Number-parity effect for confined fermions in one dimension

For $N$ spin-polarized fermions with harmonic pair interactions in a one-dimensional trap an odd-even effect is found. The spectrum of the one-particle reduced density matrix of the system's ground state differs qualitatively for $N$ odd and $N$ even. This effect only occurs for strong attractive and repulsive interactions. Since it does not exist for bosons, it must originate from the repulsive nature implied by the fermionic exchange statistics. In contrast to the spectrum, the one-particle density and correlation function for strong attractive interactions do not show any sensitivity on the number parity. This also suggests that reduced-density-matrix-functional theory has a more subtle $N$ dependency than density-functional theory.

Figure 1

Scaled effective potential as a function of $\sqrt{t}\tilde{p}$ for $N=3$ (left panel) and $N=4$ (right panel).

Figure 2

Spectrum $\left\{{\lambda }_k\right\}$ for $t={10}^{-3}$. Upper panel:   $N=3$ and $1\le k\le 40$; lower panel:   $N=4$ and $1\le k\le 62$. The solid lines present the corresponding analytical result for $\left\{{\lambda }_k\right\}$ from Eq. (11) and from the first term of Eq. (13), respectively. The insets show the final crossover from the regime of quasidegenerate to that of nonquasidegenerate eigenvalues.

Figure 3

$x$ dependency of the one-particle density for $N=3$ (left panel) and $N=4$ (right panel), where we set $l_{-}\equiv 1$. Blue dashed, black solid, and red dotted lines correspond to $t=0.1$ (attractive), $t=1$ (free fermions), and $t=10$ (repulsive), respectively [for $t=10$ it is plotted $10n\left(10x\right)]$.

Figure 4

$x$ and $y$ dependency of the two-particle density $n(x,y)$ (not normalized and we set $l_{-}\equiv 1$) for $t=0.1, N=3$ (left panel), and $N=4$ (right panel).