One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: I. One, Two, and Three Particles

Abstract

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies.The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the N particle case.

Infinite barriers and symmetries for a few trapped particles in one dimension

“…In particular, when the Hamiltonian splits into a sum of identical sub-Hamiltonians, then this symmetry can be described by the wreath product T t S N , where S N is the symmetric group on N particles, T t is the time-translation subgroup for each independent particle, and is the wreath product that interweaves S N with N copies of T t (described in more detail below). Interactions break this symmetry into the subgroup T t × S N [42,43].…”

“…In the case of totally indistinguishable bosons and fermions, these phase relations lift the degeneracy completely (i.e. the famous Bose-Fermi mapping of Girardeau [48]), otherwise it is more complicated for more than two particles [42,43,49,50].…”

“…The configuration space symmetry groups for cases IV and V only have one-dimensional irreps. The configuration space symmetry group for case VI is isomorphic to the symmetry of a square and does have a two-dimensional irrep, but explicit construction of wave functions shows that not all double-degenerate energy levels correspond to that irrep [42].…”

“…For non-interacting identical two particle systems, the kinematic group always has T P 2 as a subgroup [42]. Its subgroup T 1 × T 2 ⊂ T P 2 is the product of two commuting continuous symmetry transformations in phase space.…”

“…The eigensolutions at the unitary limit H τ ∞ can be algebraically constructed from the eigensolutions of H τ 0 by restricting the particle permutation antisymmetric states (19b) to the domains X I and X II , also called the 'snippet' basis [42,43,49,50,61]:…”