One-dimensional electron fluid at high density

We calculate the ground-state energy, pair correlation function, static structure factor, and momentum density of the one-dimensional electron fluid at high density using variational quantum Monte Carlo simulation. For an infinitely thin cylindrical wire the predicted correlation energy is found to fit nicely with a quadratic function of coupling parameter $r_s$. The extracted exponent $\alpha$ of the momentum density for $k\sim k_F$ is used to determine the Tomonaga-Luttinger parameter $K_{\rho }$ as a function of $r_s$ in the high-density regime. We find that the simulated static structure factor and pair correlation function for infinitely thin wires agree with our recent high-density theory [K. Morawetz et al., Phys. Rev. B 97, 155147 (2018)].

Figure 1

VMC energy $(E-E_{\infty })$ plotted against the reciprocal of the square of the system size for the infinitely thin wire. The energies per particle were extrapolated to the thermodynamic limit using the form $E\left(N\right)=E_{\infty }+B{N}^{-2}$.

Figure 2

Correlation energy ${\epsilon }_c$ as a function of $r_s$ for an infinitely thin wire (and, in the inset, for a harmonic wire of width $b=0.5$). The solid line is a quadratic fit as a function of $r_s$.

Figure 3

Top: PCF of an infinitely thin wire at several densities. Bottom: PCF of a harmonic wire of width $b=0.5$ at several densities. The data shown are for $N=99$. The inset shows the same data at the origin. The random errors in the data are small, as evidenced by the lack of visible noise in the curves.

Figure 4

PCFs $g\left(r\right)$ of 1D HEGs in infinitely thin wires. Results obtained from VMC simulation are compared with recent high-density theory [55] at several densities. The inset shows the oscillations of $g\left(r\right)$ at larger distances in greater detail.

Figure 5

SSF of an infinitely thin wire at several system sizes for $r_s=0.8$. The main plot shows the behavior at the peak, and the inset shows a zoomed-out view.

Figure 6

VMC SSF of an infinitely thin wire with $N=99$, compared with the high-density theory [55] (solid line). The main plot shows the SSF for $r_s=0.5$, and the inset is for $r_s=0.6$.

Figure 7

SSF peak height at $k=2k_F$ plotted against system size $N$ for infinitely thin wires with different coupling parameters $r_s$.

Figure 8

MD of an infinitely thin wire (top) and MD of a harmonic wire of width $b=0.5$ (bottom) at several densities. The data shown are for $N=99$. The statistical error bars are much smaller than the symbols and have therefore been omitted for clarity. The inset shows a zoomed-out view.

Figure 9

Tomonaga-Luttinger liquid exponent $\alpha$ in Eq. (9) extracted from our MDs against the fitting range of data ($|k-k_F|<\epsilon k_F$) for a ferromagnetic, infinitely thin wire (top) and a harmonic wire of width $b=0.5$ (bottom). The extracted exponent is linearly fitted with the solid line in the region $\epsilon >0.035$ and extrapolated to $\epsilon =0$.

Figure 10

Exponent $\alpha$, found by fitting Eq. (9) to the MDs of ferromagnetic 1D electron gases and extrapolating to $\epsilon =0$, plotted against $r_s$. The error bars on the data points approximately account for the random error due to the Monte Carlo evaluation of the momentum density of the trial wave function (approximate because the data points in the MD are correlated); however, the random noise on the exponents is clearly larger than these error bars. There is an additional uncertainty in the MD and hence $\alpha$ due to the stochastic optimization of the trial wave function, and this may be responsible for the larger noise. Nevertheless, the noise in the exponent $\alpha$ as a function of $r_s$ is at least an order of magnitude smaller than the systematic behavior.

Figure 11

Exponent $\alpha$ found by fitting Eq. (9) to the MDs of ferromagnetic systems for $r_s=0.5$ and different harmonic wire widths $b$.

Figure 12

Tomonaga-Luttinger parameter $K_{\rho }$ plotted against $r_s$ and, in the inset, plotted as a function of exponent $\alpha$ for an infinitely thin wire. The low-density DMC data are adopted from Lee and Drummond [53].