Unified Green's function approach for spectral and thermodynamic properties from algorithmic inversion of dynamical potentials

Dynamical potentials appear in many advanced electronic-structure methods, including self-energies from many-body perturbation theory, dynamical mean-field theory, electronic-transport formulations, and many embedding approaches. Here, we propose a novel treatment for the frequency dependence, introducing an algorithmic inversion method that can be applied to dynamical potentials expanded as sum over poles. This approach allows for an exact solution of Dyson-like equations at all frequencies via a mapping to a matrix diagonalization, and provides simultaneously frequency-dependent (spectral) and frequency-integrated (thermodynamic) properties of the Dyson-inverted propagators. The transformation to a sum over poles is performed introducing $n\mathrm{th}$ order generalized Lorentzians as an improved basis set to represent the spectral function of a propagator. Numerical results for the homogeneous electron gas at the $G_0{W}_0$ level are provided to argue for the accuracy and efficiency of such unified approach.

Figure 1

Numerical example for the algorithmic inversion method on sum over poles. Upper panel: Real (blue) and imaginary (green) part of a time-ordered self-energy, having a SOP with 8 poles (the occupied pole of $G_0$ is not shown). Lower panel: Dyson-inverted propagator $G$ obtained with a numerical inversion on a grid (dotted) compared with the SOP representation obtained using the algorithmic-inversion method and evaluated on the same grid (solid line). The same color code for real and imaginary parts as in the upper panel is used.

Figure 2

Numerical example of a transformation to a SOP form. The target propagator is chosen to have a spectral function as a linear combination of three Gaussians, sampled using 50 points (black dots). We represent the trial spectral function on $1\text{st}$ (orange) and $2\text{nd}$ (green) order Lorentzian basis elements—each centered on the midpoint between adjacent grid points and broadened with the size of the interval—and get real and imaginary parts of the propagator analytically afterward (see Sec. 2b for details). We plot the result in the panels (a) and (b). In panel (c) we study the mean-square error (MSE) of the fitted propagator against the target as a function of the number of basis elements used for the representation. The last value on this graph represents the MSE of the (sum of the) panels (a) and (b) between the fitted and the target values. All the other points in panel (c) are obtained placing the Lorentzians uniformly in the given interval with a broadening equal to the distance between two consecutive functions. In the panels (d)–(f) the same study is repeated for the absolute difference between the normalization of the target function and the one of the fitted propagator, its mean, and its variance. The y scales of the graphs (c)–(f) are logarithmic.

Figure 3

Flow chart representing different strategies for the calculation of the self-energy given a Green's function $G$ on SOP as input for the heg_sgm.x code. The strategy used in this article is highlighted with green lines.

Figure 4

Flow chart representing different strategies for the calculation of the total-energy given a self-energy $\mathrm{\Sigma }$ on a frequency grid as input for the heg_sgm.x code. The strategy used in this article is highlighted with green lines.

Figure 5

Convergence study for the correlation energy per particle $E_{\mathrm{corr}}$ from a $G_0{W}_0$ calculation for the HEG at $r_s=4$. The parameters to converge are described in Sec. 3c. For each convergence curve (represented by a single line), we study the value of $E_{\mathrm{corr}}$ by varying the corresponding parameter, fixing all the others at the convergence point (called baseline calculation). At each step, we vary the target parameter by $20\%$ in the convergent direction and observe a plateau in $E_{\mathrm{corr}}$.

Figure 6

Imaginary part (divided by $\pi$ and taken without sign) of the screened potential of the HEG at $r_s=4$ in a one-shot $G_0{W}_0$ calculation. Momenta are in units of the Fermi momentum and energies in units of the plasma frequency of the gas at the specified density. The color map is logarithmic, and the values $<{10}^{-6}$ and $>{10}^3$ are mapped to ${10}^{-6}$ and ${10}^3$, respectively. Left panel: Analytic result at zero broadening ($\delta =0$). The functional form for the color plot is taken from Ref. [61]. The plasmonic band (dashed line) is calculated numerically, see Sec. 4a for details. Central panel: Analytic result at finite (Lorentzian) broadening ($\delta ={10}^{-5}{\epsilon }_f$), see Sec. pp3 of the Appendix for details. Right panel: Numerical results obtained using SOPs with $2\text{nd}$ order Lorentzians, see Sec. 4a for details.

Figure 7

Spectral function of the HEG at $r_s=4$ from a $G_0{W}_0$ calculation. The Fermi energy is ${\epsilon }_f=\frac{k_f^2}{2m_e}$ with $k_f$ the Fermi momentum. $\mu ={\epsilon }_f(1-0.0545)$ is the chemical potential. The scale of the color-map is logarithmic.

Figure 8

Selected frequency integrated quantities from a $G_0{W}_0$ calculation of the HEG at $r_s=4$. Panel (a): The occupation number in arbitrary units as a function of the momentum $k$ together with the renormalization factor (discontinuity of $n_k$). Panel (b): The occupied band energy $\langle {\epsilon }_k\rangle$ as a function of $k$ (see Sec. 3b for details). Panel (c): The Galitzki-Migdal total-energy resolved over $k$ contributions $e_k$, according to the rhs of Eq. (28) as function of the momentum $k$. $k_f$ and ${\epsilon }_f$ are the Fermi momentum and energy respectively.

Figure 9

Spectral functions of the HEG at selected $k$ points (indicated on top) for $r_s=2$ and $r_s=4$. Energies are in units of ${\epsilon }_f$ and momenta of $k_f$.

Figure 10

Spectral functions of the HEG at several densities. At the top $r_s$ specifies the density. The Fermi energy is ${\epsilon }_f=\frac{{\hslash }^2{k}_f^2}{2m_e}$ with $k_f$ the Fermi momentum. $\mu$ is the chemical potential. The color map is logarithmic.

Figure 11

Occupation factors in arbitrary units for several densities ($r_s$ from 1 to 10). $k_f$ is the Fermi momentum. In the inset we display the band width $b.w.$ in units of ${\epsilon }_f$, the effective mass $m_e^{*}$ in units of $2m_e$, the plasmaron frequency ${\omega }_{pp}$ in units of ${\epsilon }_f$, and the renormalization factor in arbitrary units at each density.

Figure 12

Correlation energy in Hartree units for several densities within the $G_0{W}_0$ approximation in the HEG. The data found in this paper are marked using green crosses. In blue we show the results from Ref. [42], and in orange from Ref. [26]. In solid green we plot the correlation energy fit from Eq. (30) (same functional form as in [67]) on the present (green) data. For reference, in dashed grey we add also the quantum Monte Carlo data obtained by Ceperley and Alder [68] using the fit made by Perdew and Zunger [67].

Figure 13

2D-projection of $\mathrm{\Omega }$ onto the $(k_z,k_y)$ plane. $\mathrm{\Omega }$ can be seen as the volume of revolution around the $k_z$ axis of the region delimited by the red line.

Figure 14

Convergence studies for the total energy at $r_s=\{1,2,3,5,6,7,8,9,10\}$ following the protocol detailed in Sec. 3c.