[Paper Review] Equivalence of Matrix Models for Complex QCD Dirac Spectra

Gernot Akemann|ArXiv.org|Jul 12, 2003

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper establishes the equivalence between two matrix models for QCD with a non-zero quark chemical potential: Stephanov's model (directly tied to QCD) and Akemann's model (enabling full calculation of spectral correlations). The equivalence is proven via partition function mapping in the broken chiral symmetry phase, holding exactly in the weak non-Hermiticity limit and approximately at strong non-Hermiticity for small μ. The result confirms the universality and QCD relevance of spectral predictions from Akemann's model.

ABSTRACT

Two different matrix models for QCD with a non-vanishing quark chemical potential are shown to be equivalent by mapping the corresponding partition functions. The equivalence holds in the phase with broken chiral symmetry. It is exact in the limit of weak non-Hermiticity, where the chemical potential squared is rescaled with the volume. At strong non-Hermiticity it holds only for small chemical potential. The first model proposed by Stephanov is directly related to QCD and allows to analyze the QCD phase diagram. In the second model suggested by the author all microscopic spectral correlation functions of complex Dirac operators can be calculated in the broken phase. We briefly compare those predictions to complex Dirac eigenvalues from quenched QCD lattice simulations.

Motivation & Objective

• To establish the theoretical equivalence between two distinct matrix models proposed for QCD with non-zero chemical potential.
• To demonstrate that Akemann's model, which enables full calculation of microscopic spectral correlation functions, is physically equivalent to Stephanov's model, which is directly derived from QCD symmetry principles.
• To validate the universality of spectral predictions from Akemann's model by comparing them to quenched QCD lattice simulations.
• To clarify the conditions under which the equivalence holds, particularly in the limits of weak and strong non-Hermiticity.

Proposed method

• Mapping the partition functions of Stephanov's and Akemann's matrix models to prove their equality in the broken chiral symmetry phase.
• Using the replica trick and orthogonal polynomial techniques to analytically continue the partition function into the complex plane, enabling spectral correlation calculations.
• Applying the weak non-Hermiticity limit by rescaling μ² with volume V, keeping Vμ² fixed, to establish exact equivalence for Nf = 1, 2, 3 degenerate flavors.
• Extending the equivalence to strong non-Hermiticity via perturbative analysis in small μ, with leading-order agreement for Nf = 1.
• Comparing theoretical predictions for the microscopic Dirac spectrum to quenched lattice QCD data using rescaled eigenvalue density and level spacing.
• Employing a parameter-free comparison by fixing the microscopic spectral density parameter α = 0.19 from lattice data, with no fitting applied.

Experimental results

Research questions

• RQ1 Are the two distinct matrix models for QCD with chemical potential physically equivalent?
• RQ2 Does the spectral correlation function prediction from Akemann's model, derived via analytic continuation, match the predictions of Stephanov's QCD-inspired model?
• RQ3 In what parameter regime—weak or strong non-Hermiticity—does the equivalence hold?
• RQ4 Can the universal spectral predictions of the matrix model be validated against quenched lattice QCD simulations?

Key findings

• The partition functions of Stephanov's and Akemann's matrix models are exactly equivalent in the broken chiral symmetry phase under the weak non-Hermiticity limit, where Vμ² is held fixed.
• For strong non-Hermiticity, the equivalence holds to leading order in small μ for Nf = 1, and is expected to extend to Nf ≥ 2.
• The microscopic spectral density predicted by Akemann's model agrees excellently with quenched QCD lattice data at weak non-Hermiticity, with α = 0.19 determined from lattice level spacing and no fitting applied.
• At strong non-Hermiticity (μ = 0.2), the predicted spectral density along the real and imaginary axes matches lattice histograms with no free parameters, confirming the model's validity.
• The agreement persists across different lattice sizes when α is held constant, indicating robustness of the spectral prediction.
• The equivalence implies that the spectral correlation functions derived from Akemann's model are universal and directly relevant to QCD, even in the presence of a chemical potential.