QCD at Zero Baryon Density and the Polyakov Loop Paradox

TL;DR

The paper investigates whether the grand canonical partition function $Z_{GC}$ and the canonical partition function $Z_C$ describe the same thermodynamics at zero baryon density and explains why the Polyakov loop is nonzero in the grand canonical ensemble but zero in the canonical one. It shows that $Z_C(T,B)$ receives contributions only from integer $B$ sectors; non-integer sectors have zero weight and do not affect thermodynamics ($Z_C$ enforces $B\in \mathbb{Z}$). Lattice simulations with four degenerate staggered quarks demonstrate that the free energy density computed in the two ensembles agrees in the thermodynamic limit and matches a hadron resonance gas for $T< T_c$ and a free quark gas for $T>T_c$, with finite-volume corrections controlled and vanishing as volume grows. The work clarifies center symmetry breaking in the canonical ensemble and shows that a center-symmetric grand canonical partition function can be formed by projecting out non-zero triality, thereby reconciling the two formulations and providing a practical zero-density QCD framework. Overall, the study deepens understanding of ensemble equivalence and center symmetry in lattice QCD and suggests that non-zero triality sectors are unphysical for thermodynamic observables.

Abstract

We compare the grand canonical partition function at fixed chemical potential mu with the canonical partition function at fixed baryon number B, formally and by numerical simulations at mu=0 and B=0 with four flavours of staggered quarks. We verify that the free energy densities are equal in the thermodynamic limit, and show that they can be well described by the hadron resonance gas at T < T_c and by the free fermion gas at T>T_c. Small differences between the two ensembles, for thermodynamic observables characterising the deconfinement phase transition, vanish with increasing lattice size. These differences are solely caused by contributions of non-zero baryon density sectors, which are exponentially suppressed with increasing volume. The Polyakov loop shows a different behaviour: for all temperatures and volumes, its expectation value is exactly zero in the canonical formulation, whereas it is always non-zero in the commonly used grand-canonical formulation. We clarify this paradoxical difference, and show that the non-vanishing Polyakov loop expectation value is due to contributions of non-zero triality states, which are not physical, because they give zero contribution to the partition function.

Figures (7)

• Figure 1: Phase diagram of $Z_{GC}(i\mu_I)$ in the $(\mu_I,T)$ plane. The arrows indicate the orientation of the Polyakov loop. The vertical lines mark the "order-order" $Z_3$ transitions, which are first-order. Properties of the "order-disorder" $Z_3$ transitions (curved lines) depend on the parameters (number of flavours, quark masses) of the theory.
• Figure 2: Distribution of the complex Polyakov loop trace in the grand canonical ( top) and canonical ( bottom) ensembles. left: $4^3\times 4$, right: $6^3\times 4$. In the thermodynamic limit, the distributions agree for both ensembles, up to two additional $Z_3$-rotations in the canonical ensemble.
• Figure 3: $\frac{\Delta F(T,\mu_I)}{V T^4}$ as a function of $\frac{\mu_I}{T}$, at temperatures $\frac{T}{T_c} \sim 0.9,\; 1.0,\; 1.1$ from left to right. The free energy density varies much more upon entering the high-temperature phase, and the $Z_3$ first-order transitions become visible (right).
• Figure 4: $\frac{\Delta F(T,\mu_I)}{V T^4}$ as a function of $\frac{\mu_I}{T}$ for $\frac{T}{T_c} \sim 0.9$. The histogram method is very noisy. We show ( left) a rescaled version of the leftmost plot in Fig. \ref{['fig:results_chempot']}. We also present ( right) results based on a reweighting method with variance reduction Kratochvila:2005mk. The results are in agreement with the histogram method, but allow for a more reliable description by a Fourier expansion. One Fourier coefficient suffices to describe the data points. The reweighting method Kratochvila:2005mk calculation is computationally demanding and has not been performed yet for the $8^3 \times 4$ lattice. We thus only draw the fit, which is based on histogram data.
• Figure 5: $\frac{\Delta F(T,\mu_I)}{V T^4}$ as a function of $\frac{\mu_I}{T}$ for $\frac{T}{T_c} \sim 1.1$. ( left) The histogram method; ( right) the reweighting method Kratochvila:2005mk, supplemented by the histogram results for $8^3 \times 4$. A simple modification of the free gas expression describes all the data. As the volume increases, the data come close to the Stefan-Boltzmann limit ($T \to \infty$) even though $\frac{T}{T_c} \sim 1.1$ only.