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We investigate the nature of a $Z_2$-invariant XY ring-exchange interactionwith a frustrated Ising coupling on the triangular lattice. In the limit ofpure XY ring-exchange interaction, we show that the classical ground state isdegenerate resulting from the $Z_2$-invariance of the Hamiltonian. Quantumfluctuations lift these classical degenerate ground states and produce anunusual state whose excitation spectrum exhibits a gapped maximum quadraticdispersion near ${\bf k}=0$ and vanishes at the midpoints of each side of theBrillouin zone. This result is in contrast to a gapless quadratic dispersionnear ${\bf k}=0$ in the U(1)-invariant counterpart. We also study the effectsof frustration when competing with a classically frustrated Ising interaction.We provide a glimpse into the possible quantum phases that could emerge. Acomprehensive understanding of this Hamiltonian, however, cannot be elucidatedanalytically and requires an explicit numerical simulation.

XY ring exchange model with frustrated Ising coupling on the triangular lattice