One-particle spectral function and analytic continuation for many-body implementation in the exact muffin-tin orbitals method

We investigate one of the most common analytic continuation techniques in condensed matter physics, namely the Padé approximant. Aspects concerning its implementation in the exact muffin-tin orbitals (EMTO) method are scrutinized with special regard towards making it stable and free of artificial defects. The electronic structure calculations are performed for solid hydrogen, and the performance of the analytical continuation is assessed by monitoring the density of states constructed directly and via the Padé approximation. We discuss the difference between the k-integrated and k-resolved analytical continuations, as well as describing the use of random numbers and pole residues to analyze the approximant. It is found that the analytic properties of the approximant can be controlled by appropriate modifications, making it a robust and reliable tool for electronic structure calculations. At the end, we propose a route to perform analytical continuation for the EMTO+dynamical mean field theory method.

Figure 1

Schematic picture of the complex energy contours used in the EMTO+DMFT method. The charge density and number of states are calculated for points $z$ on a semicircle enclosing the valence band. For finite-temperature calculations (e.g., LDA+DMFT) the equidistant Matsubara points $i{\omega }_j$ (in red) along the imaginary axis relative to the Fermi level $E_F$ are used. The density of states is calculated for points $z$ (blue) on a horizontal contour defined as $\Re \left(z\right)+i\delta$ ($\delta$ being a small real number).

Figure 2

Schematic diagram of the analytic continuation before (straight path) or after (dashed path) Bloch summation. $G(\mathbf{k},z)$ in the top left corner is given for a set of points. The goal is to obtain an analytic expression $\tilde{G}\left(z^{\prime }\right)$ (lower right corner) that can be used to approximate the true function $G\left(z^{\prime }\right)$. Following the straight path, the Padé approximant needs to be constructed for each $\mathbf{k}\in$ IBZ and the Bloch sum is performed at the desired point. Along the dashed path, only one single Padé approximant is needed.

Figure 3

Density of states for solid hydrogen compared with approximants. The approximants were constructed using the $\ell m$-resolved local Green's function, using the $N$ first Matsubara points corresponding to a temperature of 500 K. Inset: Error as defined in Eq. (17).

Figure 4

Density of states for the $\Gamma$-point compared with approximants of different order. Inset (a): Same as main figure, but zoomed in around $-0.65$ to $-0.55$ Ry. Inset (b): Error as defined in Eq. (17).

Figure 5

(a) Pole-zero distribution of the $N=16$ approximant for the $\Gamma$-point, with and without randomness. The true pole stemming from the original input data is situated at $-0.5959-i3.5719\times {10}^{-6}$ Ry. Adding randomness with $\eta ={10}^{-6}$ gives the pole position at $-0.5963+i3.4499\times {10}^{-5}$ Ry. The spurious poles and zeros show a larger scattering. A zero of the true approximant at $4.8371-i0.3177$ Ry and a zero from the randomized data at $3.2222+i2.8388$ Ry not shown. Inset: Zoomed in around the imaginary axis. (b) Same as (a), but here $\eta ={10}^{-3}$. Inset: Zoomed in around the imaginary axis.

Figure 6

Density of states for solid hydrogen compared with approximants. The approximants were constructed using the $\ell m$-resolved k-dependent Green's function. Once the k-resolved Green's function was given for the horizontal density of states contour the Bloch summation was performed. Inset: Error as defined in Eq. (17).

Figure 7

Density of states and the $N=16$ approximant with defects (green) compared with approximants of the same order, but reconstructed using Eq. (16). For one approximant (blue) the imaginary part of the residue was kept, for the other (violet) only the real part was used. Inset: Error as defined in Eq. (17).