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We show how to calculate the amplitude for quantum spin tunneling with a $z$easy axis. In this case, using the usual spherical polar angles to parametrizespin coherent states, typically the tunnelling variable is the polar angle$\theta$, passing from 0 to $\pi$, while the azimuthal angle $\phi$ isimaginary and constant. As the energy is constant along the tunnellingtrajectory and normalized to vanish, and the Wess-Zumino term $S_{WZ} = is\intd\tau \thinspace\dot{\phi}(1-\cos\theta )$ also seems to vanish, it isdifficult to see how we can obtain a non-zero contribution to the tunnellingamplitude. Indeed, the instanton trajectory gives zero contribution. We showhow, the entire contribution to the tunnelling amplitude in fact comes frommoving $\phi$ from zero to its complex value at the beginning of the tunnellingtrajectory at $\theta=0$ and moving it back to zero at the end at $\theta=\pi$.The Wess-Zumino term can also be expressed in a coordinate system independentmanner as $S_{WZ} =is\int d\tau d\xi \left[\hat n\cdot (\partial_\tau\hatn\times\partial_\xi\hat n)\right]$, a format that does not appear to be wellknown or understood in the condensed matter literature. We show in detail howthis expression corresponds exactly to the more familiar, coordinate systemdependent expression, and how to obtain the tunnelling amplitude directly fromit.

Coordinate (in)dependence and quantum inteference in quantum spin tunnelling