An overview on the phase diagram of the frustrated two-leg ladder model
  Using Density-Matrix Renormalization Group, we investigate the general phase diagram of the frustrated two-leg ladder with Heisenberg interactions along legs, rungs and diagonals. We confirm that all antiferromagnetic gapped states belong to the same universality class as the Haldane phase. In a three-dimensional phase diagram, we determine a continuous surface with singularities in the string-order parameter or its first derivative, corresponding to a transition between two Haldane phases with different topological order. Some parts of this transition surface are critical with zero gap and vanishing string-order parameter. In the complementary parts, the transition is first order with finite gap and string order. The boundary of this surface with the ferromagnetic region is a critical end line, when the surface is critical and a triple line anywhere else. Part of this boundary coincides with the exactly soluble model proposed by D.V. Dmitriev, V. Ya Krivnov and A.A. Ovchinnikov [Phys. Rev. B 56, 5985 (1997)]. 
T. Hakobyan, J.H. Hetherington and M. Roger