Nonperturbative Conserving Approximation
Scheme for Self-Consistent Calculation of
the Electron Self-Energy
Takada GroupThe electron self-energy
is a fundamental
quantity in the Green's-function approach to the
many-electron problem in solids. It is directly related
to the one-electron spectral function A(k, ) that canbe observed experimentally by angle-resolved
photoemission spectroscopy (ARPES). Accurate
information on(k, ) in the wide range ofmomentum k and energy
is useful not only for
elucidating the detailed Fermi-liquid (FL) properties
in usual metals but also for discussing thenon-Fermi-liquidbehavior
inherent
totheTomonaga-Luttinger liquid (TLL) in systems with one
spatial dimension.
Usually,
is calculated with use of many-body
perturbation-theoretic (MBPT) methods, such as theHedin’s GW approximation (GWA). This GWA is aconserving approximation in the Baym-Kadanoff'ssense, obeying the conservation laws of themacroscopic quantities like the total electron number,but because of the omission of all vertex corrections it
does not satisfy the Ward identity (WI), an exactrelation between and the vertex function due togauge
invariance
representing
thelocal
electron-number conservation. Under the violation ofthe WI, an accurate enough is not obtained in FL
metals. In TLL metals, the situation becomes evenworse, because any MBPT approach is inappropriate
from the outset for describing the non-perturbative
features of the TLL.
About a decade ago, a nonpertubative approach or
the GW
scheme was developed for theself-consistent calculation of
based on arigorous algorithm to obtain the exact
as a fixed
point of the self-energy revision operator [2]. This
scheme always satisfies the WI and is applicable toboth FL and TLL in a unified manner, but it
encountered a serious problem in the electron liquidwith the densities lower than the metallic ones;
convergent results for are not obtained, if its density
specified by the dimensionless parameter rs is larger
than 5.25 at which there appears the dielectric
catastrophe associated with the divergence of thecompressibility and concomitantly that of the static
polarization function in the long wave-length limit.
In view of this situation, we have revised theoriginal GW scheme by (i) giving the better
functional form for based on a detailed analysis of
the analytical structure of the exact , (ii) modifying
the scheme itself into the one free from the difficulty
originating from the dielectric catastrophe, and (iii)
proposing an alternative scheme on the condition thatan accurate piece of information is available for byother calculations or experiments [3]. As an example

to illustrate the power of our scheme, the results forthe momentum distribution function n(k) are shown inFig. 1 in the low-density electron gas for rs up to 8 incomparison with those recently obtained by thereptation Monte Carlo (RMC) method. In contrast to
the inaccuracy in RMC, our results are so accurate thatthey all satisfy the important sum rules of k k2nn(k)for n=0 (total number), n=1 (kinetic energy), and evenn=2 (fluctuation of kinetic energy) at least up to six
digits.

Fig. 1. Momentum distribution function n(k) for the electron
liquid at rs=1 (diamond), 2 (ellipse), 4 (plus), 5 (star), 6 (cross),and 8 (rectangle) in the GW method in comparison with thosefor rs=1 (longer dashed), 2 (dashed), 3.99 (dotted-dashed), 5(dotted), and 10 (solid) in the reptation Monte Carlo method.

In the future, we shall apply this scheme to realsolids, along with (i) pursuing further analysis of theemergence of electron-hole asymmetric excitations,leading to a very characteristic n(k) and a self-induced
excitonic state for rs > 8, and (ii) exhibitingtransmutation from bosonic to fermionic natures with
the increase of excitation energies in theone-dimensional TLL.
References[1] Y. Takada, Phys. Rev. Lett. 87, 226402 (2001).
[2] Y. Takada, Phys. Rev. B 52, 12708 (1995).
[3] H. Maebashi and Y. Takada, Phys. Rev. B 84,245134 (2011).
Authors
H. Maebashi and Y. Takada