Emergence of an excitonic collective mode in the dilute electron gas

By comparing two expressions for the polarization function $\mathrm{\Pi }(\mathit{q},i\omega )$ given in terms of two different local-field factors, $G_{+}(\mathit{q},i\omega )$ and $G_s(\mathit{q},i\omega )$, we have derived the kinetic-energy-fluctuation (or sixth-power) sum rule for the momentum distribution function $n\left(\mathit{p}\right)$ in the three-dimensional electron gas. With use of this sum rule, together with the total-number (or second-power) and the kinetic-energy (or fourth-power) sum rules, we have obtained $n\left(\mathit{p}\right)$ in the low-density electron gas at negative compressibility (namely, $r_s>5.25$ with $r_s$ being the conventional density parameter) up to $r_s\approx 22$ by improving on the interpolation scheme due to Gori-Giorge and Ziesche proposed in 2002. The obtained results for $n\left(\mathit{p}\right)$ combined with the improved form for $G_s(\mathit{q},\omega +i{0}^{+})$ are employed to calculate the dynamical structure factor $S(\mathit{q},\omega )$ to reveal that a giant peak, even bigger than the plasmon peak, originating from an excitonic collective mode made of electron-hole pair excitations, emerges in the low-$\omega$ region at $\left|\mathit{q}\right|$ near $2p_{\mathrm{F}}$ ($p_{\mathrm{F}}$: the Fermi wave number). Connected with this mode, we have discovered a singular point in the retarded dielectric function at $\omega =0$ and $\left|\mathit{q}\right|\approx 2p_{\mathrm{F}}$.

Figure 1

Bird's-eye view of $S(\mathit{q},\omega )$ in the electron gas at zero temperature at (a) a usual metallic density $r_s=4$, in which the plasmon peak dominates, and (b) a very dilute density $r_s=22$, in which an excitonic collective peak, much bigger than the plasmon peak, emerges.

Figure 2

Comparison of various formulas for the on-top pair distribution function $g\left(0\right)$ plotted as a function of $r_s$ over a very wide range.

Figure 3

Parametrized $n_0, n_{-}$, and $n_{+}$ as a function of $r_s$ in both present (solid curves) and Gori-Giorgi and Ziesche (dashed curves) schemes. The input data obtained by EPX (triangles) and $\mathrm{GW}\mathrm{\Gamma }$ (solid circles) are also shown.

Figure 4

Parametrized $B\left(r_s\right)$ and $C\left(r_s\right)$ as a function of $r_s$. The former is compared with the one proposed by Moroni, Ceperley, and Senatore, together with the value in the Wigner-lattice limit.

Figure 5

An example of $n\left(\mathit{p}\right)$ calculated with use of $g\left(0\right)$ in Yasuhara (solid), GP (dotted-dashed), and Overhauser (dashed) forms at $r_s=8$. Our results are compared with that in the GZ scheme as well as that in $\mathrm{GW}\mathrm{\Gamma }$.

Figure 6

Obtained $n\left(\mathit{p}\right)$ for (a) $1\le r_s\le 16$ and (b) $16\le r_s\le 22$.

Figure 7

Parameters $A_{\pm }$ and $a_{\infty }$ determined so as to satisfy the three sum rules for $n\left(\mathit{p}\right)$ at each $r_s$. Three different forms for $g\left(0\right)$ are employed. For comparison, $A_{\pm }$ in GZ are also shown. (In GZ, $a_{\infty }=0$.) The inset displays $A_{-}^{-1}$ for the region of $r_s$ in which $A_{-}$ is very large.

Figure 8

The average kinetic energy and the fluctuation of the kinetic energy as a function of $r_s$. No difference is seen by the different choice of $g\left(0\right)$ because of the fulfillment of the three sum rules. Inset: Corresponding values of $I_n$ for $n=0,2,4$, and 6.

Figure 9

The retarded polarization function ${\mathrm{\Pi }}_{\mathrm{WI}}^R(\mathit{q},\omega )$ in comparison with ${\mathrm{\Pi }}_0^R(\mathit{q},\omega )$ at $r_s=22$ for (a) real and (b) imaginary parts.

Figure 10

Difference between the “exact” value of ${\varepsilon }_c$ due to Perdew and Wang and the one calculated through the adiabatic connection formula with using the local-field factor in the original RA (the dotted-dashed curve), a simple version of the modified RA (the dashed curve), and an improved version of the modified RA (the solid curve) schemes.

Figure 11

(a) Dispersion relation of the ghost exciton mode at various values of $r_s$. (b) Sound velocity of the ghost exciton mode in units of the Fermi velocity $v_{\mathrm{F}}$ plotted as a function of $|{\kappa }_F/\kappa |$.

Figure 12

Oscillator strength of the ghost exciton mode. Inset: Fraction of this oscillator strength to the full strengths in the limit of $q\to 0$, together with the sound velocity in units of $v_{\mathrm{F}}$, plotted as a function of $r_s$.

Figure 13

The dynamical structure factor at (a) $r_s=8$ and (b) $r_s=16$. Inset: Bird's-eye view.

Figure 14

Dispersion relation of the excitonic collective mode as determined from the peak position in $S(\mathit{q},\omega )$ in $(q,\omega )$ space. The shaded area represents the electron-hole single-pair excitation region and the dotted curve indicates the boundary of $\omega =(2p_{\mathrm{F}}q-q^2)/2m$.

Figure 15

Two-dimensional contour map of the real part of ${\varepsilon }^R(\mathit{q},\omega )$ at (a) $r_s=4$ and (b) $r_s=8$.

Figure 16

(a) Calculated $q_{\mathrm{ex}}$ in units of $p_{\mathrm{F}}, A_{\mathrm{ex}}$, and $B_{\mathrm{ex}}$ as a function of $r_s$. (b) Comparison of ${\varepsilon }^R(q,\omega )$ between the full calculation in Eq. (53) and the approximate one in Eq. (71) at $r_s=22$ for $\omega =0.24E_{\mathrm{F}}$ at which energy $S(\mathit{q},\omega )$ becomes maximum due to the excitonic collective mode.

Figure 17

Two-dimensional contour map of ${\varepsilon }^R(\mathit{q},\omega )$ at $r_s=22$ for (a) real and (b) imaginary parts.