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.

💡 Research Summary

In this paper the authors employ the diffusion quantum Monte Carlo (DMC) method to obtain the full quasiparticle energy band of the two‑dimensional homogeneous electron gas (2D HEG) and, from it, the quasiparticle effective mass m* and the occupied bandwidth ΔE. The study treats both the paramagnetic (unpolarized) and fully ferromagnetic (spin‑polarized) phases over a wide range of density parameters rₛ (the radius of a circle containing one electron on average). By adding or removing a single electron from a specific momentum state k and computing the total‑energy difference, they directly evaluate the single‑particle excitation energy E(k). This definition coincides with the quasiparticle band near the Fermi surface, allowing a reliable extraction of m* via the relation m* = ℏ²k_F / (∂E/∂k)_{k_F}.

The trial wave functions consist of Slater determinants of plane‑wave orbitals multiplied by a Jastrow correlation factor; backflow transformations are incorporated to improve the nodal surface and recover a larger fraction of the correlation energy. The wave functions are first optimized in variational Monte Carlo (VMC) using variance minimization and linear‑least‑squares energy minimization, then used in fixed‑node DMC calculations. The authors systematically test for finite‑size effects by varying the number of electrons (N = 26–202) and the simulation‑cell Bloch vector, and they demonstrate that the band data are essentially size‑independent. Time‑step convergence is also verified, with consistent results obtained for τ ranging from 0.01 to 0.4 a.u. depending on rₛ.

The DMC energy bands are fitted to a quartic polynomial E(k) = α₀ + α₂k² + α₄k⁴ over the entire occupied region. This global fit mitigates statistical noise that would otherwise dominate a local derivative at k_F, and it also smooths out the pathological Hartree‑Fock‑like behavior that can appear near the Fermi surface in finite cells. The fitted bands reveal clear non‑quadratic curvature: for the paramagnetic fluid the quartic coefficient α₄ is positive at low densities (rₛ = 5, 10 a.u.), whereas for the ferromagnetic fluid α₄ is negative at all densities. Consequently, the occupied bandwidth ΔE = E(k_F) – E(0) is smaller than the free‑electron value for the paramagnetic case at intermediate and low densities, while it exceeds the free‑electron bandwidth for the ferromagnetic case at every density studied. In all cases the DMC bandwidth is substantially smaller than the Hartree‑Fock bandwidth, reflecting the high degree (≈99 %) of correlation energy recovered by the backflow‑enhanced trial functions.

From the fitted bands the effective masses are extracted. In the paramagnetic HEG, m* grows markedly with decreasing density: at rₛ = 1 a.u. the mass is slightly below the bare electron mass, whereas at rₛ = 5 a.u. it is enhanced by roughly 40 % and at rₛ = 10 a.u. by about 60 %. By contrast, in the ferromagnetic HEG the effective mass decreases as the density is lowered, dropping to ≈0.8 mₑ at rₛ = 10 a.u. These trends are in quantitative agreement with recent experimental measurements by Tan et al. (2005), Padmanabhan et al. (2008), and Gokmen et al. (2009). The authors note that earlier GW calculations produce a wide spread of m* values depending on the choice of effective interaction (RPA versus Kukkonen‑Overhauser) and on whether the Dyson equation is solved self‑consistently; many of those GW results do not reproduce the experimentally observed density dependence. Earlier QMC work by Kwon et al. also failed to show a significant mass enhancement in the paramagnetic fluid, likely because of a lower correlation‑energy recovery, smaller system sizes, and the use of particle‑promotion excitations rather than simple addition/removal of electrons.

The paper includes a careful error analysis. Finite‑size biases are shown to be negligible by comparing results from different cell sizes and Bloch vectors. Time‑step errors are demonstrated to be small by the agreement of bands obtained with multiple τ values. The authors discuss that any overestimation of the bandwidth (by up to ≈9 % at rₛ = 10 a.u. for the paramagnetic case) would lead to a comparable underestimation of m*, but this systematic effect is much smaller than the statistical uncertainties and does not affect the main conclusions.

In summary, this work provides the first comprehensive DMC determination of the full occupied quasiparticle band of the 2D HEG, delivering highly accurate effective masses for both spin‑unpolarized and fully spin‑polarized phases across a broad density range. The results resolve longstanding discrepancies between theory and experiment, demonstrate the superiority of backflow‑enhanced DMC over GW and earlier QMC approaches for this problem, and lay a solid foundation for future calculations of other Fermi‑liquid parameters, temperature‑dependent properties, and spin‑tronic device modeling based on realistic 2D electron systems.