Quadrupolar phases and plateau states in skewed ladders

Two legged skewed spin-$\frac{1}{2}$ ladders are frustrated and exhibit exotic quantum phases in ground state due to strong quantum fluctuations and competing spin exchanges. Here, we study ground state properties of a spin-$\frac{1}{2}$ Heisenberg model on 3/4, 3/5 and 5/5 skewed ladders in the presence of a Zeeman magnetic field, $B$, using exact diagonalization and the density matrix renormalization group method. We note the existence of plateaus at $m =$ 1/3 and 2/3 for 3/4 skewed ladder, at $m =$ 1/4, 1/2, and 3/4 for 3/5 skewed ladder, and at $m =$ 0, 1/3, and 2/3 for 5/5 skewed ladder, where $m$ is the ratio of the observed magnetization ($M$) to the saturated magnetization (${M_\mathrm{max}}$). The plateau state is always a gapped state and the plateau width depends on the gap in the system. Surprisingly, the 3/4 and 5/5 skewed ladders show interesting quadrupolar or n-type spin nematic phases below the 1/3$^rd$ plateau, i.e, at very low magnetic fields. These two systems are unique as they host both a plateau and a quadrupolar phase at low magnetic fields. The linear variation of pitch angle of the spin with magnetization and behavior of binding energy of magnon pairs as function of magnetic field are also calculated in both the systems. We also study the contribution of the binding energy to two magnon condensate.