  Quantum fluctuations in the effective spin one-half layered structure
triangular-lattice quantum Heisenberg antiferromagnet Ba$_3$CoSb$_2$O$_9$ lift
the classical degeneracy of the antiferromagnetic ground state in magnetic
field, producing a series of novel spin structures for magnetic fields applied
within the crystallographic ab plane. Theoretically unresolved, however, are
the effects of interlayer antferromagnetic coupling and transverse magnetic
fields on the ground states of this system. To address these issues, we have
used specific heat, neutron diffraction, thermal conductivity, and magnetic
torque measurements to map out the phase diagram as a function of magnetic
field intensity and orientation relative to the crystallographic ab plane. For
H parallel to the ab plane, we have discovered an additional, previously
unreported magnetic-field-induced phase transition at low temperature and an
unexpected tetracritical point in the high field phase diagram, which - coupled
with the apparent second-order nature of the phase transitions - eliminates
several theoretically proposed spin structures for the high field phases. Our
calorimetric measurements as a function of magnetic field orientation are in
general agreement with theory for field-orientation angles close to plane
parallel but diverge at angles near plane perpendicular; a predicted
convergence of two phase boundaries at finite angle and a corresponding change
in the order of the field induced phase transition is not observed
experimentally. Our results emphasize the role of interlayer coupling in
selecting and stabilizing field-induced phases, provide new guidance into the
nature of the magnetic order in each phase, and reveal the need for new physics
to account for the nature of magnetic ordering in this archetypal 2D spin
one-half triangular lattice quantum Heisenberg antiferromagnet.
