PHYSICAL REVIEW D 101, 016014 (2020) Thermodynamic geometry of the quark-meson model Bonan Zhang, Shen-Song Wan , and Marco Ruggieri* School of Nuclear Science and Technology, Lanzhou University, 222 South Tianshui Road, Lanzhou 730000, China (Received 31 July 2019; published 22 January 2020) 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, hΔUΔNi, 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. DOI: 10.1103/PhysRevD.101.016014 I. INTRODUCTION sense that a large dl2 corresponds to a small probability of S1 S2 g An interesting idea of statistical mechanics is that of a a fluctuation from to . With the aid of ij it is possible ¼ 2 metric in the manifold spanned by the thermodynamic to define the thermodynamic curvature, R R1212=g ¼ ð Þ variables: This is related to the theory of fluctuations with g det gij and R1212 corresponding to the only among equilibrium states and leads to the concept of independent component of the Riemann tensor for a two- thermodynamic geometry and thermodynamic curvature dimensional manifold. As it is clear from the very definition [1–44]. For example, in the grand-canonical ensemble the of gij, the thermodynamic curvature depends on the second equilibrium state is specified as long as the intensive and third order moments of the thermodynamic variables 1 2 independent variables like temperature, chemical potential that are conjugated to ðβ ; β Þ; therefore it carries infor- and others are fixed, and physical quantities like energy and mation about the fluctuation of the physical quantities in particle number fluctuate with the probability given by the particular around a phase transition, where these fluctua- Gibbs ensemble. Considering the pair of intensive variables tions are expected to be very large. For example, if 1 2 1 2 ðβ1 β2Þ¼ð1 −μ Þ ðβ ; β Þ the probability of a fluctuation from S1 ¼ðβ ; β Þ ; =T; =T then R contains information about 1 1 2 2 the fluctuations of energy and particle number. One of the to S2 ¼ðβ þ dβ ; β þ dβ Þ is proportional to merits of the thermodynamic curvature is that in the pffiffiffi 1 proximity of a second order phase transition jRj ∝ ξd − βi βj ð Þ g exp 2 gijd d ; 1 where d denotes the spatial dimension and ξ is the correlation length: As a consequence, it is possible to where g ≡∂2 log Z=∂βi∂βj is called the thermodynamic grasp information about the correlation length by means of ij thermodynamics only. This divergence is related to the metric tensor, g ¼ detðg Þ is the determinant of g and Z is ij ij vanishing of the determinant of the metric; therefore the the grand-canonical partition function. It is therefore thermodynamic geometry gives information on the location l2 ¼ βi βj natural to define the line element d gijd d which of the phase transition in the ðβiÞ space. measures effectively a distance between S1 and S2, in the The main purpose of this article is to report on our study of the thermodynamic geometry, and in particular of the *[email protected] thermodynamic curvature, of the quark-meson (QM) model of quantum chromodynamics (QCD); see [45–48] and Published by the American Physical Society under the terms of references therein. This model is used to describe chiral the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to symmetry breaking in vacuum, as well as chiral symmetry the author(s) and the published article’s title, journal citation, restoration at finite temperature T and baryon chemical and DOI. Funded by SCOAP3. potential μ; it can be used to model, for example, the QCD 2470-0010=2020=101(1)=016014(10) 016014-1 Published by the American Physical Society ZHANG, WAN, and RUGGIERI PHYS. REV. D 101, 016014 (2020) phase transition after the big bang, as well as the cold and dense quark matter presumably present in the core of compact stellar objects. In these applications usually a positive value of μ is considered since matter with finite baryon density is modeled, while a negative μ would give a finite antibaryon density. Although the phase diagram of this model is well known within the community of quark-gluon plasma physicists, we summarize it in order to give a qualitative understanding of the phases and transition lines at finite temperature and chemical potential. In the QM model, we introduce both quarks and mesons in the Lagrangian; the meson potential is such that the Oð4Þ symmetry is broken spontaneously down to Oð3Þ at T ¼ μ ¼ 0, leaving one massive σ meson FIG. 1. Phase diagram of the quark-meson model at finite and an isotriplet of pions, as well as a mean field value for T and μ. The labels χSB and χSR denote the phases in which chiral the σ field. The latter couples to quarks via the interaction symmetry is broken spontaneously and (approximately) restored σψψ¯ , which results in a nonvanishing chiral condensate respectively. The first order and crossover lines meet at the critical hψψ¯ i ≠ 0. At finite T and μ chiral symmetry is restored due end point labeled CEP. to thermal fluctuations that destroy the mean field, and at zero chemical potential this transition is a second order one, chiral symmetry is spontaneously broken, to a high temper- with an order parameter that vanishes exactly above the ature phase in which color is deconfined and chiral critical temperature and the transition happens continu- symmetry is approximately restored [49–53]. The situation ously. In the chiral limit, the symmetry breaking pattern/ is however unclear at finite baryon chemical potential for restoration is exact. However, an explicit but soft chiral QCD with three colors, due to the sign problem that forbids symmetry breaking term is introduced in the Lagrangian to first principle calculations. Because of this, effective gives mass to pions: As a consequence, chiral symmetry is models like the QM model have been used to study the broken explicitly, the mean field never vanishes although phase structure of QCD at finite μ: Nowadays there is becoming very small at high temperature, and the transition consensus that the smooth crossover becomes a first order becomes a smooth crossover. This situation is very similar phase transition if μ is large enough, suggesting the to that encountered in statistical models of ferromagnetism, presence of the critical end point in the ðT;μÞ plane at in which a magnetic field always induces a finite value for which the crossover becomes a true phase transition with the average spin although the temperature is higher than the divergent susceptibilities, and this point marks the separa- Curie temperature: In the case of the QM model, as well as tion between the crossover on the one hand and the first in QCD, the mass term acts like the magnetic field and the order line on the other hand; see for example [54,55] for spin polarization is played by the condensate of the σ field. reviews. At high chemical potential the change of the condensate We consider here the QM model at finite T and μ, which with the temperature is no longer smooth: This is inter- has been applied many times to study the phase structure of preted as the presence of a critical end point in the phase QCD, and we study its thermodynamic geometry following diagram, at which a first order and the crossover lines meet. the lines depicted in [43] where a similar study has been This point is identified with a second order phase transition performed for the Nambu-Jona-Lasinio (NJL) model. The since at this point, and only at this point, the susceptibilities advantage of using the QM model is its renormalizability, diverge as they would at a second order phase transition; which removes the dependence of the results on the see [45–48] and references therein, as well as the results effective ultraviolet cutoff that instead appears in NJL shown in the present article. calculations. Moreover, it is interesting to check how the These considerations are summarized in Fig. 1 in which predictions of the phase structure of QCD change when we show the phase diagram of the QM model at finite T and different effective models are used: This can not only shed μ, where μ corresponds to the baryon chemical potential. light on the qualitative picture, but also put a quantitative χ χ The labels SB and SR denote the phases in which chiral statement on the theoretical uncertainty of model predic- symmetry is broken spontaneously and (approximately) tions, for example on the location of the CEP.