Topologically Nontrivial Three-Body Contact Interaction in One Dimension

It is known that three-body contact interactions in one-dimensional $n(\ge 3)$-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group $P{T}_n$. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of the path integral. Focusing on spinless particles, we construct an $n(n-1)(n-2)\mathrm{/}3!$-parameter family of $n$-body Hamiltonians that corresponds to one particular one-dimensional unitary representation of $P{T}_n$. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely-thin magnetic fluxes in the $n$-body configuration space.