  A phase transition is often accompanied by the appearance of an order
parameter and symmetry breaking. Certain magnetic materials exhibit exotic
hidden-order phases, in which the order parameters are not directly accessible
to conventional magnetic measurements. Thus, experimental identification and
theoretical understanding of a hidden order are difficult. Here we combine
neutron scattering and thermodynamic probes to study the newly discovered
rare-earth triangular-lattice magnet TmMgGaO$_4$. Clear magnetic Bragg peaks at
K points are observed in the elastic neutron diffraction measurements. More
interesting, however, is the observation of sharp and highly dispersive spin
excitations that cannot be explained by a magnetic dipolar order, but instead
is the direct consequence of the underlying multipolar order that is "hidden"
in the neutron diffraction experiments. We demonstrate that the observed
unusual spin correlations and thermodynamics can be accurately described by a
transverse field Ising model on the triangular lattice with an intertwined
dipolar and ferro-multipolar order.
