Correlation effects in one-dimensional metallic quantum wires under various confinements

TL;DR

The paper investigates how transverse confinement influences electron correlations in ferromagnetic quasi-one-dimensional quantum wires by employing first-order RPA (FRPA) to compute the static structure factor $S(q)$, pair-correlation function $g(r)$, correlation energy $ε_c$, and ground-state energy $E_g$ across six confinement models. It reveals that in the ultrathin, high-density limit, the correlation energy converges to $ε_c=-π^2/360$ for the harmonic, cylindrical, and harmonic-delta confinements ($V_1$, $V_2$, $V_5$), while other confinements yield a different high-density limit, around $-0.03002$ a.u. The study also shows that the height of the $2k_{ m F}$ peak in $S(q)$ and the oscillations in $g(r)$ depend strongly on the confinement type, with FRPA results agreeing well with QMC data in the applicable regimes and a fitted expression capturing the $2k_{ m F}$ peak behavior. Overall, the work highlights the critical role of confinement modeling in predicting ground-state properties of quasi-1D metallic wires and provides useful benchmarks for theory and potential device design.

Abstract

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 $ε_{\rm c}(r_{\rm s})= - π^2/360 \sim -0.02741$ 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 $ε_{\rm c} \sim -0.03002$ 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.

Figures (8)

• Figure 1: Effective electron-electron interaction potential $V(q)$ plotted in reciprocal space against a nondimensionalized quantity $q/2k_\text{F}$, for various confinement models. The different curves represent $V_1(q)$ to $V_6(q)$ as in Eqs. (\ref{['V1']}), (\ref{['V2']}), (\ref{['V3']}), (\ref{['V4']}), (\ref{['V5']}), and (\ref{['V6']}) discussed in Sec. \ref{['models']}, respectively. As shown in the inset, the potential $V(q)$ is plotted over a small interval of $q$, demonstrating that $V_4(q)$ and $V_5(q)$ yield almost identical results in this regime. This illustrates the variation in interaction strength and range due to the geometric confinement.
• Figure 2: SSF $S(k)$ plotted against $k/k_\text{F}$ for $r_\text{s} = 0.6$, 0.8, 1.2, and 1.5 with $b=0.1$ and 0.5 a.u. Using the FRPA, the SSFs for different confinement models $V_1(q)$ to $V_6(q)$ have been compared with the available VMC simulations of harmonic wires for $N=99$ electrons. In Figs. \ref{['SSFtheoryrs_b']}(a), \ref{['SSFtheoryrs_b']}(b), \ref{['SSFtheoryrs_b']}(c), and \ref{['SSFtheoryrs_b']}(d), the main plot shows the behavior at the $2k_\text{F}$ peak to show the variation across different confinement schemes, whereas the inset shows a zoomed-out view. In Figs. \ref{['SSFtheoryrs_b']}(e) and \ref{['SSFtheoryrs_b']}(f), the FRPA SSF is compared with VMC simulations and RPA SSFs.
• Figure 3: $2k_\text{F}$ peak height of the SSF as a function of density parameter $r_\text{s}$ for different confinement models [$V_1(q)$--$V_6(q)$] and (a) $b=0.1$ and (b) $b=0.5$ a.u. The data points representing the peak heights are joined by lines, revealing a linear relationship with $r_\text{s}$ at fixed wire width ($b<r_\text{s}$).
• Figure 4: $2k_\text{F}$ peak height of the SSF, fitted by Eq. (\ref{['Eq:SSF2kf_fit']}), for different confinement models [$V_1(q)$--$V_6(q)$]. The different symbols represent our FRPA data, and the solid lines show the corresponding fitted function for (a) $r_\text{s} = 0.6$ and (b) $r_\text{s} = 0.8$.
• Figure 5: PCF plotted against $r/r_\text{s}$ for $r_\text{s}=0.6$, 0.8, 1.2, and 1.5 with $b=0.1$ and 0.5 a.u. The PCFs for different confinement models $V_1(q)$ to $V_6(q)$ have been compared with the available VMC simulations of harmonic wires with $N = 99$ electrons. In Figs. \ref{['PCFtheory_rs_b']}(a), \ref{['PCFtheory_rs_b']}(b), \ref{['PCFtheory_rs_b']}(c), and \ref{['PCFtheory_rs_b']}(d), the main plot shows a zoomed-in view of the amplitude of oscillations around $r = 2r_\text{s}$, which varies depending on the coupling parameters ($b$, $r_\text{s}$) and the confinement model, whereas the inset shows a zoomed-out view. In Figs. \ref{['PCFtheory_rs_b']}(e) and \ref{['PCFtheory_rs_b']}(f), the FRPA PCF is compared with VMC simulations and the RPA.