Motivated by the recent proposal of realizing an SU(4) Hubbard model on triangular moire superlattices, we present a DMRG study of an $SU(4)$ spin model obtained in the limit of large repulsion for integer filling $ u_T=1,3$. We retain terms in the $t/U$ expansion up to $O(frac{t^3}{U^2})$ order, that generates nearest-neighbor exchange $J$, as well as an additional three-site ring exchange term, $K$, which is absent in the SU(2) S=1/2 case. For filling $ u_T=3$, when increasing the three-site ring exchange term $K sim frac{t^3}{U^2}$, we identify three different phases: a spin-symmetric crystal, an $SU(4)_1$ chiral spin liquid (CSL) and a decoupled one dimensional chain (DC) phase. The CSL phase exists at intermediate coupling: $U/t in [11.3,,22.9]$. The sign of $K$ is crucial to stabilizing the CSL and the DC phase. For filling $ u_T=1$ with the opposite sign of $K$, the spin-symmetric crystal phase survives to very large $K$. We propose to search for the CSL phase in moire bilayers. For example, in twisted AB stacked transition metal dichalcogenide (TMD) bilayers, the $SU(4)$ spin is formed by layer pseudospin combined with the real spin (locked to valley). The layer pseudospin carries an electric dipole moment in $z$ direction, thus the CSL is really a dipole-spin liquid, with quantum fluctuations in both the electric moment and magnetic moment . The CSL phase spontaneously breaks the time reversal symmetry and shows a quantum anomalous Hall effect in spin transport and dipole transport. Smoking gun evidence for the CSL could be obtained through measurement of the quantized dipole Hall effect in counter-flow transport.
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