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arXiv:0707.0833 · Journal of Physics A Mathematical and Theoretical

Kirzhnits gradient expansion for aD-dimensional Fermi gas

Luca Salasnich

Published Aug 1, 2007• 46 citations
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Abstract

For an ideal D-dimensional Fermi gas under generic external confinement we derive the correcting coefficient $(D-2)/3D$ of the von Weizsacker term in the kinetic energy density. To obtain this coefficient we use the Kirzhnits semiclassical expansion of the number operator up to the second order in the Planck constant $\hbar$. Within this simple and direct approach we determine the differential equation of the density profile and the density functional of the Fermi gas. In the case D=2 we find that the Kirzhnits gradient corrections vanish to all order in $\hbar$.

DOI
10.1088/1751-8113/40/33/004

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For an ideal D-dimensional Fermi gas under generic external confinement we derive the correcting coefficient (D−2)/3D(D-2)/3D(D−2)/3D of the von Weizsacker term in the kinetic energy density. To obtain this coefficient we use the Kirzhnits semiclassical expansion of the number operator up to the second order in the Planck constant ℏ\hbarℏ. Within this simple and direct approach we determine the differential equation of the density profile and the density functional of the Fermi gas. In the case D=2 we find that the Kirzhnits gradient corrections vanish to all order in ℏ\hbarℏ.
We examine the leading-order semiclassical gradient corrections to the noninteracting kinetic-energy density functional of a two-dimensional Fermi gas by applying the extended Thomas-Fermi theory at finite temperature. We find a nonzero von Weizs\"acker-like gradient correction, which in the high-temperature limit goes over to the functional form $(\ensuremath{\hbar}{}^{2}/24m)(\ensuremath{\nabla}\ensuremath{\rho}){}^{2}/\ensuremath{\rho}$. Our work provides a theoretical justification for the inclusion of gradient corrections in applications of density-functional theory to inhomogeneous two-dimensional Fermi systems at any finite temperature.
The semiclassical $\ensuremath{\hbar}$ expansion of the one-particle density matrix for a two-dimensional Fermi gas is calculated within the Wigner transform method of B. Grammaticos and A. Voros [Ann. Phys. (N.Y.) 123, 359 (1979)], originally developed in the context of nuclear physics. The method of Grammaticos and Voros has the virtue of preserving both the Hermiticity and idempotency of the density matrix to all orders in the $\ensuremath{\hbar}$ expansion. As a topical application, we use our semiclassical expansion to go beyond the local-density approximation for the construction of the total dipole-dipole interaction energy functional of a two-dimensional, spin-polarized dipolar Fermi gas. We find a finite, second-order gradient correction to the Hartree-Fock energy, which takes the form $\ensuremath{\varepsilon}{(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}/\sqrt{\ensuremath{\rho}}$, with $\ensuremath{\varepsilon}$ being small ($|\ensuremath{\varepsilon}|\ensuremath{\ll}1$) and negative. We test the quality of the corrected energy by comparing it with the exact results available for harmonic confinement. Even for small particle numbers, the gradient correction to the dipole-dipole energy provides a significant improvement over the local-density approximation.
In earlier work, expressions have been constructed for the single-particle kinetic-energy functional ${T}_{s}[\ensuremath{\rho}]$ for independent fermions subject to harmonic confinement in low dimensions, with $\ensuremath{\rho}$ the particle density. Here, the differential equation for $\ensuremath{\rho}$ is first obtained in d dimensions for an arbitrary number of closed shells. Then, by using the known Euler-Lagrange equation, the functional derivative $\ensuremath{\delta}{T}_{s}/\ensuremath{\delta}\ensuremath{\rho}(\mathbf{r})$ is constructed. ${T}_{s}[\ensuremath{\rho}]$ itself is proved to take the form of a linear combination of three pieces: (i) a von Weizs\"acker inhomogeneity kinetic energy, but with the original coefficient reduced by a dimensionality factor 1/d,1/d,1/d, (ii) a Thomas-Fermi kinetic energy, and (iii) a truly nonlocal contribution that, however, is shown to involve only the density $\ensuremath{\rho}$ itself and its first derivative. Thus, for this model, which is currently highly relevant to the interpretation of experiments on the evaporative cooling of dilute, and hence almost noninteracting, fermions, a complete density-functional theory now exists.
We consider noninteracting Fermion system in an external field, confined to one-dimensional space. Our main objective is to take advantage of low dimension to find explicitly potential-density profile as well as kinetic energy functional in the ground state. This is accomplished by deriving the complete x-space linear response and a simple recursion representation of the gradient expansion for the kinetic energy. The results can serve as a potential test for higher dimensions, e.g., in exploring convergence properties of the gradient expansion in various regions of the energy spectrum.
We determine the energy density $\ensuremath{\xi}(3∕5)n{\ensuremath{\epsilon}}_{F}$ and the gradient correction $\ensuremath{\lambda}{\ensuremath{\hbar}}^{2}{(\mathbf{\ensuremath{\nabla}}n)}^{2}∕(8mn)$ of the extended Thomas-Fermi (ETF) density functional, where nnn is the number density and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that $\ensuremath{\xi}=0.455$ and $\ensuremath{\lambda}=0.13$ give the best fit of the DMC data with an even number NNN of particles. We also study the odd-even splitting $\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\ensuremath{\omega}$ determining the constant $\ensuremath{\gamma}$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.
The local-density approximation (LDA) is the central technical tool in the modeling of quantum gases in trapping potentials. It consists in treating the gas as an assembly of independent mesoscopic fluid cells at equilibrium with a local chemical potential, and it is justified when the correlation length is larger than the size of the cells. The LDA is often regarded as a crude approximation, particularly in the ground state of the one-dimensional (1D) Bose gas, where the correlation length is ``therefore said to be'' infinite (in the sense that correlation functions decay as a power law). Here we take another look at the LDA. The local density $\ensuremath{\rho}(x)$ is viewed as a functional of the trapping potential V(x)V(x)V(x), to which one applies a gradient expansion. The zeroth order in that expansion is the LDA. The first-order correction in the gradient expansion vanishes due to reflection symmetry. At second order, there are two corrections proportional to d2V/dx2{d}^{2}V/d{x}^{2}d2V/dx2 and (dV/dx)2{(dV/dx)}^{2}(dV/dx)2, and we propose a method to determine the corresponding coefficients by a perturbative calculation in the Lieb-Liniger model. This leads to an expression for the coefficients in terms of matrix elements of the density operator, which can in principle be evaluated numerically for an arbitrary coupling constant; here we show how to efficiently evaluate the coefficient associated to the curvature of the potential d2V/dx2{d}^{2}V/d{x}^{2}d2V/dx2, which dominates the deviation to LDA near local minima or maxima of the trapping potential. Both coefficients are evaluated analytically in the limits of infinite repulsion (hard-core bosons) and small repulsion (quasicondensate). The corrected LDA density profiles are compared to density-matrix renormalization group calculations, with significant improvement compared to zeroth-order LDA.