Low-scaling GW: The space-time formalism: Difference between revisions

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== References ==
== References ==
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[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Theory]]
 
[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:Low-scaling GW and RPA]][[Category:Theory]]

Latest revision as of 10:18, 22 September 2025

Available as of VASP.6 are low-scaling algorithms for ACFDT/RPA.[1] This page describes the formalism of the corresponding low-scaling GW approach.[2] A theoretical description of the ACFDT/RPA total energies is found here. A brief summary regarding GW theory is given below, while a practical guide can be found here.

Theory

The GW implementations in VASP described in the papers of Shishkin et al.[3][4] avoid storage of the Green's function [math]\displaystyle{ G }[/math] as well as Fourier transformations between time and frequency domain entirely. That is, all calculations are performed solely on the real frequency axis using Kramers-Kronig transformations for convolutions in the equation of [math]\displaystyle{ \chi }[/math] and [math]\displaystyle{ \Sigma }[/math] in reciprocal space and results in a relatively high computational cost that scales with [math]\displaystyle{ N^4 }[/math] (number of electrons).

The scaling with system size can, however, be reduced to [math]\displaystyle{ N^3 }[/math] by performing a so-called Wick-rotation to imaginary time [math]\displaystyle{ t\to i\tau }[/math].[5]

Following the low scaling ACFDT/RPA algorithms the space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space

[math]\displaystyle{ G({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}}\phi_{n{\bf k}}^{(0)}({\bf r}) \phi_{n{\bf k}}^{*(0)}({\bf r}') e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right] }[/math]

Here [math]\displaystyle{ \Theta }[/math] is the step function and [math]\displaystyle{ f_{n{\bf k}} }[/math] the occupation number of the state [math]\displaystyle{ \phi_{n{\bf k}}^{(0)} }[/math]. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid [math]\displaystyle{ \tau_{m} }[/math], where the number of time points can be selected in VASP via the NOMEGA tag. Usually 12 to 16 points are sufficient for insulators and small band gap systems.[6]

Subsequently, the irreducible polarizability is calculated from a contraction of two imaginary time Green's functions

[math]\displaystyle{ \chi({\bf r},{\bf r}',i\tau_m) = -G({\bf r},{\bf r}',i\tau_m)G({\bf r}',{\bf r},-i\tau_m) }[/math]

Afterwards, the same compressed Fourier transformation as for the low scaling ACFDT/RPA algorithms is employed to obtain the irreducible polarizability in reciprocal space on the imaginary frequency axis [math]\displaystyle{ \chi({\bf r},{\bf r}',i\tau_m) \to \chi_{{\bf G}{\bf G}'}({\bf q},i \omega_n) }[/math].[6][2]

The next step is the computation of the screened potential

[math]\displaystyle{ W_{{\bf G}{\bf G}'}({\bf q},i\omega_m)=\left[\delta_{{\bf G}{\bf G}'}-\chi_{{\bf G}{\bf G}'}({\bf q},i\omega_m)V_{{\bf G}{\bf G}'}({\bf q})\right]^{-1}V_{{\bf G}{\bf G}'}({\bf q}) }[/math]


followed by the inverse Fourier transform [math]\displaystyle{ W_{{\bf G}{\bf G}'}({\bf q},i \omega_n) \to \chi({\bf r},{\bf r}',i\tau_m) }[/math] and the calculation of the self-energy

[math]\displaystyle{ \Sigma({\bf r},{\bf r}',i\tau_m) = -G({\bf r},{\bf r}',i\tau_m)W({\bf r}',{\bf r},i\tau_m) }[/math]

From here, several routes are possible including all approximations mentioned above, that is the single-shot, EVG0 and QPEVG0 approximation. All approximations have one point in common.

In contrast to the real-frequency implementation, the low-scaling GW algorithms require an analytical continuation of the self-energy from the imaginary frequency axis to the real axis. In general, this is an ill-defined problem and usually prone to errors, since the self-energy is known on a finite set of points. VASP determines internally a Padé approximation of the self-energy [math]\displaystyle{ \Sigma(z) }[/math] from the calculated set of NOMEGA points [math]\displaystyle{ \Sigma(i\omega_n) }[/math] and solves the non-linear eigenvalue problem

[math]\displaystyle{ \left[ T+V_{ext}+V_h+\Sigma(z) \right]\left|\phi_{n\bf k}\right\rangle = z\left| \phi_{n\bf k} \right\rangle }[/math]

on the real frequency axis [math]\displaystyle{ z=\omega }[/math].

Because preceding Fourier transformations have been carried out with exponentially suppressed errors, the analytical continuation [math]\displaystyle{ \Sigma(z) }[/math] of the self-energy can be determined with high accuracy. The analytical continuation typically yields energies that differ less than 20 meV from quasi-particle energies obtained from the real-frequency calculation.[2]

In addition, the space-time formulation allows to solve the full Dyson equation for [math]\displaystyle{ G({\bf r,r'},i\tau) }[/math] with decent computational cost.[7] This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.

Finite temperature formalism

The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.[8] This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature [math]\displaystyle{ T\gt 0 }[/math], which may be understood by an analytical continuation of the real-time [math]\displaystyle{ t }[/math] to the imaginary time axis [math]\displaystyle{ -i\tau }[/math]. Matsubara has shown that this Wick rotation in time [math]\displaystyle{ t\to-i\tau }[/math] reveals an intriguing connection to the inverse temperature [math]\displaystyle{ \beta=1/T }[/math] of the system.[9] More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability [math]\displaystyle{ \chi(-i\tau) }[/math]) over the fundamental interval [math]\displaystyle{ -\beta\le\tau\le\beta }[/math].

As a consequence, one decomposes imaginary time quantities into a Fourier series with period [math]\displaystyle{ \beta }[/math] that determines the spacing of the Fourier modes. For instance the imaginary polarizability can be written as

[math]\displaystyle{ \chi(-i\tau)=\frac1\beta\sum_{m=-\infty}^\infty \tilde \chi(i\nu_m)e^{-i\nu_m\tau},\quad \nu_m=\frac{2m}\beta\pi }[/math]

and the corresponding random-phase approximation of the correlation energy at finite temperature becomes a series over (in this case, bosonic) Matsubara frequencies

[math]\displaystyle{ \Omega_c^{\rm RPA}=\frac12\frac1\beta \sum_{m=-\infty}^\infty {\rm Tr}\left\lbrace \ln\left[ 1 -\tilde \chi(i\nu_m) V \right] -\tilde \chi(i\nu_m) V \right\rbrace,\quad \nu_m=\frac{2m}\beta\pi }[/math]

The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential [math]\displaystyle{ \epsilon_{n{\bf k}}\approx \mu }[/math], such that Matsubara series also converge for metallic systems.

Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.[10] This approach converges exponentially with the number of considered frequency points.

References