  Collective tunneling is a ubiquitous phenomenon in finite-size spin clusters
that shows up in systems as diverse as molecular magnets or spin clusters
adsorbed at surfaces. The problem we explore is to understand how small
flipping terms can cooperate to flip a large spin to the opposite direction or
a cluster of interacting elementary Ising spins into the time-reversed state.
These high order processes will involve at least two channels, a single
spin-flip channel due to a transverse field and a two-spin flip channel due to
exchange or due to single-ion anisotropies. In the present paper, we show that
high-order perturbation theory can be formulated and evaluated with the help of
simple recurrence relations, leading to a compact theory of tunnelling in
macroscopic spins, in one-dimensional clusters, as well as in small
higher-dimensional clusters. This is demonstrated explicitly in the case of the
Ising model with a transverse field and transverse exchange, and in the case of
macroscopic spins with uniaxial anisotropy. Our approach provides a transparent
theory of level crossings, where the tunneling between time reversed
configurations vanishes as a function of the external field. The crossings
result from destructive quantum interferences between competing flipping
channels. Our theory consistently predicts N crossings in chains of N Ising
spins, 2S crossings in single spins of magnitude S, and yields explicit
analytical formulae for the level crossings of open chains and macroscopic
spins. Disorder can be easily implemented in this perturbative formalism.
Leading disorder effects can be treated analytically for spin rings. At the
smallest transverse field crossing, the suppression of tunneling is most robust
with respect to disorder. We briefly discuss the implications of our findings
for the use of realistic spin clusters on surfaces to store information.
