Neutron scattering investigation of magnetic phases in Mn$_3$O$_4$ with field applied away from the easy axis

Mn$_3$O$_4$ is an orbitally ordered, magnetically frustrated spinel with strong spin-lattice coupling, which exhibits a series of low temperature magnetic and structural transitions. Recent data shows that the structural phases are radically different with \textbf{H} applied perpendicular to the easy-axis. With the current understanding of the magneto-structural coupling in this material, this data suggests the possibility of a field-tuned quantum phase transition into a tetragonal spin-disordered phase with T=0 spin frustration. In order to probe the field-direction dependence of the magnetic phase, we have performed elastic neutron scattering measurements of the magnetic properties of Mn$_3$O$_4$ with applied field at several different angles to the easy axis. We will present data suggesting that the field-temperature phase diagram of this material is radically altered by varying the applied field direction.