Exact first-order effect of interactions on the ground-state energy of harmonically-confined fermions

Abstract

We consider a system of $N$ spinless fermions, interacting with each other via a power-law interaction $\epsilon /r^n$, and trapped in an external harmonic potential $V(r)=r^2/2$, in $d=1,2,3$ dimensions. For any $0<n<d+2$, we obtain the ground-state energy $E_N$ of the system perturbatively in $\epsilon$, $E_N=E_N^{\left(0\right)}+\epsilon E_N^{\left(1\right)}+O\left({\epsilon }^2\right)$. We calculate $E_N^{\left(1\right)}$ exactly, assuming that $N$ is such that the "outer shell" is filled. For the case of $n=1$ (corresponding to a Coulomb interaction for $d=3$), we extract the $N\gg 1$ behavior of $E_N^{\left(1\right)}$, focusing on the corrections to the exchange term with respect to the leading-order term that is predicted from the local density approximation applied to the Thomas-Fermi approximate density distribution. The leading correction contains a logarithmic divergence, and is of particular importance in the context of density functional theory. We also study the effect of the interactions on the fermions' spatial density. Finally, we find that our result for $E_N^{\left(1\right)}$ significantly simplifies in the case where $n$ is even.

arXiv comment40 pages, 5 figures

References in corpus (31)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Single-atom imaging of fermions in a quantum-gas microscope
• A Quantum Gas Microscope for Fermionic Atoms
• Density Functional Theory -- an introduction
• Site-resolved Imaging of Fermionic Lithium-6 in an Optical Lattice
• Long-range interacting quantum systems
• Homogeneous Atomic Fermi Gases
• Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases
• Edge Electron Gas
• The entanglement entropy of one-dimensional gases
• Universality in the full counting statistics of trapped fermions
• Exact relations between particle fluctuations and entanglement in Fermi gases
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Noninteracting fermions in a trap and random matrix theory
• Observation of Pauli Crystals
• Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions
• Locality of correlation in density functional theory
• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Full counting statistics for interacting trapped fermions
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
• Counting statistics for non-interacting fermions in a rotating trap
• Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap
• Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential
• Atoms and Quantum Dots With a Large Number of Electrons: the Ground State Energy
• Leading correction to the local density approximation for exchange in large-$Z$ atoms
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Basic theory tools for degenerate Fermi gases
• Flat Thomas-Fermi artificial atoms