GW space-time method: Energy band-gap of solid hydrogen

TL;DR

The paper develops the GW space-time (GWST) method at finite temperature, representing $G$, $P$, $oldsymbol{ abla}$, $W$, and $oldsymbol{ riangle}$ on real-space and imaginary-time grids with Chebyshev polynomials to control systematic errors. It validates the approach with benchmark calculations on Si and Ge, and then applies one-shot $G_0W_0^{GF}$ to hexagonal solid hydrogen, extracting band gaps from the asymptotic decay of $G( au)$ and benchmarking against analytic-continuation results. The results indicate that, under pressure, the $hcp$ solid hydrogen likely cannot remain insulating around 270 GPa, and that including zero-point motion would lower the predicted gap closures to around 210 GPa, suggesting the actual metallic or semi-metallic transition may occur in a different structure. The GWST framework stores full $G$ and $W$, enabling future Diagrammatic Monte Carlo extensions to sum higher-order diagrams in a controlled, stochastic manner, with broad applicability to excited-state properties in extended systems.

Abstract

We implement the GW space-time method at finite temperatures, in which the Green's function G and the screened Coulomb interaction W are represented in the real space on a suitable mesh and in imaginary time in terms of Chebyshev polynomials, paying particular attention to controlling systematic errors of the representation. Having validated the technique by the canonical application to silicon and germanium, we apply it to calculation of band gaps in hexagonal solid hydrogen with the bare Green's function obtained from density functional approximation and the interaction screened within the random phase approximation (RPA). The band gap results, obtained from the asymptotic decay of the full Green's function without resorting to analytic continuation, suggest that the solid hydrogen above 270 GPa can not adopt the hexagonal-closed-pack (hcp) structure. The demonstrated ability of the method to store the full G and W functions in memory with sufficient accuracy is crucial for its subsequent extensions to include higher orders of the diagrammatic series by means of diagrammatic Monte Carlo algorithms.

Figures (7)

• Figure 1:
• Figure 2: Collection of (absolute)energy/error data points obtained by the fitting procedure depicted in Fig. \ref{['cheb']} repeated for Si, Ge and H across the BZ momentum points. Both absolute (top) and relative (bottom) errors are plotted as functions of band energy. Solid lines represent the fits to $f(x)= ae^{bx}$ functions of band energy $x$.
• Figure 3: Illustration of the algorithm behind the final extrapolation of $\Gamma \rightarrow \Gamma$ (left) and $\Gamma \rightarrow X$ (right) bandgaps of Si to the infinite $N_r$ and $N_k$ grid parameters. Green/red data points correspond to $N_{r/k}=\infty$ columns/rows of the Tab. \ref{['gwgap_CH']} ($\mathrm{G}_0\mathrm{W}_0^{GF}$ data) both carrying the data extrapolation error bars. The linear fit of these values to a $\sim\frac{1}{N_{k/r}}$ function is given by lines with associated fitting error bars shown by the shaded regions. The size of the shaded region is chosen to cover both the error of the fit itself and all error bars of data points.
• Figure 4: Same as figure \ref{['Si_FS']} but for Ge.
• Figure 5: Same as figure \ref{['Si_FS']} but for H with, however, a slightly different scheme for estimation of the $\Delta_\mathrm{min}$ (see Sec. \ref{['sec:ExtrapolationRes']})