Ground-state properties of electron-electron biwire systems

The correlation between electrons in different quantum wires is expected to affect the electronic properties of quantum electron-electron biwire systems. Here, we use the variational Monte Carlo method to study the ground-state properties of parallel, infinitely thin electron-electron biwires for several electron densities ($r_{\text{s}}$) and interwire separations ($d$). Specifically, the ground-state energy, the correlation energy, the interaction energy, the pair-correlation function (PCF), the static structure factor (SSF), and the momentum distribution (MD) function are calculated. We find that the interaction energy increases as $ln\left(d\right)$ for $d\to 0$ and it decreases as $d^{-2}$ when $d\to \infty$. The PCF shows oscillatory behavior at all densities considered here. As two parallel wires approach each other, interwire correlations increase while intrawire correlations decrease as evidenced by the behavior of the PCF, SSF, and MD. The system evolves from two monowires of density parameter $r_{\text{s}}$ to a single monowire of density parameter $r_{\text{s}}/2$ as $d$ is reduced from infinity to zero. The MD reveals Tomonaga-Luttinger (TL) liquid behavior with a power-law nature near $k_{\text{F}}$ even in the presence of an extra interwire interaction between the electrons in biwire systems. It is observed that when $d$ is reduced the MD decreases for $k<k_{\text{F}}$ and increases for $k>k_{\text{F}}$, similar to its behavior with increasing $r_{\text{s}}$. The TL liquid exponent is extracted by fitting the MD data near $k_{\text{F}}$, from which the TL liquid interaction parameter $K_{\rho }$ is calculated. The value of the TL parameter is found to be in agreement with that of a single wire for large separation between the two wires.

Figure 1

Cartoon representation of the EEBW model. Both wires are identical in all aspects except the spin of the electrons. The mean distance between the electrons in each wire is $2r_{\text{s}}$.

Figure 2

EEBW VMC energies (a.u./electron) offset by the extrapolated $E_{\infty }$ vs reciprocal of the square of the system size at interwire separation $d=1$ a.u. Equation (5) is linearly fitted to the VMC energy data for different system sizes $N$ to obtain the asymptotic value of the ground-state energy per electron $E_{\infty }$. Error bars are smaller than the size of the symbols.

Figure 3

Interaction energy $\mathrm{\Delta }E$ (a.u./electron) and correlation energy $E_{\text{c}}$ (a.u./electron) are plotted as functions of interwire spacing $d$ for $r_{\text{s}}\le 1$. Open symbols with dashed lines represent $\mathrm{\Delta }E$ and closed symbols with solid lines are for $E_{\text{c}}$. $\mathrm{\Delta }E$ and $E_{\text{c}}$ are calculated using $E_{\infty }$ for the wire and biwire systems.

Figure 4

Interwire interaction energy $\mathrm{\Delta }E$ vs $d$. Symbols represent our calculated data and solid lines show data obtained by fitting Eq. (10).

Figure 5

Inter- and intrawire PCFs at different wire separations $d$ for $r_{\text{s}}=0.1$ and 1. Solid lines are used for $r_{\text{s}}=0.1$ and dashed lines are for $r_{\text{s}}=1.0$. The inset shows a magnified view of the first peak of the PCF.

Figure 6

Intra- and interwire PCFs at various separations $d$. (a) $g_{11}\left(r\right)$ and $g_{12}\left(r\right)$ are plotted for the density parameter $r_{\text{s}}=2$. (b) $g_{11}\left(r\right)$ and $g_{12}\left(r\right)$ are plotted for the density parameter $r_{\text{s}}=5$. Symbols represent data for a single wire.

Figure 7

VMC inter- and intrawire PCFs at $d=1$ and 0.1 a.u. for $r_{\text{s}}=10$ are shown in the top and bottom panels, respectively. The inset shows a magnified view. Open circles are used for the single-wire PCF (top panel).

Figure 8

Top panel: Intrawire SSF $S_{11}$. Bottom panel: Interwire SSF $S_{12}$ for various values of $d$ at $r_{\text{s}}=2$. The inset shows a zoomed-in view of the peaks in the SSF.

Figure 9

SSFs of EEBWs for various values of $r_{\text{s}}$ and $d$ are shown with lines. Open circles are used for single-wire SSFs.

Figure 10

MD of the EEBW at various values of $r_{\text{s}}$ and $d$ for $N=61$. Open circles are for an infinitely thin single wire.

Figure 11

Exponent $\alpha$ calculated by fitting Eq. (18) to the MD against the range of data described by $|k-k_{\text{F}}|<\epsilon k_{\text{F}}$. The symbols are the fitted exponents and the solid lines are linear fits to the exponents in the region $\epsilon >0.05$. Here the data are shown for $d=1$ and various values of $r_{\text{s}}$.

Figure 12

Extrapolated values of the exponent $\alpha$ vs $d$ at various $r_{\text{s}}$ as symbols. Solid lines are just to guide the eyes.

Figure 13

VMC exponents $\alpha$ (symbols) plotted against $r_{\text{s}}$ at various values of $d$. The solid line shows $\alpha$ plotted using Eq. (19) for the infinitely thin single wire, which is close to the VMC EEBW data for $d=1$ a.u.

Figure 14

TL interaction parameter $K_{\rho }$ plotted against $r_{\text{s}}$ for various wire separations $d$ for EEBWs computed using Eq. (21). The solid line shows the theoretical result for an isolated, infinitely thin single wire obtained using Eq. (22), which passes close to larger-$d$ EEBW data.

Figure 15

Intra- and interwire PCFs as functions of system size $N$ for $r_{\text{s}}=2$ a.u. and $d=1$ a.u. The main plot shows the intrawire pair correlation function and the inset shows the interwire pair correlation function. The plot for $N=41$ overlaps on $N=21$ and the one for $N=61$ overlaps on $N=41$.

Figure 16

SSFs at different system sizes at $d=1$ a.u. for $r_{\text{s}}=2$ (top panel) and 5 (bottom panel). The inset in the top panel shows a zoomed-in view of the same data near $2k_{\text{F}}$.

Figure 17

Peaks of SSF at different system sizes at $d=1$ a.u. for $r_{\text{s}}=5$, 15, and 20.

Figure 18

MD as a function of system size for $r_{\text{s}}=2$ a.u. and $d=1$ a.u. The inset shows a zoomed-in view for small $k$.

Figure 19

Comparison of VMC and DMC PCFs, SSFs, and MDs at $d=1$ a.u. for some values of $r_{\text{s}}$.