Exactly Solvable System of One-Dimensional Trapped Bosons with Short- and Long-Range Interactions

Report on scipost_202107_00025v1

“…This term is the generalization to arbitrary spatial dimension of the two-body function long-range term found in the long-range Lieb-Liniger model [14,19]. We also note that this term reduces to a constant in the case of S U(1, 1) systems considered by Gambardella [32].…”

“…which is consistent with [32] (equation (27) and comment before that). Notice that for d = 1, we find that the three-body term is constant and is equal to the lower bound above [19]. From the observation above, the ground-state energy is minimized in a classical configuration in which particles are located at the apex of d-dimensional regular simplex blocks (e.g., equilateral triangles for d = 2, tetrahedron for d = 3) with edges of characteristic length a = 1/c.…”

“…and by the external potential (20). This can be seen as a higher-dimensional generalization of the confinement-induced long-range term in the modified Lieb-Liniger model [14,19].…”

Parent Hamiltonians of Jastrow wavefunctions

“…This term is the generalization to arbitrary spatial dimension of the two-body function longrange term found in the long-range Lieb-Liniger model [16,21]. We also note that this term reduces to a constant in the case of SU(1, 1) systems considered by Gambardella [34].…”

“…While in general eigenstates take the form of the Bethe ansatz, for attractive interactions the Jastrow form appears in the McGuire bright quantum soliton solution [20]. This feature is preserved upon embedding in a harmonic trap, provided the Hamiltonian is supplemented with long-range interactions [21]. In the case of hard-core repulsive interactions known as the Tonks-Girardeau gas [1], the Jastrow form is well known under harmonic confinement [22].…”