Laughlin has proposed mean-field wave functions for particles obeying (1-1/J) fractional statistics where J is an integer. We demonstrate off-diagonal long-range order in these wave functions, which is closely related to the off-diagonal long-range order in the electronic wave functions corresponding to J completely filled Landau levels. The presence of off-diagonal long-range order signifies Bose condensation of bosonlike multiplets containing J particles. For J=2, which may be relevant to high-temperature superconductivity, our result implies Cooper pairing in Laughlin's wave function for half-fermions.
We study a one-dimensional Kitaev model with uniform phase gradient pairing term. We show that the gradient constant significantly affects the phase diagram, which consists of topologically trivial and nontrivial phases, associated with Majorana edge modes. Based on the exact solution, a Bardeen-Cooper-Schrieffer-pair order parameter is introduced to characterize the phase diagram by its value and nonanalytic behavior at phase boundaries. We find that this order parameter obtains its maxima at the triple critical points, at which the pairing phase gradient suppresses the single-particle scattering process due to the coherent destructive interference. In particular, we show that the ground state at such a point possesses exact off-diagonal long-range order (ODLRO), in the thermodynamic limit. Our result provides an example of a gapless
p-wave superconducting ground state possessing ODLRO.
We consider a lattice of bosonic atoms, whose number
N may be smaller than the number of lattice sites
M. We study the Hartree-Fock wave function built up from localized wave functions
w(r) of single atoms, with nearest-neighboring overlap. The zero-momentum particle number is expressed in terms of permanents of matrices. In one dimension, it is analytically calculated to be $\ensuremath{\alpha}N(M\ensuremath{-}N+1)/M$, with $\ensuremath{\alpha}={|\ensuremath{\int}w(\mathbf{r})d\ensuremath{\Omega}|}^{2}/[(1+2a)l]$, where
a is the nearest-neighboring overlap and
l is the lattice constant. $\ensuremath{\alpha}$ is on the order of 1. The result indicates that the condensate fraction is proportional to and of the same order of magnitude as that of the vacancy concentration, hence there is off-diagonal long-range order or Bose-Einstein condensation of atoms when the number of vacancies $M\ensuremath{-}N$ is a finite fraction of the number of the lattice sites
M.