Use of the canonical approach in effective models of QCD

We discuss the canonical approach for the study of QCD phase at finite densities and temperatures in the confinement phase. The canonical approach, which is a method to extrapolate observables calculated at pure imaginary chemical potentials to those at real chemical potentials, is useful to overcome the sign problem in lattice QCD simulations at finite density. To validate the applicability of the approach, we employ the Nambu-Jona-Lasinio (NJL) and Polyakov-NJL (PNJL) models where exact solutions for the number density are available, which is the basic input of the fugacity expansion and can be compared with those of the canonical approach. We find that the number densities computed from the canonical approach are consistent with the exact solutions in most of the confinement phase. The results in the present study are applicable to the study of lattice QCD.

Figure 1

The temperature and chemical potential dependences of the baryon number density $n_B[{\mathrm{MeV}}^3]$ in the PNJL model. The star is the critical endpoint (CEP) $(T^{\mathrm{CEP}},{\mu }_B^{\mathrm{CEP}})\simeq (114,965)[\mathrm{MeV}]$.

Figure 2

The $\theta$ dependence of the imaginary number density in the PNJL model.

Figure 3

The $N_{\max }$ dependence of the number density in the PNJL model. The solid line is the exact number density calculated at the real chemical potential. The other symbols are the number densities obtained from the canonical approach for several $N_{\max }$.

Figure 4

The number densities around the phase transition or the crossover densities in the PNJL model. The solid and dashed lines are the exact number densities calculated at the real chemical potentials at $T=80$ and 160 [MeV], respectively. The open circles and open triangles are the number densities obtained from the canonical approach with $(N_{\max },N_{\sin })=(1200,3)$ and (1200,4) at $T=80$ and 160 [MeV], respectively.

Figure 5

The $N_{\sin }$ dependence of $n_B/T^3$ in the PNJL model. The solid lines are the exact number densities calculated at the real chemical potential.

Figure 6

The $N_{\sin }$ dependence of $n_B^{\text{canonical}}/n_B^{\text{exact}}$ in the PNJL model. $n_B^{\text{canonical}}$ is the number density obtained from the canonical approach and $n_B^{\text{exact}}$ is the exact number density calculated at the real chemical potential. The solid and dashed lines represent the exact value ($n_B^{\text{canonical}}/n_B^{\text{exact}}=1.0$) and the 10% difference values ($n_B^{\text{canonical}}/n_B^{\text{exact}}=0.9$ and 1.1).

Figure 7

The boundaries between the effective and ineffective regions of the canonical approach for $N_{\sin }$ in the PNJL model. The black solid and dashed lines represent the first-order phase transition and crossover lines, respectively. The points of making a 10% difference between the exact $n_B$ and the results from the canonical approach are plotted. We plot the symbols on the crossover line when the difference is less than 10% in the confinement phase.

Figure 8

The $N_{\max }$ dependence of the number density in the NJL model. The solid line is the exact number density calculated at the real chemical potential. The other symbols are the number densities obtained from the canonical approach for several $N_{\max }$.

Figure 9

The $N_{\sin }$ dependence of $n_B/T^3$ in the NJL model. The solid lines are the exact number densities calculated at the real chemical potential.

Figure 10

The $N_{\sin }$ dependence of $n_B^{\text{canonical}}/n_B^{\text{exact}}$ in the NJL model. $n_B^{\text{canonical}}$ is the number density obtained from the canonical approach and $n_B^{\text{exact}}$ is the exact number density calculated at the real chemical potential.

Figure 11

The boundaries between the effective and ineffective regions of the canonical approach for $N_{\sin }$ in the NJL model. The black solid and dashed lines represent the first-order phase transition and crossover lines, respectively. The points of making a 10% difference between the exact $n_B$ and the results from the canonical approach are plotted.