Generalized particle/hole cumulant approximation for the electron Green’s function
The cumulant expansion is a powerful approach for including correlation effects in electronic structure calculations beyond the GW approximation. However, current implementations are incomplete since they ignore terms that lead to partial occupation numbers and satellites both above and below the Fermi energy. These limitations are corrected here with a generalized cumulant approximation that includes both particle and hole contributions within a retarded Green’s function formalism. The computational effort is still comparable to GW, and the method can be extended easily to finite temperature. The approach is illustrated with calculations for the homogeneous electron gas and comparisons to experiment and other methods.
One of the major challenges in condensed matter theory is to capture the effects of electron-electron interactions. Such many-body effects are responsible for the renormalization of energies and redistribution of spectral weight, but they also lead to new features such as satellite structures in the spectral function and partial occupation numbers. These features arise from the coupling of electrons to excitations (e.g., plasmons) that mix particle and hole states and cannot be captured by any independent-particle description. While this coupling can be treated formally, e.g., using many-body perturbation theory for the electron Green’s function , such expansions often converge poorly. Thus it is often preferable to introduce some auxiliary quantity from which is obtained. This is a general strategy in many-body theory, a prominent example being the Dyson equation , where the auxiliary quantity is the electron self-energy . The self-energy is then expanded to low order, most commonly via the GW approximation of Hedin Hedin (1999); Lundqvist (1967), while the Green’s function contains contributions from diagrams of all orders. Another example is the so-called cumulant expansion Kubo (1962), based on an exponential ansatz for the Green’s function , where the cumulant is now the auxiliary quantity, and practical calculations are caried out, again with an appropriate low order approximation, this time for . However both approaches have limitations. To go beyond, one must answer three questions: (i) What is the best fundamental quantity to calculate; (ii) What is the best ansatz, e.g., what auxiliary quantity should be used?; and (iii) What is the optimal approximation for that quantity? It is desirable that the development be exact in principle, and that even a simple approximation gives good results. To answer these questions here, we show that a generalized cumulant (GC) ansatz for the retarded Green’s function including both particle- and hole-branches is particularly advantageous and improves on both the GW Hedin (1980) and previous cumulant approximations Almbladh and Hedin (1983); Hedin (1980); Aryasetiawan et al. (1996); Guzzo et al. (2011) (see Fig. 1).
The GW achieves its efficiency by expanding in the screened – rather than the bare – Coulomb interaction , and retaining only the leading term. This is accomplished by summing certain classes of diagrams, e.g., the “bubble-diagrams” in the random phase approximation (RPA) for . Yet while GW gives very good quasi-particle properties, it often gives a poor description of the spectral function and its satellites Aryasetiawan et al. (1996); Guzzo et al. (2011). The usual procedure to overcome such difficulties, is to try to improve (iii), i.e., search for higher order approximations such as vertex corrections. However, as yet, no practical direct approximation for the vertex has been found. In addition, physical properties such as positive spectral weight and normalization are violated at 2nd order in Hedin (1980); Bergersen et al. (1972); Minnhagen (1975). These shortcomings have led an exponential ansatz, i.e., with the cumulant instead of as the auxilliary quantity in (ii) Hedin (1999); Minnhagen (1975); Bergersen et al. (1972). The exponential form is motivated by analogy to the case for core-electrons coupled to bosonic excitations Langreth (1970); Ness et al. (2011), where the cumulant expansion is exact. The exponential representation is physically appealing as it systematically includes higher order diagrams that serve implicitly as dynamical vertex corrections Guzzo et al. (2011). This strategy is found to be advantageous in many cases, e.g., systems of electrons coupled to plasmons and phonons Hedin (1999), multiple plasmon-satellites in photoemission Guzzo et al. (2011, 2012); Lischner et al. (2013); Aryasetiawan et al. (1996); Gunnarsson et al. (1994); Fujikawa et al. (2008), the time-evolution of excitations Pavlyukh et al. (2013), correlation energies Holm and Aryasetiawan (2000), and dynamical mean field theory Casula et al. (2012). Exponential forms are also found in a wider context, e.g. in summations over vacuum bubbles based on the linked cluster theorem Nozières and de Dominicis (1969), coupled cluster methods Shavitt and Bartlett (2009), the Thouless theorem for determinantal wave functions Thouless (1972), and the Landau formula for energy loss Landau (1944).
