Bond-Operator Mean Field Theory for the Bilayer Heisenberg Model
  Bond-operator mean field equations for the square-lattice, S=1/2 bilayer Heisenberg model are developed and solved numerically. In the vicinity of both the zero-field critical point and the field-induced transitions, comparisons are made with T=0 and finite-temperature strong coupling expansions. The mean-field theory suggests that the quantum critical region for the field-induced transitions is restricted to significantly lower temperatures than one might have concluded based on strong-coupling expansions or other numerical studies. 
Yasuhiro Matsushita (1), Martin P. Gelfand (2) and Chikara Ishii (1)   ((1) Science Univ. of Tokyo, (2) Colorado State Univ.)