  We describe the phase diagram and thermodynamic properties of a chain of
axially-tunnel-coupled fractional quantum Hall systems realized by rotating a
series of optical dipole traps about their center. We demonstrate not only a
experimentally feasible pathway to a state describable as a Mott-insulator of
composite bosons, but also describe the nature of the coherent states at higher
tunnel coupling strength, and identify a series of new superfluid phases with
rich behavior. The phase diagram directly reveals not only characteristic
features of the few-body systems, including the effective mass of composite
particle- and hole-like excitations and their interactions, but emergent
properties of the chain also reveal a fundamental mapping between the adiabatic
dynamics of two dimensional systems governed by particle braiding and the
hydrostatic response of the gas in the conducting phases.
