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Physical Review B

Cumulant expansion of the retarded one-electron Green function

Joshua J. Kas, John J. Rehr, Lucia Reining

Published Aug 11, 2014• 106 citations
Publisher

Abstract

The cumulant expansion is a powerful approach for including correlation effects in electronic structure calculations beyond the $GW$ approximation. However, the expansion is not generally valid, as current implementations ignore terms that mix particle and hole states and lead to partial occupation numbers of one-electron states. These limitations are corrected here using a cumulant expansion of the retarded one-electron Green's function that includes both particle and hole contributions. The approach provides a consistent framework that improves on the $GW$ approximation to the spectral function without additional computational effort. The method is illustrated with results for the homogeneous electron gas and comparisons to experiment and other methods.

DOI
10.1103/physrevb.90.085112

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The cumulant expansion is a powerful approach for including correlation effects in electronic structure calculations beyond the GWGWGW approximation. However, the expansion is not generally valid, as current implementations ignore terms that mix particle and hole states and lead to partial occupation numbers of one-electron states. These limitations are corrected here using a cumulant expansion of the retarded one-electron Green's function that includes both particle and hole contributions. The approach provides a consistent framework that improves on the GWGWGW approximation to the spectral function without additional computational effort. The method is illustrated with results for the homogeneous electron gas and comparisons to experiment and other methods.
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.
A cumulant-based GWGWGW approximation for the retarded one-particle Green's function is proposed, motivated by an exact relation between the improper Dyson self-energy and the cumulant generating function. Qualitative aspects of this method are explored within a simple one-electron independent phonon model, where it is seen that the method preserves the energy moment of the spectral weight while also reproducing the exact Green's function in the weak-coupling limit. For the three-dimensional electron gas, this method predicts multiple satellites at the bottom of the band, albeit with inaccurate peak spacing. However, its quasiparticle properties and correlation energies are more accurate than both previous cumulant methods and standard G0W0{G}_{0}{W}_{0}G0​W0​. Our results point to features that may be exploited within the framework of cumulant-based methods and suggest promising directions for future exploration and improvements of cumulant-based GWGWGW approaches.
We present a finite-temperature extension of the retarded cumulant Green's function for calculations of exited-state, correlation, and thermodynamic properties of electronic systems. The method incorporates a cumulant to leading order in the screened Coulomb interaction W, and improves on the GW approximation of many-body perturbation theory. Results for the homogeneous electron gas are presented for a wide range of densities and temperatures, from cool to warm dense matter regimes, which reveal several hitherto unexpected properties. For example, correlation effects remain strong at high T while the exchange-correlation energy becomes small; also the spectral function broadens and damping increases with temperature, blurring the usual quasiparticle picture. These effects are evident, e.g., in Compton scattering which exhibits many-body corrections that persist at normal densities and intermediate T. The approach also yields exchange-correlation energies and potentials in good agreement with existing methods.
Thermodynamic properties of the interacting homogeneous electron gas are calculated using a finite-temperature cumulant Green's function approach over a broad range of densities and temperatures up to the warm dense matter regime $T\ensuremath{\sim}{T}_{F}$, where TF{T}_{F}TF​ is the Fermi degeneracy temperature. These properties can be separated into independent particle and exchange-correlation contributions, and our focus here is on the latter. Our approach is based on the Galitskii-Migdal-Koltun and electron number sum rules from the finite temperature many-body Green's function formalism, together with an extension of the cumulant Green's function to finite temperature. Previously this approach yielded exchange-correlation energies and potentials in good agreement with quantum Monte-Carlo calculations. Here the method is extended for various thermodynamic quantities including the chemical potential, total energy, Helmholtz free-energy, electronic equation of state, specific heat, and isothermal compressibility, which optionally include spin dependence. We find that the exchange-correlation contributions are weakly varying at low temperature but exhibit significant temperature dependence in the WDM regime, as well as a crossover from exchange- to correlation-dominated behavior. In contrast to the T=0+T={0}^{+}T=0+ limit, we also find that renormalization effects are largely but not completely suppressed at finite temperature. Comparisons with other approaches at various levels of approximation are also discussed.
Strong electron correlation remains difficult to describe within conventional density functional approximations, particularly when natural-orbital occupations become strongly fractional. We introduce a systematic particle-hole expansion that generalizes density functional theory (DFT) through an explicit functional of the one-electron reduced density matrix. The previously derived cumulant-based correction emerges as the leading term and the thermal (entropic) correction as an infinite-order member whose coefficients are fixed by a single parameter. The expansion can be formulated as a moment problem solved by semidefinite programming with computational scaling comparable to DFT. We show that both the first- and second-order corrections yield closed-form natural-orbital occupations. Applications to periacenes and Fe-porphyrin demonstrate that the second-order correction substantially reduces excessive orbital fractionalization, yielding natural-orbital occupations and fundamental gaps in improved agreement with correlated calculations.
Abstract We use the cumulant Green’s functions method (CGFM) to study the single-band Hubbard model. The starting point of the method is to diagonalize a cluster (‘seed’) containing N correlated sites and employ the cumulants calculated from the cluster solution to obtain the full Green’s functions for the lattice. All calculations are done directly; no variational or self-consistent process is needed. We benchmark the one-dimensional results for the gap, the double occupancy, and the ground-state energy as functions of the electronic correlation at half-filling and the occupation numbers as functions of the chemical potential obtained from the CGFM against the corresponding results of the thermodynamic Bethe ansatz and the quantum transfer matrix methods. The particle-hole symmetry of the density of states is fulfilled, and the gap, occupation numbers, and ground-state energy tend systematically to the known results as the cluster size increases. We include a straightforward application of the CGFM to simulate the singles occupation of an optical lattice experiment with lithium-6 atoms in an eight-site Fermi-Hubbard chain near half-filling. The method can be applied to any parameter space for one, two, or three-dimensional Hubbard Hamiltonians and extended to other strongly correlated models, like the Anderson Hamiltonian, the t − J, Kondo, and Coqblin-Schrieffer models.