  Motivated by the recent proposal of realizing an SU(4) Hubbard model on
triangular moir\'e superlattices, we present a DMRG study of an $SU(4)$ spin
model obtained in the limit of large repulsion for integer filling $\nu_T=1,3$.
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
$\nu_T=3$, 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 $\nu_T=1$ 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 moir\'e 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.
