  Strongly interacting one-dimensional quantum systems often behave in a manner
that is distinctly different from their higher-dimensional counterparts. When a
particle attempts to move in a one-dimensional environment it will unavoidably
have to interact and 'push' other particles in order to execute a pattern of
motion, irrespective of whether the particles are fermions or bosons. A present
frontier in both theory and experiment are mixed systems of different species
and/or particles with multiple internal degrees of freedom. Here we consider
trapped two-component bosons with short-range inter-species interactions much
larger than their intra-species interactions and show that they have novel
energetic and magnetic properties. In the strongly interacting regime, these
systems have energies that are fractions of the basic harmonic oscillator trap
quantum and have spatially separated ground states with manifestly
ferromagnetic wave functions. Furthermore, we predict excited states that have
perfect antiferromagnetic ordering. This holds for both balanced and imbalanced
systems, and we show that it is a generic feature as one crosses from few- to
many-body systems.
