The one-particle density matrices for hard-core bosons in a one-dimensional harmonic trap are computed numerically for systems with up to 160 bosons. Diagonalization of the density matrix shows that the many-body ground state is not Bose-Einstein condensed. The ground-state occupation, the amplitude of the lowest natural orbital, and the zero momentum peak height scale as powers of the particle number, and the corresponding exponents are related to each other. Close to its diagonal, the density matrix for hard-core bosons is similar to that of noninteracting fermions.
We investigate the dynamical properties of anyons confined in one-dimensional optical lattices combined with a weak harmonic trap using the exact numerical method based on a generalized Jordan-Wigner transformation. The evolving density profiles, momentum distributions, occupation distributions, and occupations of the lowest natural orbital after quench of the harmonic trap are obtained for different statistical parameters. The density profiles of anyons display the same behaviors irrespective of statistical parameter in the full evolving period, while the behaviors dependent on statistical property are shown in the momentum distributions and occupations of natural orbitals.
The exact N-particle ground-state wave function for a one-dimensional condensate of hard-core bosons in a harmonic trap is employed to obtain accurate numerical results for the one-particle density matrix, occupation number distribution of the natural orbitals, and momentum distribution. Our results show that the occupation of the lowest orbital varies as
N0.59, in contrast to
N0.5 for a spatially uniform system, and N for a true Bose-Einstein condensate.
We investigate the ground-state properties of anyons confined in one-dimensional optical lattices with a weak harmonic trap using the exact numerical method based on Jordan-Wigner transformation. It is shown that in the Bose limit $(\ensuremath{\chi}=1)$ and Fermi limit $(\ensuremath{\chi}=0)$ the momentum distributions are symmetric but in between they are asymmetric. It turns out that the origin of asymmetry comes from the fractional statistics that anyons obey. The occupation distribution and the modulus of natural orbitals show crossover behaviors from the Bose limit to the Fermi limit.
We study a one-dimensional system of interacting anyons with short-range interactions under external confinement. This model, which describes anyons with zero-range
s-wave and
p-wave scattering interactions in the strong interaction regime, continuously interpolates between spin-polarized fermions with
p-wave interactions and free bosons. At zero temperature, the correlation functions decay exponentially with distance, with oscillations governed by the statistics parameter. The decay rate is maximal for
p-wave fermions and decreases monotonically as the statistics parameter approaches the bosonic limit, where it vanishes. The momentum distribution is asymmetric, a hallmark of one-dimensional anyons, and takes the form of a shifted Lorentzian with universal power-law tails, ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{2}$. We prove analytically that, following release from a harmonic trap, the asymptotic momentum distribution converges to that of free bosons in the same trap, a phenomenon known as dynamical bosonization. We also establish the universality of the ground-state
n-particle reduced density matrices: their natural occupations are independent of the confining potential, while the associated natural
n-point functions for different confinements are related through a simple analytical transformation. In particular, for the one-particle reduced density matrix, we derive exact expressions for both the natural occupations and the natural orbitals at arbitrary particle number. These results extend and unify earlier partial findings for
p-wave fermions, and they provide a clear conceptual explanation of the double degeneracy observed in their spectrum.
Using the Anyon–Hubbard Hamiltonian, we analyze the ground-state properties of anyons in a one-dimensional lattice. To this end we map the hopping dynamics of correlated anyons to an occupation-dependent hopping Bose–Hubbard model using the fractional Jordan–Wigner transformation. In particular, we calculate the quasi-momentum distribution of anyons, which interpolates between Bose–Einstein and Fermi–Dirac statistics. Analytically, we apply a modified Gutzwiller mean-field approach, which goes beyond a classical one by including the influence of the fractional phase of anyons within the many-body wavefunction. Numerically, we use the density-matrix renormalization group by relying on the ansatz of matrix product states. As a result it turns out that the anyonic quasi-momentum distribution reveals both a peak-shift and an asymmetry which mainly originates from the nonlocal string property. In addition, we determine the corresponding quasi-momentum distribution of the Jordan–Wigner transformed bosons, where, in contrast to the hard-core case, we also observe an asymmetry for the soft-core case, which strongly depends on the particle number density.