Chiral pair fluctuation are considered near the phase boundary of the inhomogeneous chiral phase (iCP). The fluctuations are then bosonized and an effective action for the chiral pair fluctuation is basically constructed by considering the ring diagram of the polarization function. We can evaluate the self-energy and effective four point interaction among fluctuations in a consistent way. The peculiar dispersion of the fluctuation, reflecting the spatially inhomogeneous transition, gives rise to interesting and qualitative results. Thermal fluctuations prohibit the second-order transition, while the effect of the quantum fluctuations is rather modest. Quantum and thermal fluctuations changes the second-order transition to the first-order one by changing the sign of the effective four-point interaction between effective bosons.
These features may be observed by relativistic heavy-ion collisions through the analysis of the thermodynamic observables. Distinct from the second-order phase transition, the first moment such as entropy exhibits an anomalous behavior due to fluctuations, which may be one of the signals of the phase transition to iCP.
Some similar aspects are also remarked between iCP and the FFLO state in superconductivity.