Three-body exclusion principle, duality mapping, and exact ground state of a harmonically trapped, ultracold Bose gas with three-body hard-core interactions in one dimension

摘要

Motivated by previous suggestions that three-body hard-core interactions inlower-dimensional ultracold Bose gases might provide a way for creation ofnon-Abelian anyons, the exact ground state of a harmonically trapped 1D Bosegas with three-body hard-core interactions is constructed by duality mapping,starting from an $N$-particle ideal gas of mixed symmetry with three-bodynodes, which has double occupation of the lowest harmonic oscillator orbitaland single occupation of the next $N-2$ orbitals. It has some similarity to theground state of a Tonks-Girardeau gas, but is more complicated. It is provedthat in 1D any system of $N\ge 3$ bosons with three-body hard-core interactionsalso has two-body soft-core interactions of generalized Lieb-Liniger deltafunction form, as a consequence of the topology of the configuration space of$N$ particles in 1D, i.e., wave functions with \emph{only} three-body hard corezeroes are topologically impossible. This is in contrast with the case of 2D,where pure three-body hard-core interactions do exist, and are closely relatedto the fractional quantized Hall effect. The exact ground state is comparedwith a previously-proposed Pfaffian-like approximate ground state, whichsatisfies the three-body hard-core constraint but is not an exact energyeigenstate. Both the exact ground state and the Pfaffian-like approximationimply two-body soft-core interactions as well as three-body hard-coreinteractions, in accord with the general topological proof.