Correlation effects in quasi-one-dimensional quantum wires

We explore the role of electron correlation in quasi-one-dimensional quantum wires as the range of the interaction potential is changed and their thickness is varied by performing exact quantum Monte Carlo simulations at various electronic densities. In the case of unscreened interactions with a long-range $1/x$ tail there is a crossover from a liquid to a quasi-Wigner crystal state as the density decreases. When this interaction is screened, quasi-long-range order is prevented from forming, although a significant correlation with $4k_F$ periodicity is still present at low densities. At even lower electron concentration, exchange is suppressed and the electrons behave like spinless fermions. Finally, we study the effect of electron correlations in the double quantum wire experiment [Steinberg et al., Phys. Rev. B 73, 113307 (2006)] by introducing an accurate model for the screening in the experiment and explicitly including the finite length of the system in our simulations. We find that decreasing the electron density continuously drives the system from a liquid to a state with quite strong $4k_F$ correlations. This crossover takes place around $22\mu {\text{m}}^{-1}$, near the density where the electron localization occurs in the experiment. The charge and spin velocities are also in good agreement with the experimental findings in the proximity of the crossover. We argue that correlation effects play an important role at the onset of the localization transition.

Figure 1

(Color online) Structure factor for $b=0.1$ (upper panel) and $b=0.0001$ (lower panel), computed for a system with 78 electrons. The QMC (dotted) and MSA (solid lines) structure factors are reported for different densities $\left(r_s\right)$. Also the noninteracting spinless fermion (NSF) structure factor is drawn (solid black line) for comparison.

Figure 2

(Color online) Detail for the structure factor near $4k_F$ for $b=0.1$, computed for $N=182$ and $N=450$ at two densities ($r_s=0.5$ and $r_s=0.75$) in the proximity of the crossover from a liquid to a quasicrystal.

Figure 3

(Color online) Scaling of the $4k_F$ component of the structure factor with respect to the number of particles. The scaling is reported for various densities with $b=0.1$. The lines are the best fit of the function in Eq. (4) given by the LL theory.

Figure 4

(Color online) Inverse charge compressibility ${\kappa }_0/\kappa$ of the unpolarized and fully polarized wire for $b=0.1$, with both screened $\left(R=1\right)$ and unscreened interactions. Also the HF (dashed black line) charge compressibility is reported for the unpolarized wire. The solid lines are obtained from the second derivative of the energy parametrization, while the points are evaluated through the charge excitations as explained in Sec. 2. The statistical error in the determination of the charge compressibility is approximately 0.02 in these units and is comparable with the point size.

Figure 5

(Color online) Inverse spin susceptibility ${\chi }_0/\chi$ for different thicknesses. The dependence on $r_s$ is shown. The points are the QMC calculations, while the lines are the WKB estimates. The arrows indicate the liquid-to-quasi-Wigner crystal crossover which is determined by the onset of the $4k_F$ peak in the static structure factor. Our estimate of the crossover density $\left(r_s^{\ast }\right)$ has an uncertainty due to the statistical fluctuations in the calculation of the static structure factor and the actual $r_s$ values we have studied.

Figure 6

(Color online) Scaling of the $4k_F$ component of the structure factor with respect to the number of particles for $b=0.1$ and $r_s=4$. For comparison, the scaling is reported for the unscreened $V_b$ interaction and the screened potential in Eq. (10) with $R=200$.

Figure 7

(Color online) Asymptotic large $r_s$ values of the charge velocity in units of $v_F$ vs inverse screening length for ultrathin wire $\left(b=0.0001\right)$ from $R=0.05$ to $R=5$. In the inset we report the full dependence of the charge velocities on $r_s$ at different $R$.

Figure 8

(Color online) Static structure factor for the screened wire with $b=0.0001$ and $R=0.1$, plotted for three values of the density $r_s=0.2$, 0.4, and 0.8. The solid lines correspond to the MSA prediction for each density and the black line is the structure factor for noninteracting spinless fermions. The magnitude of the error bars is comparable with the point size.

Figure 9

(Color online) Static structure factor for a homogeneous wire with $b=0.707$ interacting with the effective potential in Eq. (12), which includes the screening by another homogeneous wire with $r_s=0.83$, $b=1.061$, and $R=3$. The structure factor is plotted for several values of the upper wire density, with $r_s$ ranging from 1.7 to 4.6 and also for two values of the spin polarization with $\zeta =0$ and $\zeta =0.75$. The calculations have been converged to the thermodynamic limit, requiring $N=62$ for $r_s\le 3.0$ and $N=78$ for $r_s=4.6$ subject to periodic boundary conditions. The structure factor has been plotted versus $k/k_F$, where $k_F$ is the Fermi momentum in the unpolarized electron gas. The similarities of the peak heights at $4k_F$ for both values of $\zeta$ show the relative insensitivity of the crossover to spin polarization. The shoulder in $S\left(k\right)$ for the high-density curves with $\zeta =0.75$ is due to the Fermi surface of the majority species of electrons.

Figure 10

(Color online) Static structure factor for a wire as in Fig. 9, but with finite length. $S\left(k\right)$ is plotted for 40, 60, 70, 80, 90, and 100 electrons. The corresponding effective densities ${\tilde{r}}_s$ are reported in the legend. The magnitude of the error bars is comparable with the point size.

Figure 11

(Color online) Density profile for electrons in the finite wire as in Fig. 10, plotted for half of the wire length. $N=40$, 60, 70, 80, 90, and 100 are considered.

Figure 12

(Color online) Inverse spin velocity of the infinite wire. The red circles indicate estimates from the WKB approximation, whereas the black triangles are determined using the QMC method described in Sec. 2. The two lines are estimates due to the perturbative generalized random-phase approximation (Ref. 55) $v_F/v_{\sigma }^{\text{GRPA}}=1/\sqrt{1-V\left(2k_F\right)/\left(\pi v_F\right)}$. The green line for the gated wire (with $R=50$) uses the potential described by Auslaender et al. (Ref. 10) whereas the dotted blue line uses the potential [Eq. (12)] screened by the lower wire.