  Quantum Ising model on a triangular lattice hosts a finite temperature
Berezinskii-Kosterlitz-Thouless (BKT) phase with emergent U(1) symmetry, and it
will transit into an up-up-down (UUD) phase with $C_3$ symmetry breaking upon
an infinitesimal external field along the longitudinal direction, but the
overall phase diagram spanned by the axes of external field and temperature
remains opaque due to the lack of systematic invesitgations with controlled
methodologies. By means of quantum Monte Carlo at finite temperature and ground
state density matrix renormalization group simulations, we map out the phase
diagram of triangular quantum Ising model. Stemming from the upper BKT
temperature at zero field, we obtain the phase boundary between the UUD and
paramagnetic phases with its 2D $q=3$ Potts universality at weak field and
weakly first order transition at strong field. Originated from the lower BKT
temperature at zero field, we analyze the low temperature phase boundary
between the clock phase and the UUD phase with Ising symmetry breaking at weak
fields and the quantum phase transition between the UUD and fully polarized
phases at strong fields. The accurate many-body numerical results are
consistent with our field theoretical analysis. The experimental relevance
towards the BKT magnet TmMgGaO$_4$ and programmable quantum simulators are also
discussed.
