Correlation effects in one-dimensional metallic quantum wires under various confinements
Dynamical response theory is used to investigate various transverse confinements on electron correlations in the ground state of a ferromagnetic one-dimensional quantum wire for different wire widths $b$ and density parameters $r_{\rm s}$. Using the first-order random phase approximation (FRPA), which provides the ground state structure beyond the random phase approximation, we compute the structure factor, pair-correlation function, correlation energy, and ground-state energy. The correlation energy depends on the choice of confinement model and hence effective electron-electron interaction. For the ultrathin wire ($b\rightarrow 0$) in the high-density limit, the correlation energy for transverse confinement models $V_1(q)$ (harmonic), $V_2(q)$ (cylindrical), and $V_5(q)$ (harmonic-delta) approaches $\epsilon_{\rm c}(r_{\rm s})= - \pi^2/360 \sim -0.02741$ a.u., which agrees with the exact results in this limit [J. Chem. Phys. 138, 064108 (2013); Phys. Rev. B 101, 075130 (2020)]. For at least these three confinement potentials, the one-dimensional Coulomb potential can be regularized at interparticle distance $x=0$ to yield the same correlation energy. In contrast, $V_3(q)$ (infinite square well), $V_4(q)$ (infinite square-infinite triangular well), and $V_6(q)$ (infinite square-delta well), do not approach the same high-density limit; instead, the correlation energy tends to $\epsilon_{\rm c} \sim -0.03002$ a.u. The ground-state properties obtained from the FRPA are compared with quantum Monte Carlo results. The peak height in the static structure factor at $k=2k_{\rm F}$ depends significantly on the confinement model. These peaks are fitted with a function based on our finite wire-width theory demonstrating good agreement with FRPA.