Bethe ansatz for a model violating Yang-Baxter conditions

Exact Bethe-ansatz (BA) solutions of many-body problems take the form of a sum of products of single-particle orbitals from a finite set (see [1] and references therein). They are applicable to systems with non-diffractive scattering, where a many-body collision can be represented as a sequence of two-body ones. The scattering is non-diffractive whenever the Yang–Baxter consistency conditions (YBC) are satisfied. The Lieb-Liniger-McGuire (LLMG) model of one-dimensional bosons with zero-range δ interactions in a flat potential satisfies the YBC and has a BA solution. Zero-range interaction hyperplanes divide the coordinate space into sectors. The problem is then solved in the coordinate representation by matching free-particle solutions in adjacent sectors, while the YBC ensure the unambiguity of this matching.

Fermi-Bose mapping relates the LLMG model to that of spin-polarized fermions with zero-range δ′ interactions, which describe p-wave scattering in tight atomic waveguides. Consequently, the latter model has to have a BA solution as well. In this case, the solutions in adjacent sectors can be matched, but the YBC are apparently violated, as can be checked by direct calculations.

In order to check the validity of this solution independently, we solve the Schroedinger equation in the momentum representation. While the coordinate-representation solution consists of matched wavefunctions in different sectors, the momentum-representation solution is global and matching ambiguities are avoided. Using computer algebra, we have proved that in the momentum representation, the Fermi-Bose mapping of the LLMG solution obeys the Schroedinger equation with δ′ interactions. This was done for bound states of 3, 4, and 5 atoms, as well as for 3 non-bound atoms and for atom-dimer scattering.

1. V. A. Yurovsky, M. Olshanii, and D. S. Weiss, Adv. At., Mol., Opt. Phys. 55, 61 (2008).