Wire-width and electron-density dependence of the crossover in the peak of the static structure factor from $2k_{\text{F}}$ $\to$ $4k_{\text{F}}$ in one-dimensional paramagnetic electron gases

We use the variational quantum Monte Carlo (VMC) method to study the wire-width $\left(b\right)$ and electron-density $\left(r_{\text{s}}\right)$ dependences of the ground-state properties of quasi-one-dimensional paramagnetic electron fluids. The onset of a quasi-Wigner crystal phase is known to depend on electron density and the crossover occurs in the low density regime. We study the effect of wire width on the crossover of the dominant peak in the static structure factor from $k=2k_{\text{F}}$ to $k=4k_{\text{F}}$. It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at $2k_{\text{F}}$ to $4k_{\text{F}}$ occurs as the wire width decreases. Our study suggests that the crossover is due to the interplay of both $r_{\text{s}}$ and $b<r_{\text{s}}$. The finite wire-width correlation effect is reflected in the peak height of the charge and spin structure factors. We fit the dominant peaks of the charge and spin structure factors assuming fit functions based on our finite wire-width theory and clues from bosonization, resulting in a good fit of the VMC data. The pronounced peaks in the charge and spin structure factors at $4k_{\text{F}}$ and $2k_{\text{F}}$, respectively, indicate the complete decoupling of the charge and spin degrees of freedom. Furthermore, the wire-width dependence of the electron correlation energy and the Tomonaga-Luttinger parameter $K_{\rho }$ is found to be significant.

Figure 1

Correlation energy per particle ($E_{\text{c}}$) as a function of wire width $b$ for $r_{\text{s}}$ = 0.5, 1, and 2 (top to bottom).

Figure 2

(a)–(c) Charge structure factor $S_{\rho \rho }\left(k\right)$ plotted against $k/k_{\text{F}}$ and (d)–(f) charge-charge pair correlation function plotted against $r/r_{\text{s}}$ for $r_{\text{s}}=0.5$, 2, and 5 for various $b$ values with $N=98$ electrons. The insets in (b) and (c) show the peak at $k=4k_{\text{F}}$ in greater detail.

Figure 3

(a)–(c) Spin structure factor $S_{\sigma \sigma }\left(k\right)$ plotted against $k/k_{\text{F}}$ and (d)–(f) spin-spin pair correlation function plotted against $r/r_{\text{s}}$ for $r_{\text{s}}=0.5$, 1, and 2 for various $b$ values with $N=98$ electrons. The insets (a), (b), and (c) show the peak at $k=2k_{\text{F}}$ in greater detail.

Figure 4

(Top) Charge SSF peak height at $k=4k_{\text{F}}$ as a function of wire width $b$ for $r_{\text{s}}=0.5$, 1, and 2 for $N=98$ electrons. The peak heights at $k=4k_{\text{F}}$ are fitted by Eq. (7) and the fits are shown by solid lines, whereas the $2k_{\text{F}}$ peak heights are joined by dashed lines. (Bottom) Height of spin SSF plotted against $b$ for $r_{\text{s}}=0.5$, 1, and 2 for $N=98$ electrons and fitted by Eq. (8).

Figure 5

MD against $k/k_{\text{F}}$ for different values of wire width $b$ and $r_{\text{s}}$ for $N=98$ electrons.

Figure 6

Jump in the MD at $k=k_{\text{F}}$ at finite size, plotted against system size and fitted by $\mathrm{\Delta }n\left(k_{\text{F}}\right)=aL{e}^{-c{(lnL)}^{3/2}}$. The fitted parameter values are $a=1.03\left(2\right)$ a.u. and $c=0.731\left(2\right)$ for $r_{\text{s}}=1$ and $b=1$; and $a=0.93\left(5\right)$ a.u. and $c=0.777\left(7\right)$ for $r_{\text{s}}=2$ and $b=2$.

Figure 7

TL parameter $K_{\rho }$ against wire width $b$ for different values of the density parameter $r_{\text{s}}$ as indicated in the plot. The TL parameter is obtained by fitting MD data to appropriate fitting functions to obtain the exponent $\alpha$ and $K_{\rho }$. The exponent $\alpha$ is plotted against $b$ in the inset.