  Two-dimensional (2D) materials have gained lots of attention due to the
potential applications. In this work, we propose that based on first-principles
calculations, the (2$\times$2) patterned PtTe$_2$ monolayer with kagome lattice
formed by the well-ordered Te vacancy (PtTe$_{1.75}$) hosts large spin Hall
conductivity (SHC) and excellent hydrogen evolution reaction (HER) activity.
The unconventional nature relies on the $A1@1b$ band representation (BR) of the
highest valence band without SOC. The large SHC comes from the Rashba
spin-orbit coupling (SOC) in the noncentrosymmetric structure induced by the Te
vacancy. Even though it has a metallic SOC band structure, the $\mathbb Z_2$
invariant is well defined due to the existence of the direct band gap and is
computed to be nontrivial. The calculated SHC is as large as 1.25$\times 10^3
\frac{\hbar}{e} (\Omega~cm)^{-1}$ at the Fermi level ($E_F$). By tuning the
chemical potential from $E_F-0.3$ to $E_F+0.3$ eV, it varies rapidly and
monotonically from $-1.2\times 10^3$ to 3.1$\times 10^3 \frac{\hbar}{e}
(\Omega~cm)^{-1}$. In addition, we also find the Te vacancy in the patterned
monolayer can induce excellent HER activity. Our results not only offer a new
idea to search 2D materials with large SHC, i.e., by introducing
inversion-symmetry breaking vacancies in large SOC systems, but also provide a
feasible system with tunable SHC (by applying gate voltage) and excellent HER
activity.
