Diffusion quantum Monte Carlo calculation of the quasiparticle effective mass of the two-dimensional homogeneous electron gas

The quasiparticle effective mass is a key quantity in the physics of electron gases, describing the renormalization of the electron mass due to electron-electron interactions. Two-dimensional electron gases are of fundamental importance in semiconductor physics, and there have been numerous experimental and theoretical attempts to determine the quasiparticle effective mass in these systems. In this work we report quantum Monte Carlo results for the quasiparticle effective mass of a two-dimensional homogeneous electron gas. Our calculations differ from previous quantum Monte Carlo work in that much smaller statistical error bars have been achieved, allowing for an improved treatment of finite-size effects. In some cases we have also been able to use larger system sizes than previous calculations.

Figure 1

Energy bands $\mathcal{E}\left(\mathbf{k}\right)$ for paramagnetic 2D HEGs of density parameter (a) $r_s=1$, (b) $r_s=5$, and (c) $r_s=10$ at different system sizes $N$. For the curves labeled “reopt,” the wave function was optimized separately in the ground state and excited states. The free-electron and Hartree-Fock bands have been offset so that they coincide with the DMC bands at $k_F$. The inset to (a) shows the energy band around $\mathbf{k}=\mathbf{0}$ in greater detail.

Figure 2

As Fig. 1, but for ferromagnetic HEGs.

Figure 3

Quasiparticle effective mass $m^{*}$ against range of wave vectors included in the quartic fit of the energy band. Specifically, only wave vectors in the interval $[k_F-\Delta k,k_F+\Delta k]$ are used in the fit. For the dashed lines, wave vectors within 10$\%$ of $k_F$ are excluded from the fit.

Figure 4

As Fig. 3, but for ferromagnetic HEGs. For $N=57$ electrons, excluding wave vectors within 10$\%$ of $k_F$ eliminates all the wave vectors above $k_F$ at which energy-band data are available.

Figure 5

Derivatives of the paramagnetic HEG energy bands shown in Fig. 1 at (a) $r_s=1$, (b) $r_s=5$, and (c) $r_s=10$.

Figure 6

As Fig. 5, but for ferromagnetic HEGs.

Figure 7

Quasiparticle effective mass $m^{*}$ against $N^{-3/2}$, where $N$ is the system size, for 2D HEGs.

Figure 8

Quasiparticle effective mass $m^{*}$ against density parameter $r_s$ for paramagnetic or partially spin-polarized 2D HEGs, as calculated or measured by different authors. The experimental results are due to Smith and Stiles (Ref. 27), Tan et al. (Ref. 3), and Padmanabhan et al. (Ref. 1) The $GW$ results were obtained using the random-phase-approximation (RPA) effective interaction (Ref. 9) and the Kukkonen-Overhauser (KO) effective interaction (Ref. 28) by solving the Dyson equation self-consistently (SC) or within the on-shell approximation (OSA). We show the VMC results of Kwon et al. (Ref. 29) [which were later confirmed at the same system size at $r_s=1$ a.u. using transient-estimate DMC calculations (Ref. 30)], the VMC results of Holzmann et al. (Ref. 23) and the DMC results reported in our previous work (Ref. 20) as well as the results of the present work. All the results shown are for paramagnetic HEGs with the exception of the experimental results of Ref. 1, which are for a partially spin-polarized HEG.

Figure 9

Quasiparticle effective mass $m^{*}$ against density parameter $r_s$ for ferromagnetic 2D HEGs. The $GW$ results were obtained using the Kukkonen-Overhauser (KO) effective interaction by solving the Dyson equation self-consistently (SC) or within the on-shell approximation (OSA) (Ref. 31). The experimental results are due to Padmanabhan et al. (Ref. 1). We show the DMC results reported in our earlier work (Ref. 20) in addition to our current results.