  We investigate the finite temperature charge density wave (CDW) transition of
lattice Bose gases within optical cavities in the deep Mott-insulator limit. We
find a new critical regime emerges at a temperature around one-half of the
on-site interaction energy, where the first order CDW transition at low
temperatures terminates at a critical point and changes to a second order one.
By directly calculating the critical exponents and constructing the effective
theory in the corresponding critical regime, we find the emergent criticality
belongs to the five-dimensional Ising universality class. Direct experimental
observation of the emergent criticality can be readily performed by current
experimental set-ups operated in the temperature regime around half the on-site
interaction energy.
