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arXiv:2201.02854 · Physical review. B./Physical review. B

Time-dependent exchange-correlation potential in lieu of self-energy

F. Aryasetiawan

Published Feb 2, 2022• 7 citations
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Abstract

The equation of motion of the Green's function of a many-electron system is shown to be governed by a multiplicative exchange-correlation potential, in contrast to the self-energy. Crucially, this exchange-correlation potential is simply the Coulomb potential of an exchange-correlation hole and fulfills a sum rule. The formulation offers a simple physical interpretation for the propagation of an added electron or hole and may provide a route for density-functional-like approximations for calculating the Green's function.

DOI
10.1103/physrevb.105.075106

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The equation of motion of the Green's function of a many-electron system is shown to be governed by a multiplicative exchange-correlation potential, in contrast to the self-energy. Crucially, this exchange-correlation potential is simply the Coulomb potential of an exchange-correlation hole and fulfills a sum rule. The formulation offers a simple physical interpretation for the propagation of an added electron or hole and may provide a route for density-functional-like approximations for calculating the Green's function.
The one-particle Green function of a many-electron system is traditionally formulated within the self-energy picture. A different formalism was recently proposed, in which the self-energy is replaced by a dynamical exchange-correlation field, which acts on the Green function locally in both space and time. It was found that there exists a fundamental quantity, referred to as the dynamical exchange-correlation hole, which can be interpreted as effective density fluctuations induced in a many-electron system when a hole or an electron is introduced into the system, as in photoemission and inverse photoemission experiments. The dynamical exchange-correlation potential is simply the Coulomb potential of this exchange-correlation hole, which fulfils a sum rule and an exact constraint, identical to those satisfied by the static exchange-correlation hole in density-functional theory. The proposed formalism has been applied to a number of model systems such as the half-filled one-dimensional Hubbard model, the one-dimensional antiferromagnetic Heisenberg model, and the single-impurity Anderson model. The dynamical exchange-correlation hole and field of the homogeneous electron gas have also been studied with the view of constructing a density-functional approximation such as the local-density approximation. The availability of simple but accurate approximations for the exchange-correlation potential would circumvent costly computations of the traditional self-energy. The formalism may also provide new perspectives and insights into the many-body problem.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
We propose a new approach to the self-consistency equation, which arises in the problem of the motion of a hole in a quantum antiferromagnet, appropriate to the case of small exchange energy JJJ. The functional equation for the Green function is transformed into a differential equation; its solutions are analyzed and compared to the existing numerical calculations. This method allows one to study the limit of J→0J\to 0J→0. Application to other strongly correlated electron systems is discussed.
We propose a description of nonequilibrium systems via a simple protocol that combines exchange-correlation potentials from density functional theory with self-energies of many-body perturbation theory. The approach, aimed to avoid double counting of interactions, is tested against exact results in Hubbard-type systems, with respect to interaction strength, perturbation speed and inhomogeneity, and system dimensionality and size. In many regimes, we find significant improvement over adiabatic time dependent density functional theory or second Born nonequilibrium Green's function approximations. We briefly discuss the reasons for the residual discrepancies, and directions for future work.
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.