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Exact firstorder effect of interactions on the groundstate energy of harmonicallyconfined fermions
by Pierre Le Doussal, Naftali R. Smith, Nathan Argaman
Submission summary
Submission information 
Preprint Link: 
scipost_202311_00047v1
(pdf)

Date submitted: 
20231128 12:43 
Submitted by: 
Smith, Naftali 
Submitted to: 
SciPost Physics 
Ontological classification 
Academic field: 
Physics 
Specialties: 
 Quantum Physics
 Statistical and Soft Matter Physics

Approach: 
Theoretical 
Abstract
We consider a system of $N$ spinless fermions, interacting with each other via a powerlaw 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 groundstate 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 a Coulomb interaction $n=1$, 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 leadingorder term that is predicted from the local density approximation applied to the ThomasFermi 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.