GW Method for the Self-Energyof a Many-Electron System
In the last two decades, we have witnessed
the power of quantum Monte Carlo (QMC)
simulations that allow us to obtain virtuallyexact information on the static properties
associated with the ground-state wave
function of a quantum many-body system.
However, QMC cannot be used to evaluate
dynamical properties, because dynamics
requires information on excited-state wave
functions as well. Thus we need to develop
an alternative algorithm for dynamics.
Four decades ago, Baym and Kadanoff
proposed a conserving approximationscheme by formulating the dynamical
problem in terms of an energy functional[G] with G the full Green's function. The
self-energy is obtained by = [G] / G.
One can choose [G] at one’s disposal.
Although it is found to be very useful inmany cases, this scheme never provides
the exact , because no algorithm is known
to give exact [G].
In 1995, I developed a conceptually new
scheme to obtain the exact , named theself-energy revision operator theory [1];instead of pursuing [G], I paid attention tothe exact functional relations betweenand the vertex function , obeying themicroscopic conservation law. In particular,
is determined through the Bethe-Salpeter
equation with an irreducible electron-holeinteraction given by the functional
derivative, / . Starting from an arbitrary
input, both and are iteratively revised
towards self-consistency through thefunctional relations. Note that the numberof terms representing
generated in thisiterative
process
rapidlyincreases,
eventually covering all terms derivable fromthe exact [G] when the self-consistency isachieved.
In a numerical algorithm, however, thefunctional differentiation/
is notfeasible, prompting me to invent analternative scheme to revise
on acomputer in an accurate and efficient way.

Recently such a scheme is provided, giving
an accurate enough result for , in which
the Ward identity is satisfied automatically[2,3]. This method allows us to obtainaccurate results for the dynamicalquantities, promising us to open a new
dimension to the quantum many-bodyproblems.
My calculation scheme is schematicallyshown in Figure 1, which indicates that thismethod may be regarded as an extensionof the Hedin’s GW method by includingthat is determined self-consistently with .
Thus it is named the GW method.

Figure 1: Schematic representation of myself-consistent calculation scheme for theself-energy , where V is the bare Coulombinteraction, G(0) is the bare Green’s function,is the polarization function, and fxc is theexchange-correlation kernel appearing in thetime-dependent density functional theory.

As an illustration, my method isimplemented in the homogeneous electron
gas at the density parameter rs = 4
corresponding to sodium. The obtainedsingle-electron spectral function A(p, )(which is given by –ImG(p, )/ ) is shown inFigure 2. The main peak represents thestructure of a quasiparticle. Its width is very
sharp at and near the Fermi surface [Case(a)], indicating the existence of awell-defined quasiparticle. However, it isvery broad far away from the Fermi surface[Case (b)]. In fact, its tail connects
continuously with the plasmon-side band or
the plasmaron (an entity composed of thereal plasmon and the quasiparticle),
implying a strongly decaying nature of thequasiparticle. Note that without satisfying
the Ward identity or the microscopic
conservation law, this plasmaron peakdoes not come to the right energy position.

Figure 2: A(p, ) for the homogeneous electron
gas at the sodium density (a) at the Fermi
surface (p=pF) and (b) at the band bottom(p=0).

My result of A(p, ) discloses the intriguing
complexity of the single-electron spectral
function even in such a simple system as
the electron gas, urging us to make a moredetailed analysis of the experimental dataobtained by angle-resolved photoemission
spectroscopy (ARPES) in various materials
widely ranging from simple to transition
metals.
References[1] Y. Takada, Phys. Rev. B 52, 12708(1995).
[2] Y. Takada, Int. J. Mod. Phys. B 15, 2595
(2001).
[3] Y. Takada, Phys. Rev. Lett. 87, 226402(2001).
AuthorY. Takada.