Quantum Monte Carlo calculation of the energy band and quasiparticle effective mass of the two-dimensional Fermi fluid

We have used the diffusion quantum Monte Carlo method to calculate the energy band of the two-dimensional homogeneous electron gas (HEG), and hence we have obtained the quasiparticle effective mass and the occupied bandwidth. We find that the effective mass in the paramagnetic HEG increases significantly when the density is lowered, whereas it decreases in the fully ferromagnetic HEG. Our calculations therefore support the conclusions of recent experimental studies [Y.-W. Tan et al., Phys. Rev. Lett. 94, 016405 (2005); M. Padmanabhan et al., Phys. Rev. Lett. 101, 026402 (2008); T. Gokmen et al., Phys. Rev. B 79, 195311 (2009)]. We compare our calculated effective masses with other theoretical results and experimental measurements in the literature.

Figure 1

(Color online) VMC variance against energy for different trial wave functions for a 58-electron paramagnetic HEG of density parameter $r_s=5\text{a}\text{.u}\text{.}$ Plots are shown for the GS and an excited state in which an electron is removed from the highest-occupied shell. The slanted lines show fits to the VMC data. The vertical lines show the fixed-node DMC energies obtained with Slater-Jastrow-backflow wave functions. The HF ground-state and excited-state energies are $-0.100222$ and $-0.099632\text{a}\text{.u}\text{.}$ per electron, respectively. The difference between the slanted and vertical lines at zero variance gives an approximation to the correlation energy missing in the DMC calculation. This is clearly small compared with the difference between the HF and DMC energies.

Figure 2

(Color online) Energy bands of paramagnetic $N$-electron 2D HEGs at $r_s=1$ (top), 5 (middle), and 10 a.u. (bottom). The free-electron and HF bands are offset to coincide with the fitted DMC band at $k=k_F$. The curve labeled “SJ” used a Slater-Jastrow trial wave function; the others used a Slater-Jastrow-backflow trial wave function. Except where indicated otherwise, DMC time steps $\tau$ of 0.01, 0.2, and 0.4 a.u. were used at $r_s=1$, 5, and 10 a.u. The solid lines show quartic fits to the DMC data for $N=74$, 114, and 114 electrons at $r_s=1$, 5, and 10 a.u., respectively. (These are the largest system sizes for which we have sufficient data to perform an adequate fit.)

Figure 3

(Color online) As Fig. 2, but for ferromagnetic HEGs. The solid lines show quartic fits to the DMC data for $N=57$, 101, and 57 electrons at $r_s=1$, 5, and 10 a.u., respectively.

Figure 4

(Color online) Effective mass $m^{\ast }$ against density parameter $r_s$ for paramagnetic or partially spin-polarized 2D HEGs, as calculated or measured by different authors. Our DMC results were obtained from the fitted curves shown in Fig. 2. The $GW$ results were obtained using the random-phase-approximation effective interaction (Ref. 2) and the Kukkonen-Overhauser (KO) effective interaction (Ref. 8) by solving the Dyson equation self-consistently (SC) or within the on-shell approximation (OSA). All the results shown are for paramagnetic HEGs with the exception of the experimental results of Ref. 6, which are for a partially spin-polarized HEG.

Figure 5

(Color online) Effective mass $m^{\ast }$ against density parameter $r_s$ for ferromagnetic 2D HEGs. Our DMC results were obtained from the fitted curves shown in Fig. 3. The $GW$ results were obtained using the KO effective interaction by solving the Dyson equation SC or within the on-shell OSA. (Ref. 13)

Figure 6

(Color online) DMC effective mass $m^{\ast }$ against system size $N$ for paramagnetic 2D HEGs.

Figure 7

(Color online) HF energy bands (a) and their derivatives (b) for $N$-electron paramagnetic 2D HEGs of density parameter $r_s=10\text{a}\text{.u}\text{.}$

Figure 8

(Color online) Derivatives of the energy bands of paramagnetic $N$-electron 2D HEGs at $r_s=1$ (top), 5 (middle), and 10 a.u. (bottom). The curve labeled SJ used a Slater-Jastrow trial wave function; the others used a Slater-Jastrow-backflow trial wave function. Except where indicated otherwise, DMC time steps $\tau$ of 0.01, 0.2, and 0.4 a.u. were used at $r_s=1$, 5, and 10 a.u. The central difference approximation was used to evaluate the numerical derivatives. The solid lines show the derivatives of quartic fits to the DMC energy bands for $N=74$, 114, and 114 electrons at $r_s=1$, 5, and 10 a.u., respectively.

Figure 9

(Color online) As Fig. 8, but for ferromagnetic HEGs. The solid lines show the derivatives of quartic fits to the DMC energy bands for $N=57$, 101, and 57 electrons at $r_s=1$, 5, and 10 a.u., respectively.

Figure 10

(Color online) VMC renormalization factor $Z$ against number of electrons $N$ for paramagnetic 2D HEGs at $r_s=10\text{a}\text{.u}\text{.}$

Figure 11

(Color online) VMC momentum density $\rho \left(k\right)$ relative to the ground-state momentum density ${\rho }_{\text{GS}}\left(k\right)$ for different excitations to a paramagnetic 58-electron HEG at $r_s=10\text{a}\text{.u}\text{.}$ The momentum densities are averaged over reciprocal-lattice vectors of the same length, so the height of the spike at the $\left|\mathbf{k}\right|$ at which the electron is added looks smaller when there are many reciprocal-lattice vectors with the same length as $\left|\mathbf{k}\right|$.