Geometric-phase interference in a Mn$_{12}$ single-molecule magnet with four-fold rotational symmetry*

We study the magnetic relaxation rate $\Gamma$ of the single-molecule magnet Mn$_{12}$-tBuAc as a function of magnetic field component $H_T$ transverse to the molecule's easy axis. When the spin is near a magnetic quantum tunneling resonance, we find that $\Gamma$ increases abruptly at certain values of $H_T$. These increases are observed just beyond values of $H_T$ at which a geometric-phase interference effect suppresses tunneling between two excited energy levels. The effect is washed out by rotating $H_T$ away from the spin's hard axis, thereby suppressing the interference effect. Detailed numerical calculations of $\Gamma$ using the known spin Hamiltonian accurately reproduce the observed behavior. These results are the first experimental evidence for geometric-phase interference in a single-molecule magnet with true four-fold symmetry. Furthermore, the results demonstrate that geometric-phase-interference effects can play a role in the thermally assisted tunneling regime.