Despite these successes, many difficulties remain. Formal proofs of the validity of the cumulant expansion are often lacking, and its behavior can be pathological Mahan (2000). Indeed, as discussed below, none of the formulations proposed for valence spectra Almbladh and Hedin (1983); Hedin (1980); Gunnarsson et al. (1994); Aryasetiawan et al. (1996); Guzzo et al. (2011) is fully satisfactory. Neither is the method appropriate for interactions that cannot be treated in terms of bosonic excitations Gunnarsson et al. (1994). In order to motivate our GC approximation we briefly describe the standard time-ordered (TO) cumulant and its limitations. The cumulant ansatz for the zero-temperature TO Green’s function is given by
where is the non-interacting Green’s function, and are Fermionic creation and destruction operators, and is the TO cumulant. A serious problem with Eq. (1) is that vanishes for negative (positive) times for particles (holes), and hence gives no contribution to the spectral function below (above) the quasi-particle peak. Consequently the occupation numbers remain unchanged from their non-interacting values Holm and Aryasetiawan (2000). This unphysical behavior follows from the form of the non-interacting TO Green’s function , where the upper (lower) sign refers the particle (hole) branch. These defects can be traced to the neglect of diagrams with negatively propagating intermediate states Quinn and Ferrell (1958), e.g., due to recoil Hedin (1980). The missing terms account for partial occupation numbers , and are a general property of interacting Fermi systems, as observed e.g., in Compton scattering. Such terms are crucial to understanding correlation effects since is typically above in condensed matterHedin (1999).
To overcome these difficulties, a different strategy for question (i) is needed. Instead of , we take the fundamental quantity of interest to be the retarded Green’s function , with a cumulant ansatz analogous to Eq. (1) but with a generalized cumulant that includes particle and hole branches on an equal footing. This strategy is referred to here as the generalized cumulant (GC) approximation. Remarkably, many of the difficulties with the TO form disappear with this formulation, yet the approximate expression for remains simple. Thus a seemingly small change in the starting point has dramatic quantitative and qualitative consequences. In particular the GC permits calculations of electronic properties that depend on both branches, including occupation numbers, density matrices, and correlation energies. To achieve a practical method we approximate by expanding to first order in [cf. Ref. Gunnarsson et al., 1994],
where is the retarded self energy. This approximation is the dominant many-body correction in the theory and can be related to a quasi-boson treatment of the excitations of the system. The integral in Eq. (2) is easily evaluated in the lower-half frequency plane with and the spectral representation of the self-energy
where the static Hartree-Fock self-energy is separated out. Carrying out the integrations then yields
where , and is the dynamic part of is found by replacing with in Eq. (2). Finally, the spectral function is
While the above equations are similar to the TO formulae Aryasetiawan et al. (1996), a major difference lies in the excitation spectrum , where . While the GC contains all frequencies and builds in particle-hole symmetry, the TO forms only contain or for particles or holes, respectively. Consequently the spectral functions are also substantially different (see Fig. 1). The simplicity of the GC allows one to check that the basic requirements and sum-rules are fulfilled. Thus , so that is normalized to unity, and , so is always positive. In addition, so the spectral function has a centroid at the unperturbed Hartree-Fock energy , consistent with a one-shot calculation of . One also easily obtains the renormalization constant , quasi-particle energy shift , and occupation numbers ,
where the chemical potential is fixed by enforcing total occupation . The primary many-body ingredient in the GC is , the imaginary part of the retarded self energy , where is defined by a given screening approximation and has a structure that reflects peaks in the loss function . Thus the computational effort in the GC is comparable to that in . Going to higher order is technically difficult and not necessarily an improvement, since higher order terms can lead to non-physical behavior in Gunnarsson et al. (1994); Hedin (1980). Practical calculations of can be carried out using methods based on dielectric response and fluctuation-potentials Hedin (1999), e.g., by analytical continuation of the Matsubara self-energy Schöne and Eguiluz (1998), or at zero temperature from the TO self-energy. The constant describes the reduction in strength of the quasi-particle peak and agrees to 1st order in with that for where .
Physically the behavior of the GC in Eq. (4) can be interpreted as a transfer of spectral weight away from the quasi-particle peak by quasi-boson excitations of frequency . The “shake-up” counts correspond to the mean number of bosons coupled to the electron (or hole), and account for the satellite strengths in above () and below () the quasi-particle peak. In cases where it is necessary to introduce a principle-value integral or convolution procedure to avoid singular contributions in quasiparticle properties, leading to a Fano-lineshape of the quasi-particle peak Aryasetiawan et al. (1996). To further interpret the GC and compare to previous approaches, it is useful examine various limits. Due to the separation , a complete calculation of requires a Fourier transform of , where are the TO cumulants which contain instead of . Since one of the branches is always small (except close to ), and vanishes far from , one can estimate the contributions separately using the identity
For example, for hole spectra (), the leading term corresponds to the TO cumulant Hedin (1999); Aryasetiawan et al. (1996), to which the GC reduces when (Fig. 1). Interestingly, the cumulant is identical to that found within the recoil approximation of Hedin Hedin (1980). The next terms correspond to the minor branch of Ref. Gunnarsson et al., 1994. However, that approximation does not conserve spectral weight, and the remaining terms in Eq. (7) that mix particles and holes are needed to preserve normalization.
