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.
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