In this paper we study the quantum phase transition between the insulating
and the globally coherent superfluid phases in the Bose-Hubbard model with T_3
structure, the "dice lattice". Even in the absence of any frustration the
superfluid phase is characterized by modulation of the order parameter on the
different sublattices of the T_3 structure. The zero-temperature critical point
as a function of a magnetic field shows the characteristic "butterfly" form. At
fully frustration the superfluid region is strongly suppressed. In addition,
due to the existence of the Aharonov-Bohm cages at f=1/2, we find evidence for
the existence of an intermediate insulating phase characterized by a zero
superfluid stiffness but finite compressibility. In this intermediate phase
bosons are localized due to the external frustration and the topology of the
T_3 lattice. We name this new phase the Aharonov-Bohm (AB) insulator. In the
presence of charge frustration the phase diagram acquires the typical
lobe-structure. The form and hierarchy of the Mott insulating states with
fractional fillings, is dictated by the particular topology of the T_3 lattice.
The results presented in this paper were obtained by a variety of analytical
methods: mean-field and variational techniques to approach the phase boundary
from the superconducting side, and a strongly coupled expansion appropriate for
the Mott insulating region. In addition we performed Quantum Monte Carlo
simulations of the corresponding (2+1)D XY model to corroborate the analytical
calculations with a more accurate quantitative analysis. We finally discuss
experimental realization of the T_3 lattice both with optical lattices and with
Josephson junction arrays.Comment: 16 pages, 17 figure