New approach to canonical partition functions computation in $N_f=2$ lattice QCD at finite baryon density

We propose and test a new approach to computation of canonical partition functions in lattice QCD at finite density. We suggest a few steps procedure. We first compute numerically the quark number density for imaginary chemical potential $i{\mu }_{qI}$. Then we restore the grand canonical partition function for imaginary chemical potential using the fitting procedure for the quark number density. Finally we compute the canonical partition functions using high precision numerical Fourier transformation. Additionally we compute the canonical partition functions using the known method of the hopping parameter expansion and compare results obtained by two methods in the deconfining as well as in the confining phases. The agreement between two methods indicates the validity of the new method. Our numerical results are obtained in two flavor lattice QCD with clover improved Wilson fermions.

Figure 1

Schematical figure of Roberge-Weiss phase structure in the pure imaginary chemical potential regions. $T_c$ is the confinement/deconfinement crossover point at zero chemical potential, $L$ is the Polyakov loop. The vertical line $(T\ge T_{\mathrm{RW}},{\mu }_I/T=\pi /3)$ shows the first order phase transition. The dashed line is a crossover which can change to the first order phase transition for large or small quark masses.

Figure 2

Imaginary density as a function of $\theta$ in the deconfinement phase at temperatures $T/T_c=1.35$, 1.20, 1.08. The curves show fits to function (10).

Figure 3

The absolute value of the Polyakov loop (multiplied by factor 3) and its susceptibility vs ${\mu }_{qI}$ at $T/T_c=1.08$.

Figure 4

$Z_n$ vs $n$ computed using two methods at $T/T_c=1.35$.

Figure 5

Relative deviation of results for $Z_n$ obtained by new and HPE methods vs $n$ at $T/T_c=1.35$.

Figure 6

Imaginary density as a function of $\theta$ at three temperatures in the confining phase. The curves show fits to function (11) with $n_{\max }=1$ for $T/T_c=0.84$, 0.93 and $n_{\max }=2$ for $T/T_c=0.99$.

Figure 7

$Z_n$ vs $n$ computed using two methods at $T/T_c=0.93$.

Figure 8

$Z_n$ vs $n$ for three temperatures in the confinement phase. The curves show results of fitting to asymptotic behavior for $T/T_c=0.93$ and 0.84.

Figure 9

Analytical continuation for the number density vs ${\mu }_q^2$ for all temperature values. The curves show respective fits. The width of the curves shows the statistical error of extrapolation to ${\mu }_q^2>0$.