Electron correlation and confinement effects in quasi-one-dimensional quantum wires at high density

Abstract

We study the ground-state properties of ferromagnetic quasi-one-dimensional quantum wires using the quantum Monte Carlo (QMC) method for various wire widths $b$ and density parameters $r_\text{s}$. The correlation energy, pair-correlation function, static structure factor, and momentum density are calculated at high density, $r_\text{s}=0.5$. It is observed that the peak in the static structure factor at $k=2k_\text{F}$ grows as the wire width decreases. We obtain the Tomonaga-Luttinger liquid parameter $K_ρ$ from the momentum density. It is found that $K_ρ$ increases by about $10$\% between wire widths $b=0.01$ and $b=0.5$. We also obtain ground-state properties of finite thickness wires theoretically using the first-order random phase approximation (RPA) with exchange and self-energy contributions, which is exact in the high-density limit. Analytical expressions for the static structure factor and correlation energy are derived for $b \ll r_\text{s}<1$. It is found that the correlation energy varies as $b^2$ for $b \ll r_\text{s}$ from its value for an infinitely thin wire. It is observed that the correlation energy depends significantly on the wire model used (harmonic versus cylindrical confinement). The first-order RPA expressions for the structure factor, pair-correlation function, and correlation energy are numerically evaluated for several values of $b$ and $r_\text{s} \leq 1$. These are compared with the QMC results in the range of applicability of the theory.

Figures (13)

• Figure 1: PCFs for harmonic wires with $N = 99$ and $r_\text{s}=0.5$ at various wire widths $b=0.1$ to $b=0.5$ a.u. (top to bottom). The inset shows a zoomed-in view of the peak at $r=1$. The data shown are extrapolated estimates ${[2g_{\rm DMC}(r)-g_{\rm VMC}(r)]}$, where $g_\text{DMC}$ and $g_\text{VMC}$ are the DMC and VMC PCFs, respectively.
• Figure 2: SSFs for harmonic wires with $r_\text{s}=0.5$ for various wire widths $b$. The inset shows the $b$ dependence of the $2 k_\text{F}$ peak. The data shown are for $N = 99$ and are extrapolated estimates ${[2S_{\rm DMC}(k)-S_{\rm VMC}(k)]}$, where $S_\text{DMC}$ and $S_\text{VMC}$ are the DMC and VMC SSFs, respectively.
• Figure 3: MDs for harmonic wires with $N = 99$ at $r_\text{s}=0.5$ for various wire widths. The data shown are extrapolated estimates ${[2n_{\rm DMC}(k)-n_{\rm VMC}(k)]}$, where $n_\text{DMC}$ and $n_\text{VMC}$ are DMC and VMC MDs, respectively. It is observed that as $b\rightarrow0$, the harmonic wire MD agrees with the infinitely thin wire MD. The statistical error bars are omitted for clarity as they are smaller than the symbols.
• Figure 4: TL parameter $K_\rho$ as a function of $b$, obtained from QMC calculations. The exponent $\alpha$ is plotted against $b$ in the inset. The corresponding values for infinitely thin wires at $r_\text{s}=0.5$ are indicated on the vertical axes by the symbol '$\bm{\triangle}$'.
• Figure 5: SSF difference $\Delta S(k) = S^\text{Hr.}_1(x, b) - S^\text{Hr.}_1(x)$ as a function of $k/k_\text{F}$ for several harmonic wire widths $b=0.01$--0.09 for $r_\text{s}=0.5$, evaluated using the first-order RPA. The greatest value of $|\Delta S(k)|$ is for $b=0.09$ in this plot.