Phase Diagram of the Spin-$1\mathrm{/}2$ Triangular $J_1$-$J_2$ Heisenberg Model on a 3-leg Cylinder

We study the phase diagram of the frustrated Heisenberg model on the triangular lattice with nearest and next-nearest neighbor spin exchange coupling, on 3-leg ladders. Using the density-matrix renormalization-group method, we obtain the complete phase diagram of the model, which includes quasi-long-range ${120}^{\circ }$ and columnar order, and a Majumdar-Ghosh phase with short-ranged correlations. All these phases are non-chiral and planar. We also identify the nature of phase transitions.