Exclusion statistics: density matrix and its implications

Alexander A. Klyachko

The Pauli exclusion principle can be stated as inequality $\le 1$ for the electron density matrix $ρ$. Nowadays it is replaced by skew symmetry of the multi-electron wave function. The replacement leads to numerous additional constraints on $ρ$, which are discussed in this letter together with their physical implications, in particular for spin magnetic moment of a multi-electron system.

Douglas Lundholm, Jan Philip Solovej

A local exclusion principle is observed for identical particles obeying intermediate and fractional exchange statistics in one and two dimensions, leading to bounds for the kinetic energy in terms of the density. This has implications for models of Lieb-Liniger and Calogero-Sutherland type and implies a nontrivial lower bound for the energy of the anyon gas whenever the statistics parameter is an odd numerator fraction. We discuss whether this is actually a necessary requirement.

Zachary MacDonald, Brandon P. van Zyl

Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a d -dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile distribution continues to provide a good description of the system, with deviations only arising well outside of the degenerate regime. Our results illustrate that the exceedingly simple functional form of the Gentile distribution is an excellent alternative to the generally only implicit form of the Haldane distribution at low temperatures.