Leading gradient correction to the kinetic energy for two-dimensional fermion gases

Density-functional theory (DFT) is notorious for the absence of gradient corrections to the two-dimensional (2D) Thomas-Fermi kinetic-energy functional; it is widely accepted that the 2D analog of the 3D von Weizsäcker correction vanishes, together with all higher-order corrections. Contrary to this long-held belief, we show that the leading correction to the kinetic energy does not vanish, is unambiguous, and contributes perturbatively to the total energy. This insight emerges naturally in a simple extension of standard DFT, which has the effective potential energy as a functional variable on equal footing with the single-particle density.

Figure 1

Energy of $N$ noninteracting fermions in an isotropic 2D harmonic trap with angular frequency $\omega$. Main plot: Energy per particle. Inset: Zoom on the energy divided by the TF scaling $N^{3/2}$. Circles connected by dotted lines are the exact energies [63]; dark circles denote the closed-shell values $N=2,6,12,20,30$. The dashed blue line is the TF approximation, Eq. (28), consistently below the exact energies. The solid red line includes the leading quantum correction Eq. (29), interpolating the oscillations of the exact values above the TF result.

Figure 2

Densities $n\left(r\right)$ of noninteracting spin-$\frac{1}{2}$ particles in an external harmonic-oscillator potential in the vicinity of the border between the classically allowed and forbidden regions. The abscissa is the radial distance $r=\left|\mathit{r}\right|$ in oscillator units. For five filled shells ($N=30$) and for the fifth shell half-full ($N=25$), the plot shows the exact densities $n_{\mathrm{ex}}^{}$, their TF approximations $n_{tf}^{}$, and the densities $n_{\mathrm{Ai}}^{}$ obtained with Airy-averaging techniques.