We discuss the nature of the QCD phase transition in the heavy quark high-density region by considering an effective theory in which Polyakov loops are dynamical variables.
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.
The critical points are determined by finite volume scaling analysis, and in the region where the sign problem is severe, the tensor renormalization group is used to investigate.
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.