Abstract
We study the thermodynamic curvature, $R$, around the chiral phase transition at finite temperature and chemical potential, within the quark-meson model augmented with meson fluctuations. We study the effect of the fluctuations, pions and $\sigma$-meson, on the top of the mean field thermodynamics and how these affect $R$ around the crossover. We find that for small chemical potential the fluctuations enhance the magnitude of $R$, while they do not affect substantially the thermodynamic geometry in the proximity of the critical endpoint. Moreover, in agreement with previous studies we find that $R$ changes sign in the pseudocritical region, suggesting a change of the nature of interactions at the mesoscopic level from statistically repulsive to attractive. Finally, we find that in the critical region around the critical endpoint $|R|$ scales with the correlation volume, $|R| =K\;\xi^3$, with $K = O(1)$, as expected from hyperscaling; far from the critical endpoint the correspondence between $|R|$ and the correlation volume is not as good as the one we have found at large $\mu$, which is not surprising because at small $\mu$ the chiral crossover is quite smooth; nevertheless, we have found that $R$ develops a characteristic peak structure, suggesting that it is still capable to capture the pseudocritical behavior of the condensate.
Highlights
The thermodynamic theory of fluctuations allows one to define a manifold spanned by intensive thermodynamic variables, fβkg with k 1⁄4 1; 2; ...; N, and equip this with the notion of a distance, dl2 1⁄4 gijðβ1; β2; ...; βNÞdβidβj, where gij is the metric tensor, that depends in general on the fβkg and measures the probability of a fluctuation between two equilibrium states
We find that fluctuations enhance jRj at the crossover at small μ, and we interpret this as the fact that the fluctuations make the chiral broken phase more unstable and favor chiral symmetry restoration at finite temperature; near the critical end point (CEP) we do not find substantial effects of the fluctuations on R, and we interpret this as the fact that even without fluctuations, the mean field thermodynamic potential predicts a secondorder phase transition at the CEP with divergent susceptibilities and a divergent curvature [43,44], and the fluctuations cannot change this picture but can only alter the values of the critical exponents
We have studied the thermodynamic geometry around the chiral phase transition at finite temperature and chemical potential
Summary
The thermodynamic theory of fluctuations allows one to define a manifold spanned by intensive thermodynamic variables, fβkg with k 1⁄4 1; 2; ...; N, and equip this with the notion of a distance, dl2 1⁄4 gijðβ; β2; ...; βNÞdβidβj, where gij is the metric tensor, that depends in general on the fβkg and measures the probability of a fluctuation between two equilibrium states. The metric tensor can be computed from the derivatives of the thermodynamic potential; the knowledge of the latter is enough to define the metric on the manifold. Thermodynamic stability requires g > 0, where g is the determinant of the metric; the condition g 1⁄4 0 determines a phase boundary in the fβkg space and g < 0 corresponds to regions of thermodynamic instability
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