Fractional energy states of strongly-interacting bosons in one dimension

We study two-component bosonic systems with strong inter-species and vanishing intra-species interactions. A new class of exact eigenstates is found with energies that are {\it not} sums of the single-particle energies with wave functions that have the characteristic feature that they vanish over extended regions of coordinate space. This is demonstrated in an analytically solvable model for three equal mass particles, two of which are identical bosons, which is exact in the strongly-interacting limit. We numerically verify our results by presenting the first application of the stochastic variational method to this kind of system. We also demonstrate that the limit where both inter- and intra-component interactions become strong must be treated with extreme care as these limits do not commute. Moreover, we argue that such states are generic also for general multi-component systems with more than three particles. The states can be probed using the same techniques that have recently been used for fermionic few-body systems in quasi-1D.