Frequency dependence in the GW approximation

In this tutorial we present and compare the methods available in Yambo to treat the frequency dependence of key quantities for a GW calculation. We will use bulk hBN as example system.

Requirements

• SAVE folder of bulk hBN: download and extract hBN.tar.gz from the Tutorial files page.

• yambo executable (version \(\geq\)5.2)

• gnuplot for plotting some of the results

It is assumed that you are familiar with the concepts related to the SAVE folder and handling files with Yambo.

See also

• Quasiparticles corrections in the GW approximation

In Yambo, the frequency integration appearing in the expression of the GW self-energy can be handled in different ways:

• Plasmon Pole Approximation (PPA): the dynamic screening matrix \(\epsilon^{-1}_{\boldsymbol{GG}'}(\boldsymbol{q},\omega)\) is expressed as a single-pole function. The parameters for this representation are obtained through a fit enabled by the explicit evaluation of the inverse dielectric matrix at two frequencies, the static limit \(\omega = 0\) and another imaginary frequency \(\omega = i\omega_p\) - given in the input file - which is ideally comparable with the plasma frequency of the material. With this representation the solution of the frequency integral is analytical.

• Full Frequency Real-Axis (FF-RA): here the frequency integral is calculated numerically by explicitely evaluating \(\epsilon^{-1}_{\boldsymbol{GG}'}(\boldsymbol{q},\omega)\) for a number of real frequencies - given in the input, typically in the order of 100 - uniformly distrbuted in the energy range given by the range of bands included in the calculation of the response function.

• Multi-Pole Approximation (MPA): this can be thought as a generalization of the PPA able to provide FF-RA accuracy at a much lower computational cost. This approach builds from the representation of the polarizability \(\chi\) as the sum of a finite and small number of poles. The parameters for this representation are obtained solving a nonlinear system of equations defined by the explicit evaluation of the polarizability on a number of frequencies that is twice the number of poles. This provides an analytical representation of \(\chi\) over the whole complex plane which in turn allows to compute the self-energy analytically.

For a more detailed description of these methods see the Yambo code papers[1] (PPA, FF-RA) and the papers presenting the MPA method[2].

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[1]

A. Marini, et al., “yambo: An ab initio tool for excited state calculations”, Computer Physics Communications 180.8 (2009): 1392-1403; D. Sangalli, et al., “Many-body perturbation theory calculations using the yambo code”, J.Phys.:Condens. Matter 31 325902 (2019).

[2]

D. A. Leon, et al., “Frequency dependence in GW made simple using a multipole approximation”, Phys. Rev. B 104 115157 (2021); D. A. Leon, et al., “Efficient full frequency GW for metals using a multipole approach for the dielectric screening”, Phys. Rev. B 107 155130 (2023).