A9)At large p , the term κ p /σ p dominates the argument of the logarithm, i.e. we can write ln(1+ κ p /σ p ) ≃ ln( κ p /σ p ).Then, we arrive at equation (A1), given in section IV. APPENDIX B: PADE APPROXIMATION
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