[Paper Review] The 3-state Potts model as a heavy quark finite density laboratory

Seyong Kim, Ph. de Forcrand+2 more|ArXiv.org|Oct 12, 2005

Theoretical and Computational Physics4 references19 citations

TL;DR

This paper investigates the 3D three-state Potts model as a finite-density laboratory for heavy quark QCD, demonstrating that a real chemical potential weakens the first-order deconfinement transition, leading to a second-order point and eventually a crossover. Using analytic continuation from imaginary to real chemical potential, the study confirms the method's viability and reveals that the phase transition in heavy quark QCD is suppressed by finite density, contrasting with light quark behavior.

ABSTRACT

The 3-D Z(3) Potts model is a model for finite temperature QCD with heavy quarks. The chemical potential in QCD becomes an external magnetic field in the Potts model. Following Alford et al.\cite{Alford_et_al}, we revisit this mapping, and determine the phase diagram for an arbitrary chemical potential, real or imaginary. Analytic continuation of the phase transition line between real and imaginary chemical potential can be tested with precision. Our results show that the chemical potential weakens the heavy-quark deconfinement transition in QCD.

Motivation & Objective

• To map the 3D three-state Potts model to finite-temperature QCD with heavy quarks, leveraging shared Z(3) global symmetry.
• To investigate the phase diagram of the Potts model under arbitrary real or imaginary chemical potential, simulating the effect of finite quark density in QCD.
• To test the feasibility and accuracy of analytic continuation from imaginary to real chemical potential in the presence of a complex action.
• To determine how the chemical potential affects the nature of the deconfinement phase transition in heavy quark QCD via the Potts model as a proxy.
• To compare the transition behavior in heavy quark QCD (Potts model) with that in light quark QCD, revealing a key difference in response to chemical potential.

Proposed method

• The Potts model Hamiltonian is formulated with external fields coupled to Polyakov lines Φ and Φ*, representing the chemical potential effect in QCD.
• The partition function is re-expressed in terms of bond configurations and clusters, enabling a real-valued effective action despite a complex original action.
• Monte Carlo simulations are performed using cluster algorithms to sample bond configurations, avoiding the sign problem through exact summation over spin states within clusters.
• Critical points are located via the Binder cumulant of magnetization and the crossing of B4 = 1.604, ensuring precise identification of phase transitions.
• Analytic continuation from imaginary chemical potential (μ → iμ_I) to real μ is tested by comparing critical lines across both regimes on large lattices (e.g., 72³).
• A high-order polynomial fit (8th order) is used to describe the critical line in the (μ/T)² parameter space, capturing curvature effects from Roberge-Weiss symmetry.

Experimental results

Research questions

• RQ1 How does the inclusion of a real chemical potential affect the phase structure of the 3D three-state Potts model, particularly the nature of the deconfinement transition?
• RQ2 Can analytic continuation from imaginary to real chemical potential be reliably performed in a model with a complex action, and how accurate is it on large lattices?
• RQ3 What is the quantitative relationship between the critical temperature and chemical potential in the Potts model, and how does it compare to QCD expectations?
• RQ4 Does the Potts model exhibit a second-order phase transition at finite chemical potential, and if so, what is the critical value of μ/T?
• RQ5 How does the behavior of the heavy quark QCD phase diagram under finite density differ from that of light quark QCD, as revealed by the Potts model mapping?

Key findings

• The critical line in the (μ/T)² parameter space is well described by an 8th-order polynomial: M/T = 8.273 + 0.585(μ/T)² - 0.174(μ/T)⁴ + 0.160(μ/T)⁶ - 0.071(μ/T)⁸.
• The Potts model exhibits a first-order transition at zero chemical potential, which weakens to a second-order transition at a critical μ/T, and becomes a crossover beyond that point.
• Analytic continuation from imaginary to real chemical potential is feasible and accurate even on large lattices (72³), with small real μ results smoothly connected to small imaginary μ_I results.
• The sign problem is absent in the Potts model due to a change of variables to bond configurations, which yields a real partition function despite a complex action.
• The phase diagram of heavy quark QCD, as modeled by the Potts system, shows a shrinking first-order transition region with increasing chemical potential, contrasting with the behavior in light quark QCD.
• The study supports the possibility that recent light-quark simulations may challenge the conventional wisdom that chemical potential strengthens the transition in light quark systems, suggesting a similarity to the heavy quark case.