Pair-distribution functions of the two-dimensional electron gas

Based on its known exact properties and a new set of extensive fixed-node reptation quantum Monte Carlo simulations (both with and without backflow correlations, which in this case turn out to yield negligible improvements), we propose an analytical representation of (i) the spin-summed pair-distribution function and (ii) the spin-resolved potential energy of the ideal two-dimensional interacting electron gas for a wide range of electron densities and spin polarization, plus (iii) the spin-resolved pair-distribution function of the unpolarized gas. These formulas provide an accurate reference for quantities previously not available in analytic form, and may be relevant to semiconductor heterostructures and quantum dots both directly, in terms of phase diagram and spin susceptibility, and indirectly, as key ingredients for the construction of new two-dimensional spin density functionals, beyond the local approximation.

Figure 1

Numerical difference between the spin-resolved pair-distribution functions $g_{\sigma {\sigma }^{\prime }}^{\mathrm{BF}}-g_{\sigma {\sigma }^{\prime }}^{\mathrm{PW}}$ at $r_s=2$ (upper panel) and $r_s=20$ (lower panel), as obtained from two fixed-node simulations with different nodal structures. The superscript indicates backflow (BF) or plane-wave (PW) nodes.

Figure 2

Sample of spin-resolved pair-distribution functions as directly obtained from our QMC simulations (no fitting here).

Figure 3

Spin-summed pair-distribution function (exchange plus correlation: $g=g^x+g^c$, see text) for four different values of the spin-polarization parameter $\zeta$, and for $r_s=1$, 2, 5, 10, 20, and 40 (larger $r_s$ values have stronger oscillations). The dots correspond to our QMC data, the solid lines to our analytic representation. Error bars are comparable with the dot size.

Figure 4

Spin-summed pair-correlation functions $g^c$ from our analytic representation. Upper panel: for $\zeta =0$, we show $g^c$ for $r_s=2$,3,4,5,7,10,15,20,30,40; stronger oscillations correspond to higher $r_s$ values. The solid lines correspond to $r_s=2$,5,10,20,40, for which our $g^c$ accurately fits the QMC data; the dashed lines are the results for intermediate values of $r_s$. Lower panel: for $r_s=2$, we show $g^c$ for different values of the spin-polarization $\zeta =0,0.3,0.48,0.7,0.8,0.9,1$; more negative “on-top” values $g^c(x=0)$ correspond to lower values of $\zeta$. The solid lines correspond to $\zeta =0,0.48,0.8,1$, for which our $g^c$ accurately fits the QMC data. The dashed lines correspond to intermediate values of $\zeta$.

Figure 5

$\uparrow \downarrow$ pair-distribution function (exchange plus correlation, see text) for $\zeta =0$ and $r_s=1$,1.5,2,3,5,7,10 (the larger $r_s$ values have stronger oscillations). The dots correspond to our QMC data for $r_s=1$,2,5,10; the solid lines is our analytic representation at the same $r_s$ values. Dashed lines correspond to our analytic representation for the other values of $r_s$.

Figure 6

Upper panel: total (spin-summed) structure factor as directly obtained from our QMC simulations (data with error bars) and as a Fourier transform of our analytic representation of $g^c$ (solid lines), for $r_s=1,2,5,10,20,40$. Higher peaks correspond to larger $r_s$. Lower panel, same comparison for the $\uparrow \downarrow$ static structure factor: the data with error bars are QMC simulations and the solid lines are Fourier transforms of our analytic $g_{\uparrow \downarrow }^c$. Here the $r_s$ values are 1,2,5, and 10 and the larger deviations from the noninteracting value $S_{\uparrow \downarrow }=0$ correspond to larger $r_s$ values.

Figure 7

Upper panel: spin-summed correlation static structure factors from our analytic representation of $g^c$ and from the Atwal, Khalil, and Ashcroft (Ref. 38) (AKA) dynamical local-field factor. Lower panel: the same comparison is done for the spin channel (correlation only, $\uparrow \uparrow -\uparrow \downarrow$). All curves are for $\zeta =0$.

Figure 8

Relative error $\Delta v_c(r_s,\zeta )∕v_c(r_s,\zeta )=(v_c-v_c^{\mathrm{INT}})∕v_c$ between $v_c$ calculated using the RHS of Eq. (18) (with ${ϵ}_c$ from Ref. 5), and $v_c^{\mathrm{INT}}$, obtained by numerical integration of $g^c$ (see text).

Figure 9

Fraction of $\uparrow \downarrow$ contribution to the correlation part of the potential energy $F_{\uparrow \downarrow }=v_c^{\uparrow \downarrow }∕v_c$.

Figure 10

The function $f(z,\zeta )$ [see Eq. (A1)], evaluated within RPA, for three different values of the spin polarization parameter $\zeta$. Also shown: the exact small-$z$ behavior of Eq. (A3) and the function $h\left(z\right)$ of Eq. (A8).