Phase structure of lattice QCD in the heavy quark high-density region and the three-state Potts model

TL;DR

The paper addresses the phase structure of lattice QCD in the heavy quark high-density region by mapping the heavy dense QCD effective theory to a 3D $Z_3$ spin model, i.e., the three-state Potts model with a complex external field. It derives the effective theory via a hopping parameter expansion, introduces the parameter $C=(2\kappa)^{N_t} e^{\mu/T}$, and shows how the quark determinant reduces to a product that maps to real and imaginary external fields $h$ and $q$. Finite-size scaling and Binder cumulant analysis locate a critical endpoint with $B_{4c}=1.601(6)$ and $h_c=0.000479(3)$, $C_c=1.195(7)\times 10^{-4}$, indicating Ising universality; a second high-density critical point appears at large $C$. Tensor renormalization group (HOTRG) enables sign-problem-free analysis in 3D, revealing that at large $h$ the transition disappears, and confirming the first-order–crossover–first-order scenario across density; the high-density limit corresponds to a constant determinant, akin to quenched QCD.

Abstract

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 $μ=0$ to $μ=\infty$, 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.

Figures (7)

• Figure 1: The corresponding parameters $(h,q)$ of the spin model when changing $C$ in heavy dense QCD for $N_{\rm f}=2$Ejiri:2026ijj.
• Figure 2: Probability distribution of magnetization in the complex plane for the three-state Potts model at $(\beta, h, q)=(0.36697, 0.0, 0.0)$ on a $40^3$ lattice.
• Figure 3: Distribution of the local Polyakov loop at each point in one configuration of quenched QCD. The left panel shows the distribution for the symmetric phase $(\beta =5.60)$, and the right panel shows the distribution for the broken phase $\beta = 6.00)$Ejiri:2022jai.
• Figure 4: Distribution of spins $\tilde{s}(\vec{x}, t)$ at each point in one configuration $(\beta=0.30)$ of three-state Potts model, after coarse-graining with the diffusion equation with $Dt/a^2 =0.5$. The left panel is $\beta=0.30$ (symmetric phase), and the right panel is $\beta=0.40$ (broken phase) Ejiri:2026ijj.
• Figure 5: Binder cumulant at the $\beta_c$ as a function of $h$ on lattices with $L$ from 40 to 90.