Phase structure of heavy dense lattice QCD and the three-state Potts model

The nature of the finite temperature phase transition of QCD depends on the particle density and the mass of the dynamical quarks. We discuss the properties of the phase transition at high density, considering an effective theory describing the high-density heavy-quark limit of QCD. This effective theory is a simple model in which the Polyakov loop is a dynamical variable, and the quark Boltzmann factor is controlled by only one parameter, $C(\mu,m_q)$, which is a function of the quark mass $m_q$ and the chemical potential $\mu$. The Polyakov loop is an order parameter of $Z_3$ symmetry, and the fundamental properties of the phase transition are thought to be determined by the $Z_3$ symmetry broken by the phase transition. By replacing the Polyakov loop with $Z_3$ spin, we find that the effective model becomes a three-dimensional three-state Potts model ($Z_3$ spin model) with a complex external field term. We investigate the phase structure of the Potts model and discuss QCD in the heavy-quark region. As the density varies from $\mu=0$ to $\mu=\infty$, we find that the phase transition is first order in the low-density region, changes to a crossover at the critical point, and then becomes first-order again. This strongly suggests the existence of a first-order phase transition in the high density heavy-quark region of QCD.