On the combined use of GW approximation and cumulant expansion in the calculations of quasiparticle spectra: The paradigm of Si valence bands

Abstract

PACS numbers: 71.15.Qe, 71.45.Gm Appendix A: Integral representation of cumulant seriesThe purpose of this appendix is to provide a rigorous mathematical foundation for the transformation (37) and its inverse (39) of the main text. We omit subscripts and superscripts and write simply ρ and C. Our goal is to define the mapping ρ(ν) → C(t), specify its domain and range, prove that it is one-to-one and onto, and describe its inverse map C(t) → ρ(ν). Moreover, at the end of the section we restrict our attention to a rather generic particular case that is sufficient for the application explained in the main part of this paper.Let M + (R) and M(R) respectively denote the space of finite positive measures and the space of finite signed measures on the real line (cf. Sections 1.2 and 4.1 in [1]). Indeed, an element of M(R) is simply a difference of two elements from M + (R). This space contains all absolutely integrable real-valued functions, but it also contains the "true" measures, such as absolutely summable linear combinations of Dirac delta-functions at certain points of R. For any finite signed measure ρ ∈ M(R) we interpret (37)of the main text asThis transformation is well-defined since the function (e −iνt − 1 + iνt)/ν 2 can be extended at ν = 0 in a way that it becomes a bounded continuous function on the whole real line and thus it can be integrated against the measure ρ. The basics of integration theory with respect to abstract measures can be found in the classical textbooks on measure theory, such as [1] and [2]. Let us remark that integration with respect to a general measure ρ is usually written with ρ(ν)dν replaced by dρ(ν) or ρ(dν) in mathematical literature in order to emphasize that ρ need not have a density (i.e. it could be a true measure). We will stick to the former notation, since it is also common to understand signed (or even complex) measures as particular cases of distributions (cf. Section 6.11 in [3]). * Corresponding author. Email: branko@ifs.hr Differentiation of (A1) is justified using the dominated convergence theorem (cf. Theorem 2.27(b) from [2]) and it yieldsĊ In the literature on probability theory (such as Chapter 3 of [7]), ρ is usually called the characteristic function of ρ, even though the normalization is slightly different there. Substituting t = 0 into (A1) and (A2) givesFrom (A3)-(A6) we see that an equivalent way of expressing C in terms of ρ iswith the usual convention thatIt is easy to see that C(t) is always two times continuously differentiable, but it is not true that every such function can be obtained from some ρ ∈ M(R), as we shall soon see.Now we claim that the transform defined by (A1) is a one-to-one map from M(R) to some collection of twice continuously differentiable functions, i.e. that different measures ρ map to different functions C. In order to verify that we assume ρ 1 → C and ρ 2 → C. From (A3) we get2 so the well-known fact that the Fourier-Stieltjes transform ρ → ρ is one-to-one (see Subsection VI. for any choice of points t 1 , . . . , t n ∈ R an...

Vertex Corrections for Positive-Definite Spectral Functions of Simple Metals

“…It was then proposed that self-energy (SE) diagrams with vertex corrections may cancel the spurious SC effects [25,[30][31][32][33]. This fueled a number of notable attempts to include the vertex function in a model fashion: using the plasmon model for the screened interaction [34], neglecting the incoherent part of the electron spectral function [35], and employing the Ward identities and a model form of the exchange-correlation kernel [27,[36][37][38], or the cumulant expansions [39][40][41][42][43][44][45]. Although these methods clarified a number of issues, they do not provide an exhaustive picture [46].…”

“…It is conceivable that higher order satellites can be accurately described by the combination of our approach with the cumulant expansion scheme outlined in Refs. [41,45]. A related approach based on the equation of motion coupledcluster theory has been put forward recently [89].…”

Cumulant Green's function calculations of plasmon satellites in bulk sodium: Influence of screening and the crystal environment

“…[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] The CEA was inspired by the exact Green's function of a electron-boson model hamiltonian for a core level 21 and has been hence extensively used for core-level photoemission (see e.g. [22][23][24][25], and also in other contexts, as for the electron-phonon interaction and the polaron problem (see e.g.…”

SternheimerGW: A program for calculating GW quasiparticle band structures and spectral functions without unoccupied states

“…Progress in high-energy resolution angle-resolved photoelectron spectroscopy (ARPES) had led to a renewed interest in the spectral function of materials, in particular the signatures of electron correlations and bosonic excitations. These features were originally examined for the homogeneous electron gas, 9,[27][28][29][30] but have recently been investigated both experimentally and theoretically for charge carriers in semiconductors coupling to plasmons, 1,[31][32][33][34][35][36][37][38][39][40][41][42][43] Fröhlich polarons [44][45][46][47][48][49][50][51][52][53][54][55] or hybridizations of these excitations. 56,57 While the GW approximation does produce satellite features in the spectral function, their strength is overestimated and their energy is blue-shifted.…”

“…For these results the frequencydependent self-energy obtained with SternheimerGW is used as a starting point for a cumulant expansion, to obtain both quasiparticle band structure and plasmon replicas with the correct energy and intensity. 1,40 In Fig. 1b, we show the use of SternheimerGW to evaluate the Fermi surface of the transition metal dichalcogenide NbS 2 .…”