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arXiv:1605.08014 · Physical review. B./Physical review. B

Manifestly Hermitian semiclassical expansion for the one-particle density matrix of a two-dimensional Fermi gas

K. Bencheikh, Brandon P. van Zyl, K. Berkane

Published Aug 15, 2016• 3 citations
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Abstract

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.

DOI
10.1103/physrevb.94.075423

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The semiclassical \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ 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 \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ 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 \ensuremathε(\ensuremath∇\ensuremathρ)2/\ensuremathρ\ensuremath{\varepsilon}{(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}/\sqrt{\ensuremath{\rho}}\ensuremathε(\ensuremath∇\ensuremathρ)2/\ensuremathρ​, with \ensuremathε\ensuremath{\varepsilon}\ensuremathε being small (∣\ensuremathε∣\ensuremath≪1|\ensuremath{\varepsilon}|\ensuremath{\ll}1∣\ensuremathε∣\ensuremath≪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.
Abstract The one-body density matrix (ODM) for a zero temperature non-interacting Fermi gas can be approximately obtained in the semiclassical regime through different ℏ -expansion techniques. One would expect that each method of approximating the ODM should yield equivalent density matrices which are both Hermitian and idempotent to any order in ℏ . However, the Kirzhnits and Wigner–Kirkwood methods do not yield these properties, while the Grammaticos–Voros method does. Here we show explicitly, for arbitrary d-dimensions through an appropriate change into symmetric coordinates, that each method is indeed identical, Hermitian, and idempotent. This change of variables resolves the inconsistencies between the various methods, showing that the non-Hermitian and non-idempotent behavior of the Kirzhnits and Wigner–Kirkwood methods is an artifact of performing a non-symmetric truncation to the semiclassical ℏ -expansions. Our work also provides the first explicit derivation of the d-dimensional Grammaticos–Voros ODM, originally proposed by Redjati et al (2019 J. Phys. Chem. Solids 134 313–8) based on their d = 1 , 2 , 3 , 4 expressions.
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.
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.
We present a density-matrix functional within the recently introduced framework for tensor-product expansions of the two-particle density matrix. It performs well both for the homogeneous electron gas as well as atoms. For the homogeneous electron gas, it performs significantly better than all previous density-matrix functionals, becoming very accurate for high densities and outperforming the Hartree-Fock method at metallic valence electron densities. For isolated atoms and ions, it is on par with generalized-gradient approximations to the density-functional theory. We also present analytic results for the correlation energy in the low-density limit of the free-electron gas for a broad class of such functionals.
The quantum corrections to the two-dimensional Wigner crystal, for filling \ensuremath{\nu}\ensuremath{\le}1/3, are discussed by using a Hartree-Fock expansion based on wave functions which are (i) related to one another by magnetic translations, (ii) orthonormal, and (iii) strongly localized. Such wave functions are constructed in terms of Gaussians that are localized at the sites of a triangular (Wigner) lattice and have a small overlap c. The ground-state energy per particle is calculated by an expansion in \ensuremath{\surd}\ensuremath{\nu} and in \ensuremath{\delta}\ensuremath{\equiv}c1/4{\mathit{c}}^{1/4}c1/4, which is rapidly convergent and stable under the thermodynamical limit. In particular, in this limit the cancellation of the infrared divergences occur order by order in the above expansion. The accurate control on the approximations allows a clear-cut comparison with the energy obtained by the Laughlin ansatz on the ground state and the numerical results confirm that the Wigner-crystal picture is energetically favored with respect to the Laughlin state for \ensuremath{\nu}1/9.
Density-functional theory is utilized to investigate the zero-temperature transition from a Fermi liquid to an inhomogeneous stripe, or Wigner crystal phase, predicted to occur in a one-component, spin-polarized, two-dimensional dipolar Fermi gas. Correlations are treated semiexactly within the local-density approximation using an empirical fit to quantum Monte Carlo data. We find that the inclusion of the nonlocal contribution to the Hartree-Fock energy is crucial for the onset of an instability to an inhomogeneous density distribution. Our density-functional theory supports a transition to both a one-dimensional stripe phase and a triangular Wigner crystal. However, we find that there is an instability first to the stripe phase, followed by a transition to the Wigner crystal at higher coupling.