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arXiv:2311.09013 · SciPost Physics

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

Pierre Le Doussal, Naftali R. Smith, Nathan Argaman

Published Aug 7, 2024• 1 citation
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Abstract

We consider a system of N N spinless fermions, interacting with each other via a power-law interaction \epsilon/r^n ϵ / r n , and trapped in an external harmonic potential V(r) = r^2/2 V ( r ) = r 2 / 2 , in d=1,2,3 d = 1 , 2 , 3 dimensions. For any 0 < n < d+2 0 < n < d + 2 , we obtain the ground-state energy E_N E N of the system perturbatively in \epsilon ϵ , E_{N}=E_{N}^{≤ft(0)}+\epsilon E_{N}^{≤ft(1)}+O≤ft(\epsilon^{2}) E N = E N ≤ f t ( 0 ) + ϵ E N ≤ f t ( 1 ) + O ≤ f t ( ϵ 2 ) . We calculate E_{N}^{≤ft(1)} E N ≤ f t ( 1 ) exactly, assuming that N N is such that the “outer shell” is filled. For the case of n=1

DOI
10.21468/scipostphys.17.2.038

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We calculate the ground state energies of a system of two dipolar fermions trapped in a harmonic oscillator potential. The dipoles are assumed to be aligned parallel to each other. We perform the calculations of ground state energy as a function of strength of interaction between two fermions by employing variational method with Hylleraas-like explicitly correlated wave function. Furthermore, we perform calculations of ground state energy within Hartree-Fock approximation and the magnitude of correlation energy is estimated by subtracting these results from the corresponding wave function based results. We also carry out calculations of ground state energies within the realm of density functional theory by using recently reported expressions for exchange and correlation energies under local density approximation. By comparing correlated wave function based results with those obtained using density functional theory approach we examine the role of fermion-fermion correlation and assess the accuracy of local density approximation based expression for the correlation energy functional.
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We provide a detailed study of the properties of a few interacting spin 1 / 2 fermions trapped in a one-dimensional harmonic oscillator potential. The interaction is assumed to be well represented by a contact delta potential. Numerical results obtained by means of direct diagonalization techniques are combined with analytical expressions for both the non-interacting and strongly interacting regime. The N = 2 case is used to benchmark our numerical techniques with the known exact solution of the problem. After a detailed description of the numerical methods, in a tutorial-like manner, we present the static properties of the system for N = 2 , 3 , 4 and 5 particles, e.g., low-energy spectrum, one-body density matrix, ground-state densities. Then, we consider dynamical properties of the system exploring first the excitation of the breathing mode, using the dynamical structure function and corresponding sum-rules, and then a sudden quench of the interaction strength.
We propose a perturbative-variational approach to interacting fermion systems on 1D and 2D lattices at half-filling. We address relevant issues such as the existence of Long Range Order, quantum phase transitions and the evaluation of ground state energy. In 1D our method is capable of unveiling the existence of a critical point in the coupling constant at (t/U)c=0.7483(t/U)_c=0.7483(t/U)c​=0.7483 as in fact occurs in the exact solution at a value of 0.5. In our approach this phase transition is described as an example of Bifurcation Phenomena in the variational computation of the ground state energy. In 2D the van Hove singularity plays an essential role in changing the asymptotic behaviour of the system for large values of t/Ut/Ut/U. In particular, the staggered magnetization for large t/Ut/Ut/U does not display the Hartree-Fock law (t/U)e−2πt/U(t/U) e^{-2 π\sqrt{t/U}}(t/U)e−2πt/U​ but instead we find the law (t/U)e−π23t/U(t/U) e^{- \frac{π^2}{3} t/U}(t/U)e−3π2​t/U. Moreover, the system does not exhibit bifurcation phenomena and thus we do not find a critical point separating a CDW state from a fermion "liquid" state.