  We systematically explore and show the existence of finite-temperature
continuous quantum phase transition (CTQPT) at a critical point, namely, during
solidification or melting such that the first-order thermal phase transition is
a special case within CTQPT. Infact, CTQPT is related to chemical reaction
where quantum fluctuation (due to wavefunction transformation) is caused by
thermal energy and it can occur maximally for temperatures much higher than
zero Kelvin. To extract the quantity related to CTQPT, we use the ionization
energy theory and the energy-level spacing renormalization group method to
derive the energy-level spacing entropy, renormalized Bose-Einstein
distribution and the time-dependent specific heat capacity. This work
unambiguously shows that the quantum phase transition applies for any finite
temperatures.
