Canonical approach to finite density QCD with multiple precision computation

In this study, we calculated the dependence of thermodynamic observables, namely, pressure, baryon number density, and baryon susceptibility, on the baryon chemical potential ${\mu }_B$ through lattice quantum chromodynamics using the canonical approach. We compare the results with those obtained using the multiparameter reweighting (MPR) method. The results of these methods were found to be in very good agreement in the regions where the errors of the MPR method are under control. The canonical method yields reliable results up to ${\mu }_B/T=3$, where $T$ is the temperature. Multiple precision operations play important roles in the evaluation of canonical partition functions.

Figure 1

Canceled significant digits in the DFT calculations at temperatures above (upper red points) and below (lower green points) $T_c$.

Figure 2

Relationship between the behavior of $Z_C(B,T)$ and the precision of the variables in the DFT at temperatures above (right) and below (left) $T_c$. Both include 16 (double precision, upper red points), 32 (second green points), 48 (third blue points), and 64 (lowest cyan points) significant figures. Errors are estimated by the Jackknife method.

Figure 3

Chemical potential dependence of pressure. The red, green, blue, cyan, magenta, and brown points are the results at $T/T_c=1.35$, 1.20, 1.08, 0.99, 0.93, and 0.83, respectively. The upper bound of the baryon chemical potential is determined by Eq. (27).

Figure 4

Comparison of pressure calculated by the canonical approach and the MPR method. The colors of the data points are the same as in Fig. 3 with some additional colors. The data points plotted in the additional colors of dark red, dark green, dark blue, dark cyan, dark magenta, and dark brown points are the results at $T/T_c=1.35$, 1.20, 1.08, 0.99, 0.93, and 0.83, respectively, as calculated by the MPR method.

Figure 5

Chemical potential dependence of baryon number density. The colors of the data points are the same as in Fig. 3. The upper bound of the baryon chemical potential is determined by Eq. (27).

Figure 6

Comparison of the baryon number densities calculated by the canonical approach and the MPR method. The colors of the data points are the same as in Fig. 4.

Figure 7

Chemical potential dependence of baryon susceptibility. The colors of the data points are the same as in Fig. 3. The upper bound of the baryon chemical potential is determined by Eq. (27).

Figure 8

Comparison of baryon susceptibilities calculated by the canonical approach and the MPR method. The colors of the data points are the same as in Fig. 4.

Figure 9

Calculation of winding number coefficients, $W_n$: Schematical overview.

Figure 10

Pure imaginary chemical potential dependence of fermion determinants. Red and blue points are calculated by the winding number expansion with 16 noise vectors and LU decomposition, respectively.

Figure 11

The baryon number density as a function of its real baryon chemical potential for several $N_{\max }$. Here, $N_{\max }$ indicates the truncation of the fugacity expansion in Eq. (a1). Also, $\beta =1.90$ $(T/T_c=1.08$). The line with $N_{\max }=120$ corresponds to the data in Fig. 6.