Thermodynamic Geometry of the Quark-Meson Model

We study the thermodynamic geometry of the quark-meson model, focusing on the curvature, R, around the chiral crossover at finite temperature and baryon chemical potential. We find a peculiar behavior of R in the crossover region, in which the sign changes and a local maximum develops; in particular, the height of the peak of R in the crossover region becomes large in the proximity of the critical end point and diverges at the critical end point. The appearance of a pronounced peak of R close to the critical end point supports the idea that R grows with the correlation volume around the phase transition. Moreover, the change of sign of R in the crossover region supports the idea that an attractive interaction develops at the mesoscopic level in that range of temperature. We also analyze the mixed fluctuations of energy and baryon number, which grow up substantially in the proximity of the critical end point: In the language of thermodynamic geometry these fluctuations are responsible for the vanishing of the determinant of the metric, which results in thermodynamic instability and are thus related to the appearance of a critical end point.