The Canted Spiral: An Exact Ground State of XXZ Zigzag Ladders

We derive the exact ground states for a one dimensional family of $S=1\mathrm{/}2$ XXZ Hamiltonians on the zigzag ladder. These states exhibit true long range spiral order that spontaneously breaks the U(1) invariance of the Hamiltonian. Besides breaking a continuous symmetry in $d=1$, this spiral ordering has a ferromagnetic component along the symmetry axis that can take any value between zero and full saturation. In this sense, our canted spiral solutions are a generalization of the SU(2) Heisenberg ferromagnet to non-zero ordering wave-vectors of the transverse spin components. We extend this result to the $d=2$ anisotropic triangular lattice.