Quasiparticle Effective Mass of the Three-Dimensional Fermi Liquid by Quantum Monte Carlo

According to Landau’s Fermi liquid theory, the main properties of the quasiparticle excitations of an electron gas are embodied in the effective mass $m^{*}$, which determines the energy of a single quasiparticle, and the Landau interaction function, which indicates how the energy of a quasiparticle is modified by the presence of other quasiparticles. This simple paradigm underlies most of our current understanding of the physical and chemical behavior of metallic systems. The quasiparticle effective mass of the three-dimensional homogeneous electron gas has been the subject of theoretical controversy, and there is a lack of experimental data. In this Letter, we deploy diffusion Monte Carlo (DMC) methods to calculate $m^{*}$ as a function of density for paramagnetic and ferromagnetic three-dimensional homogeneous electron gases. The DMC results indicate that $m^{*}$ decreases when the density is reduced, especially in the ferromagnetic case. The DMC quasiparticle energy bands exclude the possibility of a reduction in the occupied bandwidth relative to that of the free-electron model at density parameter $r_s=4$, which corresponds to Na metal.

Figure 1

First row: DMC energy bands for paramagnetic 3D HEGs at $r_s=1$, 5, and 10 obtained using system sizes of $N=54$, 130, 178, 226, and 274 electrons. Padé functions are fitted to the DMC data. Second row: DMC energy bands for ferromagnetic 3D HEGs at $r_s=1$, 5, and 10, obtained using system sizes of $N=137$, 181, and 229 electrons. Padé functions are fitted to the DMC data. DMC energy bands for paramagnetic and ferromagnetic 3D HEGs at $r_s=2$, 3, and 4 are plotted in the Supplemental Material [33].

Figure 2

Occupied bandwidths of paramagnetic and ferromagnetic 3D HEGs as a function of the density parameter $r_s$ calculated using the HF and DMC methods. All bandwidths are given relative to the corresponding free-electron value.

Figure 3

Quasiparticle effective masses $m^{*}$ of paramagnetic and ferromagnetic 3D HEGs as functions of $1/N$, where $N$ is the system size. The Monte Carlo statistical error bars are included, but they are smaller than the symbols. It should be noted that these error bars only quantify the noise in our data due to the Monte Carlo process. Unquantified noise includes (i) the effects of stochastically optimized backflow functions on the fixed-node DMC energy and (ii) quasirandom finite-size effects, such as Friedel oscillations being forced to be commensurate with the simulation cell.

Figure 4

Quasiparticle effective masses $m^{*}$ of paramagnetic (Para) and ferromagnetic (Ferro) 3D HEGs at the infinite-system-size limit as functions of density parameter $r_s$. Padé functions were fitted to the DMC quasiparticle energy bands to determine the effective mass. The many-body $G{W}_x$ and variational diagrammatic Monte Carlo (VDMC) results are from Refs. [52] and [53], respectively. The $GW$ SS and $GW$ SRPA results are from Refs. [54] and [55], respectively. The “standard $GW$” data and “modified $GW$” data are taken from Ref. [56]. All the $GW$ results are for paramagnetic 3D HEGs.