Microscopic Phase-Transition Theory of Charge Density Waves: Revealing Hidden Crossovers of Phason and Amplitudon

We develop a self-consistent phase-transition theory of charge density waves (CDWs), starting from a purely microscopic model. Specifically, we derive a microscopic CDW gap equation |Δ_{0}(T)|, taking into account the thermal phase fluctuations (i.e., thermal excitation of phason) and their influence on CDW pinning (i.e., the phason mass) and the CDW gap. We demonstrate that, as the temperature increases from zero, the phason gradually softens, leading to a thermal depinning crossover (where the phason becomes gapless) at T_{d} and a subsequent first-order CDW phase transition at T_{c}>T_{d}. The predicted values of T_{d}, T_{c} as well as the large ratio of |Δ_{0}(T=0)|/(k_{B}T_{c}) for the quasi-1D CDW material (TaSe_{4})_{2}I show quantitative agreements with experimental measurements and explain many of the previously observed key thermodynamic features and unresolved issues in the literature. To further validate the theory, we calculate the energy gap of CDW amplitudon and its lifetime and reveal a crossover of amplitudon from a lightly damped to a heavily damped excitation during pinning-depinning crossover while its energy gap is nearly unchanged throughout the entire CDW phase. This finding quantitatively captures and explains the recently observed coherent signal in ultrafast THz emission spectroscopy on (TaSe_{4})_{2}I.