The simple relationship between the wavefunctions of a system of impenetrable one-dimensional bosons and impenetrable one-dimensional fermions is exploited to derive an expansion of the boson reduced density matrices in terms of the fermion reduced density matrices and vice versa. This expansion is independent of the statistical ensemble used, and of the interparticle potential (subject to the impenetrability condition). The special case of zero impenetrability radius with no other forces is treated in detail, using the grand canonical ensemble. This leads to a natural generalization of a formula previously found for zero temperature only, for the one-particle reduced density matrix.The simple relationship between the wavefunctions of a system of impenetrable one-dimensional bosons and impenetrable one-dimensional fermions is exploited to derive an expansion of the boson reduced density matrices in terms of the fermion reduced density matrices and vice versa. This expansion is independent of the statistical ensemble used, and of the interparticle potential (subject to the impenetrability condition). The special case of zero impenetrability radius with no other forces is treated in detail, using the grand canonical ensemble. This leads to a natural generalization of a formula previously found for zero temperature only, for the one-particle reduced density matrix.

Journal of Mathematical PhysicsJul 1, 1966183 citations

Two-dimensional boson–fermion mixtures in harmonic traps

B. Tanatar, E. Erdemir

Physica B Condensed MatterMay 1, 20032 citations

Finite one-dimensional impenetrable Bose systems: Occupation numbers

Peter J. Forrester, Norman E. Frankel, Timothy M. Garoni +1 more

Bosons in the form of ultracold alkali-metal atoms can be confined to a one-dimensional (1D) domain by the use of harmonic traps. This motivates the study of the ground-state occupations ${\ensuremath{\lambda}}_{i}$ of effective single-particle states ${\ensuremath{\varphi}}_{i},$ in the theoretical 1D impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The ${\ensuremath{\lambda}}_{i}$ and ${\ensuremath{\varphi}}_{i}$ are the eigenvalues and eigenfunctions, respectively, of the one-body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations ${\ensuremath{\lambda}}_{i}$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.Bosons in the form of ultracold alkali-metal atoms can be confined to a one-dimensional (1D) domain by the use of harmonic traps. This motivates the study of the ground-state occupations ${\ensuremath{\lambda}}_{i}$ of effective single-particle states ${\ensuremath{\varphi}}_{i},$ in the theoretical 1D impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The ${\ensuremath{\lambda}}_{i}$ and ${\ensuremath{\varphi}}_{i}$ are the eigenvalues and eigenfunctions, respectively, of the one-body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations ${\ensuremath{\lambda}}_{i}$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.

arXiv:cond-mat/0211126Physical Review AApr 17, 2003139 citations

\sqrt{N}

in both the circular and harmonically trapped cases.