Collective dynamics of interacting Ising spins: Exact results for the Bethe lattice

Abstract

We study the low temperature dynamics in films made of molecular magnets, i. e. crystals composed of molecules having large electronic spin S in their ground state. The electronic spin dynamics is mediated by coupling to a nuclear spin bath; this coupling allows transitions for a small fraction of electronic spins between their two energy minima, Sz=+- S, under resonant conditions when the change of the Zeeman energy in magnetic dipolar field of other electronic spins is compensated by interaction with nuclear spins. Transitions of resonant spins can result in opening or closing resonances in their neighbors leading to the collective dynamics at sufficiently large density P0 of resonant spins. We formulate and solve the equivalent dynamic percolation problem for the Bethe lattice (BL) of spins interacting with z neighbors and find that depending on the density of resonant spins P0 and the number of neighbors z the system has either one (2 5) kinetic transitions at P_{0}=Pc1 ~ exp(-1/3)/(3z) and P0=Pc2 ~ exp(-1)/z. The former transition is continuous and associated with the formation of an infinite cluster of coupled resonant spins similarly to the static percolation transition occurring at P0 ~ 1/z. The latter transition, z>5, is discontinuous and associated with the instantaneous increase in the density of resonant spins from the small value ~ 1/z to near unity. Experimental implications of our results are discussed.