Thermodynamics and quark susceptibilities: A Monte Carlo approach to the Polyakov–Nambu–Jona-Lasinio model

The Monte-Carlo method is applied to the Polyakov-loop extended Nambu–Jona-Lasinio model. This leads beyond the saddle-point approximation in a mean-field calculation and introduces fluctuations around the mean fields. We study the impact of fluctuations on the thermodynamics of the model, both in the case of pure gauge theory and including two quark flavors. In the two-flavor case, we calculate the second-order Taylor expansion coefficients of the thermodynamic grand canonical partition function with respect to the quark chemical potential and present a comparison with extrapolations from lattice QCD. We show that the introduction of fluctuations produces only small changes in the behavior of the order parameters for chiral symmetry restoration and the deconfinement transition. On the other hand, we find that fluctuations are necessary in order to reproduce lattice data for the flavor nondiagonal quark susceptibilities. Of particular importance are pion fields, the contribution of which is strictly zero in the saddle point approximation.

Figure 1

Comparison between the trajectory generated by a cooling algorithm (left) and the field configurations produced by the Metropolis algorithm (right) applied to the Polyakov loop effective potential at temperature $T=T_c=0.27\mathrm{GeV}$. The lines represent equipotential curves of $\mathcal{U}$ [Eq. (5)] in the $A_3\mathrm{\text{−}}{A}_8$-plane.

Figure 2

Left: Results for pressure $p$, energy density $\epsilon$ and entropy density $s$ from the Monte-Carlo evaluation of the Polyakov loop effective potential fitted to lattice results (lines), compared to lattice QCD data from 43 (open symbols). Right: Comparison of the Polyakov loop expectation value from the saddle point approximation (solid line) and the Monte-Carlo evaluation (open points) with lattice QCD data taken from 44 (solid points).

Figure 3

Dependence of the chiral condensate, $⟨\overline{\psi }\psi {⟩}_T/⟨\overline{\psi }\psi {⟩}_0$, and of the Polyakov loop expectation value $⟨\Phi ⟩$ on the parameter $k=(LT{)}^3$ in a finite volume. Deviations from the mean-field result in the infinite-volume limit are manifest only for the chiral condensate. The behavior of the Polyakov loop is completely unchanged.

Figure 4

The chiral susceptibility as a function of temperature for different volume aspect ratios. We use the peak position as a measure for the chiral transition temperature $T_{\sigma }$. The solid line shows the trajectory of the chiral transition temperature as it rises with increasing volume.

Figure 5

Chiral susceptibility as a function of $T/T_{\sigma }$: With increasing volume the Monte-Carlo results (open symbols) approach the saddle-point mean field result (solid curve).

Figure 6

Flavor-diagonal second order expansion coefficient ${\chi }_{uu}=2{\chi }_{20}$ for different ratios $k$ compared with the saddle point approximation. The Stefan-Boltzmann limit is marked by SB.

Figure 7

Different contributions to the off-diagonal susceptibility ${\chi }_{11}={\chi }_{ud}$ for different volume ratios $k$ computed in the Monte-Carlo approach. Left panel: Contribution from pionic fluctuations, for which the volume dependence is large. Right panel: Contribution from fluctuations of the $A_8$ field, which show a negligible volume dependence.

Figure 8

Scaled pionic contribution to the off-diagonal susceptibility compared with the ChPT prediction. All Monte-Carlo PNJL results are multiplied by the volume $V_k/V_{64}$ and therefore scale with the $k=64$ curve, using Eq. (37).

Figure 9

Temperature dependence of the flavor off-diagonal susceptibility ${\chi }_{ud}$ in the Monte-Carlo approach to the PNJL model, using $k=64$ ($LT=4$). Lattice data [34, 35] with the same volume aspect ratio $LT$ and different pion masses are also shown for orientation.