Ground state phases of the two-dimension electron gas with a unified variational approach

TL;DR

The paper investigates ground-state phases of the two-dimensional electron gas (2DEG) across densities using a unified variational approach based on a Slater-Jastrow-backflow neural quantum state with multi-planewave orbitals. They introduce the (MP)$^2$NQS ansatz, combining MP-NQS backflow with multiple plane-wave components and a shared Jastrow term, optimized via Metropolis-Langevin sampling and SPRING SR. Their variational energies beat previous diffusion Monte Carlo benchmarks and the transition to a Wigner crystal occurs automatically at $r_s=37\pm1$, with an intermediate nematic spin-correlated liquid (NSCL) phase featuring short-range anisotropic spin correlations. The results demonstrate that a single, highly expressive variational ansatz can describe liquid, intermediate, and crystal phases within a unified framework, offering new insights into microemulsion-like behavior in the 2DEG and implications for strongly correlated 2D materials.

Abstract

The two-dimensional electron gas (2DEG) is a fundamental model, which is drawing increasing interest because of recent advances in experimental and theoretical studies of 2D materials. Current understanding of the ground state of the 2DEG relies on quantum Monte Carlo calculations, based on variational comparisons of different ansatze for different phases. We use a single variational ansatz, a general backflow-type wave function using a message-passing neural quantum state architecture, for a unified description across the entire density range. The variational optimization consistently leads to lower ground-state energies than previous best results. Transition into a Wigner crystal (WC) phase occurs automatically at rs = 37 +/- 1, a density lower than currently believed. Between the liquid and WC phases, the same ansatz and variational search strongly suggest the existence of intermediate states in a broad range of densities, with enhanced short-range nematic spin correlations.

Figures (12)

• Figure 1: Accurate and automatic detection of the Wigner crystal transition. The ground-state energies from variational optimization of the unified (MP)$^2$NQS ansatz systematically outperform state-of-the-art results from diffusion Monte Carlo (DMC) using separate variational ansatze as trial wave functions (top panel). Statistical error of energies from the (MP)$^2$NQS ansatz is smaller than the line width. The inset shows a study near the WC transition using transfer learning, which preserves the character of the state while changing its density. The bottom panel shows the Bragg peak values in the static structure factor $S(\boldsymbol{k})$ calculated from (MP)$^2$NQS. The shaded region indicate uncertainty from the optimization process as discussed in the main text. Results here are obtained from simulation cells containing $N=56$ electrons.
• Figure 2: Charge-charge correlations for representative densities, shown as (a) the pair correlation function $\tilde{g}(\boldsymbol{r})$, and (b) the static structure factor $S(\boldsymbol{k})$. The main panels show the radially averaged quantities, while the insets show the full two dimensional versions. Each inset is divided into two halves, with the left showing $r_s=10$ and the right showing $r_s=40$. At $r_s=40$, long-range correlation is evident from the persistent oscillations in the tail of $\tilde{g}(r)$ and the corresponding Bragg peak (pink star) in $S(k)$. Results shown in the insets are from $N=120$.
• Figure 3: Signatures of the intermediate NSCL states. In (a), line cuts of the spin correlation function, $g_s(\boldsymbol{r})$, are shown scanning the densities between a prototypical liquid ($r_s=5$) and a WC ($r_s=45$). Significant short-range anisotropic spin correlations are present starting from $r_s \sim 10$. The anisotropy grows with $r_s$, with the nearest spin showing stronger (anti-parallel) correlation and drifting further toward the near-neighbor position in the WC, as shown in the inset, which plots the location of the first minimum position in the left half of the main panel, as a function of $r_s$. The 2D $g_s(\boldsymbol{r})$ of representative states are shown in (b) $r_s=5$, (c) $r_s=45$ and (d) $r_s=34$. At $r_s=34$, we found many local minima having total energy within $0.01$% of the lowest state. The spin densities of the six lowest-energy states are shown in (e), ordered by their energies starting with the lowest at top left, whose $g_s(\boldsymbol{r})$ is shown in (d), to the highest at bottom right of the panel.
• Figure S1: (a) Compiled optimization trials showing energy (top) and $S(k)$ peak (bottom) across different values of $r_s$. Grey open circles and a dotted line mark the lowest energy trials for a given density, while a lighter/darker color corresponds to a smaller/larger $S(k)$ peak. Shaded region corresponds to same of fig 1. (b) Energy difference to the lowest obtained state vs $r_s$. The different colors denote liquid states vs those that are WC.
• Figure S2: Fitted values of the peak location of $g_s(\boldsymbol{r})$ along the AFM direction as a function of $r_s$ for both rectangular and square cell of liquid phase and rectangular cell of crystal phase. The dotted horizontal line shows the averaged value of measured WC distance.