Grant No. DMR-1350237 (W.-J.H. and H.-H.L.), a Smalley Postdoctoral Fellowship of the Rice Center for Quantum Materials (H-H. L.), the National High Magnetic Field Laboratory through the NSF Grant No. DMR-1157490 and the State of Florida (S.-S.G.). The majority of the computational calculations have been performed on the Shared University Grid at Rice funded by NSF under Grant EIA-0216467, a partnership between Rice University, Sun Microsystems, and Sigma Solutions, Inc., the Big-Data Private-Cloud Research Cyberinfrastructure MRI-award funded by NSF under Grant No. CNS-1338099 and by Rice University, the Extreme Science and Engineering Discovery Environment (XSEDE) by NSF under Grants No. DMR160003 and DMR160057. Computational support has also been provided by XSEDE from the NSF under Grant No. DMR160004 (S.-S.G.).

Supplemental Material
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Ring exchanges and spin operators {#App:ring}
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The $3$-site and $4$-site ring exchanges $P_{123}$ and $P_{1234}$ are equivalently expressed in terms of the physical spin operators. In general ground, the ring exchange $P_{ijk\cdots \ell}$ can be expressed in terms of the products of two-site exchange operator $P_{ijk\cdots \ell m} = P_{ij} P_{jk} \cdots P_{\ell m}$. The two-site ring exchange operator can be written in terms of physical spin operators. Defining $X\equiv {\bf S}_i \cdot {\bf S}_j$, we can 