Unconventional symmetry breaking due to nonlocal order parameters has attracted considerable attention in many strongly correlated metals. Famous examples are the nematic order in Fe-based superconductors (SCs) and the star-of-David charge density order in kagome metals. Such exotic symmetry breaking in metals is a central issue of modern condensed matter physics, while its theoretical foundation is still unclear in comparison with the well-established theory of superconductivity. To overcome this difficulty, here, we introduce the form factor that generalizes the nonlocal order parameter into the Luttinger-Ward (LW) Fermi liquid theory. We then construct a rigorous formalism of the density-wave equation that gives the thermodynamically stable form factor, like the SC gap equation. In addition, a rigorous expression of the Ginzburg-Landau free energy for the unconventional order is presented to calculate various thermodynamic properties. In the next stage, we apply the derived formalism to a typical Fe-based SC FeSe, by using the one-loop LW function that represents the free-energy gain due to the interference among paramagnons. The following key experiments are naturally explained: (i) Lifshitz transition (=disappearance of an electron pocket) due to the bond + orbital order below T_c; (ii) Curie-Weiss (CW) behavior of the nematic susceptibility at higher T, and the deviation from the CW behavior at lower T near the nematic quantum critical point; and (iii) scaling relation of the specific heat jump at T_c, ΔC/T_c∝T^b_c with b∼3. (Note that b=0 in the Bardeen-Cooper-Schrieffer theory.) These results lead to a conclusion that the nematicity in FeSe is the bond + orbital order due to the paramagnon interference mechanism. The present theory paves the way for solving various unconventional phase transition systems.

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