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

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. 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 $K_{\rho }$ from the momentum density. It is found that $K_{\rho }$ 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 $r_{\text{s}}\le 1$. These are compared with the QMC results in the range of applicability of the theory.

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 $[2g_{\mathrm{DMC}}\left(r\right)-g_{\mathrm{VMC}}\left(r\right)]$, 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 $2k_{\text{F}}$ peak. The data shown are for $N=99$ and are extrapolated estimates $[2S_{\mathrm{DMC}}\left(k\right)-S_{\mathrm{VMC}}\left(k\right)]$, 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 $[2n_{\mathrm{DMC}}\left(k\right)-n_{\mathrm{VMC}}\left(k\right)]$, where $n_{\text{DMC}}$ and $n_{\text{VMC}}$ are DMC and VMC MDs, respectively. It is observed that as $b\to 0$, 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 $\mathbf{▵}$.

Figure 5

SSF difference $\mathrm{\Delta }S\left(k\right)=S_1^{\text{Hr}}(x,b)-S_1^{\text{Hr}}\left(x\right)$ 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 $\left|\mathrm{\Delta }S\right(k\left)\right|$ is for $b=0.09$ in this plot.

Figure 6

First-order RPA SSF as a function of $k/k_{\text{F}}$ for various harmonic wire widths $b=0.01$–1 (top to bottom) at $r_{\text{s}}=1$. The inset shows a zoomed-in view of the $2k_{\text{F}}$ peak.

Figure 7

First-order RPA for harmonic wires (solid line) compared with extrapolated estimates $[2S_{\mathrm{DMC}}\left(k\right)-S_{\mathrm{VMC}}\left(k\right)]$ of SSFs for $N=99$ at $r_{\text{s}}=0.5$. The main plot shows the SSF for $b=0.01$, and the inset is for $b=0.6$.

Figure 8

Theoretical PCF compared with extrapolated estimates of PCFs $[2g_{\mathrm{DMC}}\left(r\right)-g_{\mathrm{VMC}}\left(r\right)]$ for harmonic wires with $N=99$ at $r_{\text{s}}=0.5$. The main plot is for $b=0.01$ and the inset is for $b=0.6$.

Figure 9

Correlation energy per particle for a harmonic wire and a cylindrical wire calculated using first-order RPA (shown as solid and dashed lines) valid in the high-density limit, compared with VMC and DMC data obtained for a harmonic wire. The main plot shows the comparison for $r_{\text{s}}=0.5$ and the inset is for $r_{\text{s}}=0.1$. The value of the first-order RPA correlation energy for an infinitely thin wire is represented by $\mathbf{▵}$. The disagreement of the QMC data with first-order RPA at smaller values of $b$ is due to the approximation used in Eq. (13), which restricts its applicability to the high-density limit.

Figure 10

Change in the first-order RPA correlation energy per particle from its value for the infinitely thin wire as a function of $b$ for a harmonic wire (upper) and cylindrical wire (lower) for various $r_{\text{s}}$ values (0.1–1) from top to bottom.

Figure 11

Analytical expressions for the wire-width-dependent correlation energy [Eqs. (38) and (39)], which are valid only for small $b$, plotted as a function of wire width $b$ for $r_{\text{s}}=0.5$. The solid curves are for analytical results and dashed curves are result of numerical calculations of correlation energy as in Eq. (32). The plot shows the applicability of the analytical results in the range $b<0.1$.

Figure 12

Total ground-state energy $E_{\text{g}}$ as a function of wire width $b$. In the main plot, the first-order RPA for $r_{\text{s}}=0.5$ is compared with DMC simulations and in the inset the analytical first-order RPA results are plotted for $r_{\text{s}}=0.2$–1 in steps of 0.1 (top to bottom). The statistical error bars on the DMC results are omitted as they are much smaller than the symbols used.

Figure 13

Top panel: Total ground-state energy ($E_g$) as a function of density parameter $r_{\text{s}}$. In the main plot, analytical first-order RPA results are plotted for $b=0.1$, 0.3, 0.5, 0.7, and 0.9 (bottom to top), and in the inset the first-order RPA results for $b=0.5$ are compared with VMC simulations. Bottom panel: Correlation energy per particle for a harmonic wire calculated using first-order RPA (shown as solid lines) compared with the available QMC simulation data. The LRDMC data shown are taken from Ref. [31].