[Paper Review] The QCD phase diagram at finite density

Christian Schmidt, Zoltán Fodor+1 more|arXiv (Cornell University)|Oct 18, 2005

High-Energy Particle Collisions Research4 citations

TL;DR

This paper applies the density of states (DOS) method to map the QCD phase diagram at finite quark chemical potential and temperature using four flavor staggered fermions on small lattices. By constraining the plaquette and reweighting via the phase of the fermion determinant, it identifies a triple point at μₜᵣᵢ ≈ 300 MeV and Tₜᵣᵢ ≈ 140 MeV, signaling a first-order transition between three distinct phases: a chirally symmetric phase, a hadronic phase, and a dense, strongly interacting matter phase with baryon density up to 20× nuclear density.

ABSTRACT

We study the density of states method to explore the phase diagram of the chiral transition on the tempeature and quark chemical potential plane. Four quark flavours are used in the analysis. Though the method is quite expensive small lattices show an indication for a triple-point connecting three different phases on the phase diagram.

Motivation & Objective

• To overcome the complex action problem in finite-density lattice QCD by applying the density of states (DOS) method.
• To explore the QCD phase diagram in the temperature–quark chemical potential plane for four degenerate quark flavors.
• To determine the existence and location of a triple point connecting three distinct phases: chirally symmetric, hadronic, and dense matter phases.
• To investigate the quark number density and baryon density in the dense phase, especially near the critical chemical potential.

Proposed method

• The DOS method reorders the partition function by fixing the plaquette expectation value P as a constraint, enabling simulation with a real, positive measure g(U) = |det M| exp(–S_G).
• The density of states ρ(x) is computed via the fluctuation-dissipation theorem from plaquette fluctuations around the constrained value x.
• Observables are extrapolated to different (μ, β) points using reweighting via the ratio R(μ, μ₀, β, β₀) = |det M(μ)| / |det M(μ₀)| × exp(S_G(β) – S_G(β₀)).
• A Gaussian potential V(x) = ½γ(x – P)² is used to enforce the plaquette constraint in hybrid Monte Carlo simulations.
• The phase of the fermion determinant, cos(θ), is included in the reweighting to account for the complex action problem.
• Critical lines and the triple point are identified by analyzing the plaquette susceptibility and quark number density across multiple lattice sizes and chemical potentials.

Experimental results

Research questions

• RQ1 Does a triple point exist in the QCD phase diagram at finite chemical potential and temperature for four-flavor QCD?
• RQ2 What is the nature of the phase transition between the chirally symmetric, hadronic, and dense matter phases?
• RQ3 How does the quark number density evolve across the phase transition, and what is its maximum value?
• RQ4 To what extent does the phase diagram depend on quark mass, and is the critical line mass-independent?

Key findings

• A triple point is identified at μₜᵣᵢ ≈ 300 MeV and Tₜᵣᵢ ≈ 148 MeV on the 4⁴ lattice, with Tₜᵣᵢ decreasing to ≈137 MeV on the 6⁴ lattice.
• The phase diagram shows two nearly perpendicular transition lines joining at the triple point, indicating a first-order transition between three distinct phases.
• The quark number density exhibits a sharp rise at the critical chemical potential μ_c ≈ 0.4–0.5a⁻¹, corresponding to a baryon density of (2–3)×n_N, rising to (10–20)×n_N in the dense phase.
• The critical coupling β_c = 4.938(4) is in excellent agreement with multi-parameter reweighting results, validating the DOS method with phase reweighting.
• The phase boundary is found to be independent of quark mass within statistical uncertainties, as shown for am = 0.03.
• The inclusion of the phase factor cos(θ) shifts the transition to lower β, consistent with the expected suppression of the chiral condensate at finite μ.