Mean-field theory for Heisenberg zigzag ladder: Ground state energy and   spontaneous symmetry breaking
  The spin-1/2 zig-zag Heisenberg ladder (J_1 - J_2 model) is considered. A new representation for the model is found and a saddle point approximation over the spin-liquid order parameter < \vec \sigma_{n-1}(\vec \sigma_{n}\times \vec \sigma_{n+1}) > is performed. Corresponding effective action is derived and analytically analyzed. We observe the presence of phase transitions at values J_2/J_1=0.231 and J_2/J_1=1/2. 
V. V. Mkhitaryan, T. A. Sedrakyan