Universal Composite Boson Formation in Strongly Interacting One-Dimensional Fermionic Systems

Attractive $p$-wave one-dimensional fermions are studied in the fermionic Tonks-Girardeau regime in which the diagonal properties are shared with those of an ideal Bose gas. We study the off-diagonal properties and present analytical expressions for the eigenvalues of the one-body density matrix. One striking aspect is the universality of the occupation numbers which are independent of the specific shape of the external potential. We show that the occupation of natural orbitals occurs in pairs, indicating the formation of composite bosons, each consisting of two attractive fermions. The formation of composite bosons sheds light on the pairing mechanism of the system orbitals, yielding a total density equal to that of a Bose-Einstein condensate.

Figure 1

Eigenvalues ${\lambda }_{k\pm }$ as a function of the pair number $k$ for different number of fermions $N$ with symbols. The approximation to Lorentzian shape, see Eq. (14), is shown with a continuous line for each number of fermions.

Figure 2

OBDM for $N=2$, $N=4$, and $N=10$ for the case of the untrapped case, left column, and a harmonic trap, right column.

Figure 3

Fermionic natural orbitals and composite boson density profile of an untrapped FTG gas for an even number of particles. First and second columns, the first six natural orbitals, ${\chi }_{k+}(x)$ and ${\chi }_{k-}(x)$, respectively, corresponding to the three largest doubly degenerate eigenvalues $k=1$, 2, 3 of the OBDM. Third column, the first three composite boson density profiles $P_k(x)$.

Figure 4

Fermionic natural orbitals and composite boson density profile in a harmonic trap potential for an even number of particles. First and second columns, the first six natural orbitals, ${\chi }_{k+}(x)$ and ${\chi }_{k-}(x)$, respectively, corresponding to the three largest doubly degenerate eigenvalues $k=1$, 2, 3 of the OBDM. Third column, the first three composite boson density profiles $P_k(x)$.