First-principles estimation of the superconducting transition temperature of a metallic hydrogen liquid

We present a full density-functional theory-based implementation of the stochastic path-integral approach proposed by Liu et al. [H. Liu, Y. Yuan, D. Liu, X.-Z. Li, and J. Shi, Phys. Rev. Research 2, 013340 (2020)] for estimating the superconducting transition temperature ($T_c$) of a liquid. The implementation is based on the all-electron projector augmented-wave (PAW) method. We generalize Liu et al.'s formalism to accommodate the pseudodescription of electron states in the PAW method. A formula for constructing the overlap operator of the PAW method is proposed to eliminate errors due to the incompleteness of a pseudobasis set. We apply the implementation to estimate $T_c$'s of metallic hydrogen liquids. It confirms Liu et al.'s prediction that metallic hydrogen could form a superconducting liquid.

Figure 1

Flowchart for calculating Green's function and the $\mathcal{T}$ matrix. It substitutes the corresponding part of the workflow shown in Fig. 1 of Ref. [18].

Figure 2

Comparison between the full DFT-based implementation (PAW method) and the LSA in (a) pair-scattering amplitude ${\mathrm{\Gamma }}_m\left(q\right)$, (b) effective pair interaction $W_m\left(q\right)$, and (c) interaction parameters $\lambda \left(n\right)$. The results are for $T=450$ K and $P=500$ GPa. ${\mathrm{\Gamma }}_m\left(q\right)$ and $W_m\left(q\right)$ are obtained by recasting the matrix elements of $\mathrm{\Gamma }$ and $W$ as a function of $q=|{\mathit{k}}_1-{\mathit{k}}_2|$ for ${\mathit{k}}_1, {\mathit{k}}_2$ in $0.8k_F\le |{\mathit{k}}_1|,|{\mathit{k}}_2|\le 1.14k_F$.

Figure 3

Dependence of $T_c$ on (a) different numbers of atoms and (b) different numbers of beads for $P=700$ GPa. The error bars indicate the numerical uncertainties of the estimations of $T_c$'s, i.e., the values shown in parentheses in Table 1.