Exact dimer ground state and quantum phase transitions in a coupled spin ladder

Spin ladders are key models that act as intermediaries between one-dimensional and two-dimensional spin systems. In this study, we examine a coupled spin-$1/2$ ladder, where frustrated ladders with leg, rung, and diagonal interactions are linked through a horizontal coupling. By introducing a spatially anisotropic third-nearest-neighbor interaction along the horizontal direction, the model was found to possess an exact dimer ground state, characterized by a product of singlets forming a columnar dimer phase. The model is analyzed using bond-operator mean-field theory (BOMFT) and the density matrix renormalization group (DMRG). BOMFT reveals three distinct phases: a double-stripe ordered phase, a N\'eel ordered phase, and a quantum disordered dimerized phase. The critical points for the transitions are $J_1 = -0.81$ (double-stripe to dimerized) and $J_1 = 2.81$ (dimerized to N\'eel phase). DMRG results corroborate the exact ground state and refine the critical points to $J_1 = -0.79$ and $J_1 = 2.29$ for the respective transitions. Additionally, another transition is identified as the N\'eel order vanishes for $J_1 > 4.5$. The static spin structure factor further corroborates the nature of the ordered phases.