GW approximation

The GW approximation (GWA) is an approximation made in order to calculate the self-energy of a many-body system of electrons. The approximation is that the expansion of the self-energy Σ in terms of the single particle Green's function G and the screened Coulomb interaction W (in units of \hbar=1)

\Sigma = iGW - GWGWG + \cdots

can be truncated after the first term:

\Sigma \approx iG W

Another way to say the same thing is that that self-energy is expanded in a formal Taylor series in powers of the screened interaction W and the lowest order term is kept in the expansion in GWA.

The above formulae are schematic in nature and show the overall idea of the approximation. More precisely, if we label an electron coordinate with its position, spin, and time and bundle all three into a composite index (the numbers 1, 2, etc.), we have

\Sigma(1,2) = iG(1,2)W(1^+,2) - \int d3 \int d4 \, G(1,3)G(3,4)G(4,2)W(1,4)W(3,2) + ...

where the "+" superscript means the time index is shifted forward by an infinitesimal amount. The GWA is then

\Sigma(1,2) \approx iG(1,2)W(1^+,2)

To put this in context, if one replaces W by the bare Coulomb interaction (i.e. the usual 1/r interaction), one generates the standard perturbative series for the self-energy found in most many-body textbooks. The GWA with W replaced by the bare Coulomb yields nothing other than the Hartree–Fock exchange potential (self-energy). Therefore, loosely speaking, the GWA represents a type of dynamically screened Hartree–Fock self-energy.

In a solid state system, the series for the self-energy in terms of W should converge much faster than the traditional series in the bare Coulomb interaction. This is because the screening of the medium reduces the effective strength of the Coulomb interaction: for example, if one places an electron at some position in a material and asks what the potential is at some other position in the material, the value is smaller than given by the bare Coulomb interaction (inverse distance between the points) because the other electrons in the medium polarize (move or distort their electronic states) so as to screen the electric field. Therefore, W is a smaller quantity than the bare Coulomb interaction so that a series in W should have higher hopes of converging quickly.

To see the more rapid convergence, we can consider the simplest example involving the homogeneous or uniform electron gas which is characterized by an electron density or equivalently the average electron-electron separation or Wigner-Seitz radius r_s . (We only present a scaling argument and will not compute numerical prefactors that are order unity.) Here are the key steps:

\epsilon(q) = 1 + \lambda^2/q^2

where \lambda is the screening wave number that scales as r_s^{-1/2}

Thus for the bare Coulomb interaction, the ratio of Coulomb to kinetic energy is of order r_s which is of order 2-5 for a typical metal and not small at all: in other words, the bare Coulomb interaction is rather strong and makes for a poor perturbative expansion. On the other hand, the ratio of a typical W to the kinetic energy is greatly reduced by the screening and is of order r_s/(1+r_s) which is well behaved and smaller than unity even for large r_s : the screened interaction is much weaker and is more likely to give a rapidly converging perturbative series.

Software implementing the GW approximation

References