Here we illustrate the GC with explicit results for the homogeneous electron gas at zero temperature. The case with corresponds to bcc Na, which is widely used in theoretical comparisons Lundqvist (1967); Aryasetiawan et al. (1996); Holm and Aryasetiawan (2000); Ness et al. (2011). For consistency we use the RPA approximation for the screened Coulomb interaction as in Ref. Lundqvist, 1967, and we checked that our results agree to high accuracy with previous calculations. The integrations involved in calculating the cumulants, occupation numbers, and total energies were performed using the trapezoidal rule, except near where the integrand was expanded to avoid the singular point. Fourier transforms from time to frequency were performed with minimal Gaussian broadening. Integrals were converged with respect to range and spacing of points to sufficient accuracy for all values reported here. Fig. 1 shows from the GC for a range of compared to the standard TO and (thin solid line) approximations. The largest discrepancy between the GC and TO forms is near , where the GC exhibits a nearly symmetrical particle-hole spectrum, consistent with a reduction of the jump in at the Fermi surface from its non-interacting value. As expected, the TO agrees with the GC far from , so that previous cumulant treatments are preserved in that limit. Note too that the quasiparticle peak has substantial broadening at large due to the onset of particle-hole and plasmon excitations. In all cases, differs markedly from the approximation, thus demonstrating the importance of vertex corrections. The differences are especially noticable at , where GC and TO exhibit multiple plasmon peaks; in contrast has only one sharp “plasmaron” peak, in qualitative disagreement with experiment Aryasetiawan et al. (1996); Guzzo et al. (2011); Lischner et al. (2013). The inset in Fig. 1 shows a nearly dispersionless satellite at not predicted by TO or GW. This feature may be experimentally observable, e.g., via ARPES, and would provide an additional measure of correlation effects.
Values of and are important diagnostics of the quality of a given many-body approximation Holzmann et al. (2011); Vogt et al. (2004); Ness et al. (2011). The are also central ingredients in the one-body density matrix. Fig. 2 shows from GC compared to GW for an electron gas with , together with values extracted from Compton scattering data for Na Huotari et al. (2010) and QMC Huotari et al. (2010).
The GC gives values of in reasonable agreement with GW and quantum Monte Carlo (QMC) though slightly lower for . They are also consistent with, though somewhat higher than, Compton data above . The calculated renormalization constant is shown in the inset to Fig. 2, and Table 1 summarizes results at compared to , , and QMC for a range of Holzmann et al. (2011); Holm and von Barth (1998). We find reasonable agreement between the GC and QMC at higher densities and a larger discrepancy at smaller values. Interestingly, results compare well with QMC, while those for self consistent are too large, confirming that correlation effects are underestimated by GW.
Finally we present a GC calculation of electron correlation energies using the Galitskii-Migdal formula. This formula was previously applied to the TO cumulant approximation in Ref. Holm and Aryasetiawan, 2000. Assuming a paramagnetic system the total energy is given by
The correlation energy per particle is defined as , where the total Hartree-Fock energy is
where . For example, for the electron gas at , and , where here and below we use Hartree atomic units with energies in Hartrees = 27.2 eV. Table 2 presents correlation energies calculated from Eq. (Generalized particle/hole cumulant approximation for the electron Green’s function) and (9) for from to . For comparison we also show results for TO, , , and QMC. For all cases GC yields improved correlation energies compared to , and also improves over TO for . Interestingly some correlation energies reported in Ref. Holm and Aryasetiawan, 2000 are also close to QMC; however, this agreement may be fortuitous as their their prescription uses some approximations beyond TO.
In conclusion, we have presented a generalized cumulant approximation based on a retarded one-particle Green’s function formalism with a cumulant exact to first order in . The GC provides a consistent framework for the electron Green’s function that yields partial occupations, multiple satellites in the spectral function on both sides of the Fermi energy, and total energies, all in reasonable agreement with available theoretical and experimental data. This improves on the GW and previous TO cumulant approximations, each of which fails to account for one or more of those properties. The method gives an improved treatment of the and other one-electron properties, especially near , and thereby provides insights into the nature of vertex corrections. Moreover, the approach is easily extended to finite temperature Faleev et al. (2006); Schöne and Eguiluz (1998). Thus the GC provides an attractive approach for going beyond GW without additional computational complexity, and points to the utility of the retarded Green’s function formalism. Results for the homogeneous electron gas show that this level of theory gives correlation energies that quantitatively improve on compared to QMC. However, they are still slightly large, and the renormalization constants are too small. However, the GC also permits some freedom in the choice of initial one-particle states which could be used to include self-consistency, as in the quasiparticle self-consistent GW method van Schilfgaarde et al. (2006). Based on differences between self-consistent and von Barth and Holm (1996); Takada (2001), it is plausible that part of the remaining discrepancy between GC and QMC can be explained by the present lack of self-consistency, a point to be investigated in the future. Other extensions, e.g., the cumulant expansion for phonons and two-particle excitations Hedin (1999); Guzzo et al. (2011) are also reserved for the future.
Acknowledgements.We thank F. Aryasetiawan, G. Bertsch, S. Biermann, M. Casula, M. Gatti, E.K.U. Gross, M. Guzzo, V. Pavlyukh, D.J. Thouless, and others in the European Theoretical Spectroscopy Facility, for useful comments. This work was supported by DOE Grant DE-FG03-97ER45623 (JJR and JJK), by ERC advanced grant SEED (LR), and was facilitated by the DOE Computational Materials Science Network. One of us (JJR) thanks the Laboratories des Solides Irradíes at the École Polytechnique, Palaiseau for hospitality during part of this work.
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