Quantum Monte Carlo study of the quasiparticle effective mass of the two-dimensional uniform electron liquid

TL;DR

This work addresses quasiparticle renormalization in the 2D-UEL by computing energy bands and the effective mass $m^*$ using real-space VMC and DMC for paramagnetic and ferromagnetic spin states in the metallic regime $1\leq r_s\leq 5$. It demonstrates that electron-electron correlations and nodal topology—captured via Slater-Jastrow and Slater-Jastrow-backflow wave functions—critically affect $m^*$, with backflow elevating $m^*$ at low density in the paramagnetic case and density-dependent suppression in the ferromagnetic case. Key methodological findings include the importance of DMC time-step control, per-$k$ wave-function optimization, and robust quartic fitting to extract band slopes near $k_F$. The results show $m^*$ near unity at $r_s=1$ for paramagnetic 2D-UEL, with $m^*$ increasing at lower density for paramagnetic and decreasing for ferromagnetic cases, reflecting a balance between correlation-enhanced mass and screening effects. These QMC benchmarks provide insight into Fermi-liquid renormalization in low-dimensional electron systems and inform comparisons with GW calculations and experimental trends.

Abstract

The real-space variation quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) are used to calculate the quasiparticle energy bands and the quasiparticle effective mass of the paramagnetic and ferromagnetic two-dimensional uniform electron liquid (2D-UEL)\@. The many-body finite-size errors are minimized by performing simulations for three system sizes with the number of electrons $N=146$, 218, and 302 for paramagnetic and $N=151$ for ferromagnetic systems. We consider 2D-UEL to be within the metallic density range $1\leq r_s \leq 5$. The VMC and DMC results predict that the quasiparticle effective mass $m^*$ of the paramagnetic 2D-UEL at high density $r_s=1$ is very close to 1, suggesting that effective mass renormalization due to electron-electron interaction is negligible. We find that $m^*$ of the paramagnetic 2D-UEL obtained by the VMC and DMC methods increases by $r_s$ but with different slopes. Our VMC and DMC results for ferromagnetic 2D-UEL indicate that $m^*$ decreases rapidly by reducing the density due to the strong suppression of the electron-electron interaction.

Figures (8)

• Figure 1: HF energy bands in (a.u.) for $N$-electron ($N=146$, 218, and 302) paramagnetic ($\zeta=0$) 2D-UEL with density parameter $r_s=1, 5$. HF energy band in the infinite system size limit (N$\rightarrow \infty$) defined in Eq. (\ref{['eq:HF']}) is also plotted. Legend for plot $r_s=5$ is the same as $r_s=1$.
• Figure 2: DMC and VMC energy bands (in a.u.) for paramagnetic and ferromagnetic 2D-UEL at $r_s=1$ and 5 obtained using system sizes $N=146$, 218, and 302 and $N=151$ for $\zeta=0$ and $\zeta=1$, respectively, and SJB wave function. A Padé function is fitted to the VMC and DMC data. The free-electron and HF bands are offset to coincide with the fitted VMC bands at $k=k_F$.
• Figure 3: VMC and DMC energy bands of N-electron (N=146) paramagnetic 2D-UEL obtained by SJ and SJB wave functions with density parameters $r_s=1,5$. The effective mass $m^*$ is obtained using a quartic function fit.
• Figure 4: VMC and DMC energy bands of paramagnetic 2D-UEL obtained using the same WF for all $k$ vectors ($0$ index) and the optimized WF at each $k$ vector ($1$ index). The energy bands are calculated for $r_s=1,5$ with $N=146$ electrons in the simulation cell and SJB wave function. Quartic function fitting is used to calculate the effective mass $m^*$. The free-electron and HF bands are offset to coincide with the fitted DMC bands at $k=k_F$.
• Figure 5: VMC and DMC energy bands of $N$-electron paramagnetic 2D-UEL ($N=302$) with density parameter $r_s=5$. The DMC energies are calculated using two time steps of 0.005 and 0.2 a.u. The effective mass is obtained using a quartic fitting.