Quantum Monte Carlo study of the phase diagram of the two-dimensional uniform electron liquid

Abstract

We present a study of spin-unpolarized and spin-polarized two-dimensional uniform electron liquids using variational and diffusion quantum Monte Carlo (VMC and DMC) methods with Slater-Jastrow-backflow trial wave functions. Ground-state VMC and DMC energies are obtained in the density range $1 \leq r_\text{s} \leq 40$. Single-particle and many-body finite-size errors are corrected using canonical-ensemble twist-averaged boundary conditions and extrapolation of twist-averaged energies to the thermodynamic limit of infinite system size. System-size-dependent errors in Slater-Jastrow-backflow DMC energies caused by partially converged VMC energy minimization calculations are discussed. We find that, for $1 \leq r_\text{s} \leq 5$, optimizing the backflow function at each twist lowers the twist-averaged DMC energy at finite system size. However, nonsystematic system-size-dependent effects remain in the DMC energies, which can be partially removed by extrapolation from multiple finite system sizes to infinite system size. We attribute these nonsystematic effects to the close competition between fluid and defected crystal phases at different system sizes at low density. The DMC energies in the thermodynamic limit are used to parameterize a local spin density approximation correlation functional for inhomogeneous electron systems. Our zero-temperature phase diagram shows a single transition from a paramagnetic fluid to a hexagonal Wigner crystal at $r_\text{s}=35(1)$, with no region of stability for a ferromagnetic fluid.

Figures (4)

• Figure 1: SJB VMC energy against energy minimization cycle for paramagnetic 2D UELs with density parameters $r_\text{s}=30$ (top panel) and 35 (bottom panel). The systems studied had $N=218$, 242, 254, and 302 electrons. The initial WF was optimized by minimizing the mean absolute deviation from the median local energy. The real WF was optimized at zero twist. The energies of the 2D UELs at $r_\text{s}=30$ were obtained using hexagonal and square simulation cells.
• Figure 2: TA DMC energies for paramagnetic ($\zeta=0$) 2D UELs with density parameter $r_\text{s}=30$ obtained using SJ and SJB wave functions. The differences between the SJB and SJ energies for $N=218$, 242, 254, and 302 are $-0.041(2)$, $-0.042(4)$, $-0.068(1)$, and $-0.037(2)$ mHa/el., respectively.
• Figure 3: Extrapolation of TA DMC fluid energies to the thermodynamic limit of infinite system size for paramagnetic ($\zeta=0$) 2D UELs with density parameter $r_\text{s}=30$. The red data points are assumed to be trapped in nonglobal minima during optimization and do not correspond to the fluid ground state.
• Figure 4: DMC energy extrapolated to infinite system size as a function of $r_\text{s}$ for 2D UELs with spin polarizations $\zeta=0$, 0.5, and 1. Our results ("Prs-wrk") are compared with those of Drummond and Needs Neilprl09, which include paramagnetic ($\zeta=0$) and ferromagnetic fluids ($\zeta=1$), and ferromagnetic and antiferromagnetic hexagonal Wigner crystals. The Madelung energy of a hexagonal lattice, $-v_\text{M0}/r_\text{s}$, where $v_\text{M0}=-0.50751467391482663$ is the Madelung constant at $r_\text{s}=1$, has been subtracted from all the DMC energies.