Phase diagram of frustrated ladder and 2D antiferromagnets

We investigate the low energy properties of the frustrated two leg diagonal ladder exhibiting both intra- and inter-chain frustration. The renormalization group is used to obtain the phase diagram while varying the microscopic lattice parameters. We particularly emphasize the role of the in-chain marginal operators, which is tuned by the in-chain frustration and can promote a dimer phase in the system. Finally, the physics of the quasi one dimensional diagonal ladder is incorporated into a two dimensional square lattice since the former is used as the primary structure to build up the square lattice. Within the validity of our method, the classical phases---a N\'{e}el antiferromagnet and a collinear antiferromagnet---are predicted. The results are compared to numerical DMRG calculations.