Quantum phase diagram of a frustrated spin-1/2 system on a Trellis Ladder

We study an isotropic Heisenberg spin-1/2 model on a trellis ladder which is composed of two $J_1-J_2$ zigzag ladders interacting through anti-ferromagnetic rung couplings $J_3$. The $J_1$ and $J_2$ are ferromagnetic zigzag spin interaction between two legs and anti-ferromagnetic interaction along each leg of a zigzag ladder. A quantum phase diagram of this model is constructed using the density matrix renormalization group (DMRG) method and linearized spin wave analysis. In small $J_2$ limit a short range stripe collinear phase is found in the presence of $J_3$, whereas, in the large $J_2/J_3$ limit non-collinear quasi-long range phase is found. The system shows a short range non-collinear state in large $J_3$ limit. The short range order phase is the dominant feature of this phase diagram. We also show that the results obtained by DMRG and linearized spin wave analysis show similar phase boundary between stripe collinear and non-collinear short range phases, and the collinear phase region shrinks with increasing $J_3$. We apply this model to understand the magnetic properties of CaV$_2$O$_5$ and also fit the experimental data of susceptibility and magnetization. The variation of magnetic specific heat capacity as function of external magnetic field is also predicted. We note that $J_3$ is a dominant interaction in this system, whereas $J_1$ and $J_2$ are approximately half of $J_3$.