Lee-Yang zero distribution of high temperature QCD and the Roberge-Weiss phase transition

Canonical partition functions and Lee-Yang zeros of QCD at finite density and high temperature are studied. Recent lattice simulations confirm that the free energy of QCD is a quartic function of quark chemical potential at temperature slightly above pseudocritical temperature $T_c$, as in the case with a gas of free massless fermions. We present analytic derivation of the canonical partition functions and Lee-Yang zeros for this type of free energy using the saddle point approximation. We also perform lattice QCD simulation in a canonical approach using the fugacity expansion of the fermion determinant and carefully examine its reliability. By comparing the analytic and numerical results, we conclude that the canonical partition functions follow the Gaussian distribution of the baryon number, and the accumulation of Lee-Yang zeros of these canonical partition functions exhibit the first-order Roberge-Weiss phase transition. We discuss the validity and applicable range of the result and its implications both for theoretical and experimental studies.

Figure 1

Comparison of the saddle point approximation (SPA) and exact result for the free energy density. We use values of $c_2$, $c_4$, and $V{T}^3$ used in our simulations in the next section ($c_2=2.20$, $c_4=0.27$).

Figure 2

Second- and fourth-order coefficients of Taylor expansion of the free energy, $c_2$ and $c_4$, for $N_s=8$ and $N_s=16$. The data for $N_s=8$ and $N_s=16$ are taken from [51, 52], respectively.

Figure 3

Canonical partition function as a function of the baryon number for $T/T_c=0.99$ (top) and 1.20 (bottom). The data are obtained from the canonical formalism, while the solid curves are obtained from the saddle point approximation. Note that only positive $Z_n$’s are shown.

Figure 4

The difference of the canonical partition functions between the data $(Z_{n_{\mathrm{B}}}^{(\text{lat})})$ and Gaussian fit $(Z_{n_{\mathrm{B}}}^{(\text{fit})})$. Circle and square symbols denote results for $N_s=8$ and 10, respectively. Open (red) and closed (blue) symbols are for $T/T_c=1.20$ and 0.99, respectively.

Figure 5

Lee-Yang zeros on the complex fugacity plane for $\beta =1.95,T/T_c=1.20$. Top panel: Zeros inside unit circle. Bottom panel: Zeros on or in the vicinity of the negative real axis. Red, green, and blue symbols denote Lee-Yang zeros for (a) $n_0=37$ and $N_s=10$, (b) $N_s=8$ and $n_0=32$, and (c) $N_s=8$ and $n_0=19$, respectively. Note that zeros also exist outside the unit circle with symmetry $\xi \leftrightarrow 1/\xi$.

Figure 6

Top: $|\mathrm{Re}\xi |$ for zeros on the negative real axis near the unit circle, where $\xi =\exp (\mu /T)$. Red, green, and blue symbols denote Lee-Yang zeros for (a) $n_0=37$ and $N_s=10$, (b) $N_s=8$ and $n_0=32$, and (c) $N_s=8$ and $n_0=19$, respectively. The curves represent predictions from the Jacobi theta function $\exp (-3(2\ell +1)/(4T^{3}V{c}_2))$ for $N_s=10$ (red) and $N_s=8$ (green and blue). Bottom: Volume-independent combination $|\mathrm{Re}\xi {|}^{V{T}^3}$. The curve represents $\exp (-3(2\ell +1)/(4c_2))$.

Figure 7

Schematic figures for the distribution of Lee-Yang zeros in several cases. (a) Ising models on the complex plane for $e^h$, where $h$ is the external magnetic field in Ising models, (b) QCD on the complex plane for baryon fugacity, and (c) QCD on the complex quark fugacity plane. Dotted circles denote the unit circle. Case (b) can be generalized to free fermion theories.