Deconfinement phase transition in a two-dimensional model of interacting $2\times 2$ plaquettes

A two-dimensional model of interacting plaquettes is studied by means of the real space renormalization group approach. Interactions between the plaquettes are mediated solely by spin excitations on the plaquettes. Depending on the plaquette-plaquette coupling $J$, we find two regimes: "confinement" $J_c< J\leq 1$, where the singlet ground state forms an infinite ("confined") cluster in the thermodynamical limit. Here the singlet-triplet gap vanishes, which is the signature for long range spin-spin correlators. "deconfinement" $0\leq J< J_c$, where the singlet ground state "deconfines" - i.e. factorizes - into finite $n$-clusters of size $2^n\times 2^n$, with $n\leq n_c(J)$. Here the singlet-triplet gap is finite. The critical value turns out to be $J_c=0.473528..$.