We present exact wave functions for the ground state of a lattice gas of spinless one-dimensional fermions, at filling factors of
1/M. The one-body hopping operator decreases exponentially. The two-body interaction is Coulomb-like: A neutralizing background is required for the system to be extensive. There is a parameter in the problem, which can be taken to be the strength of the charge. All correlation functions are evaluated, and the system is shown to have long-ranged crystalline order. For filling factor
1/2, the momentum distribution and pair correlation function are explicitly evaluated. There is no Fermi surface. The crystalline order parameter is evaluated.
The one-particle density matrix in the coordinate representation is calculated for the ideal Fermi gas in general D dimensions and a slow decay form of off-diagonal long-range order (ODLRO) is demonstrated therein. The pair distribution function and the structure factor are deduced from the one-particle density matrix. Their deviations from the classical values are viewed as a measure of the quantum phase, manifesting the existence of ODLRO. An upper bound on the structure factor due to Price [Phys. Rev. 94, 257 (1954)] is tested with our general solution and is also applied to the interacting electron gas.
The fermionic Tonks-Girardeau (FTG) gas is a one-dimensional spin-polarized Fermi gas with infinitely strong attractive zero-range odd-wave interactions, arising from a confinement-induced resonance reachable via a three-dimensional
p-wave Feshbach resonance. We investigate the off-diagonal long-range order (ODLRO) of the FTG gas subjected to a longitudinal harmonic confinement by analyzing the two-particle reduced density matrix for which we derive a closed-form expression. Using a variational approach and numerical diagonalization we find that the largest eigenvalue of the two-body density matrix is of order
N∕2, where
N is the total particle number, and hence a partial ODLRO is present for a FTG gas in the trap.