Magnetic phase diagram of the spatially anisotropic spin-$1/2$ zigzag ladder

We study the magnetic phase diagram of a spatially anisotropic zigzag ladder with exchange interactions $J_2$, and $J_2'$ along the legs. This system interpolates between the conventional $J_1$-$J_2$ chain, and the ``sawtooth,'' or ``delta'' chain, described by a linear arrangement of triangles. Both systems display dimerization in a region of the zero-field phase diagram, but the anisotropy leads to two kinds of excitations with different velocities. We study the magnetic phase diagram of the system as a function of $J_2$ and $J_2'$ and find commensurate plateaus at $m=1/2$ and $m=1/3$, and study the competition between different orders, including incommensurate, chiral, polar, and two-component liquid phase.