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We investigate the thermodynamic geometry of the quark-meson model at finitetemperature, $T$, and quark number chemical potential, $\mu$. We extendprevious works by the inclusion of fluctuations exploiting the functionalrenormalization group approach. We use recent developments to recast the flowequation into the form of an advection-diffusion equation. We adopt the localpotential approximation for the effective average action. We focus on thethermodynamic curvature, $R$, in the $(\mu,T)$ plane, in proximity of thechiral crossover, up to the critical point of the phase diagram. We find thatthe inclusion of fluctuations results in a smoother behavior of $R$ near thechiral crossover. Moreover, for small $\mu$, $R$ remains negative, signalingthe fact that bosonic fluctuations reduce the capability of the system tocompletely overcome the fermionic statistical repulsion of the quarks. Weinvestigate in more detail the small $\mu$ region by analyzing a system inwhich we artificially lower the pion mass, thus approaching the chiral limit inwhich the crossover is actually a second order phase transition. On the otherhand, as $\mu$ is increased and the critical point is approached, we find that$R$ is enhanced and a sign change occurs, in agreement with mean field studies.Hence, we completely support the picture that $R$ is sensitive to a crossoverand a phase transition, and provides information about the effective behaviorof the system at the phase transition.

Functional Renormalization Group Study of Thermodynamic Geometry Around the Phase Transition of Quantum Chromodynamic