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Lessons from Random Matrix Theory for QCD at Finite Density

K. Splittorff, J. J. M. Verbaarschot

arXiv:0809.4503v1hep-ph

TL;DR

QCD at nonzero chemical potential is difficult because the nonhermitian Dirac operator creates a complex determinant and a severe sign problem. The lecture uses chiral random matrix theory, equivalent to QCD in the microscopic domain, to compare full and phase-quenched theories and analyze the determinant phase and Dirac spectrum. It concludes that this phase dramatically changes the phase diagram and that quenching fails at nonzero chemical potential.

  • Problem

    QCD at nonzero chemical potential has a nonhermitian Dirac operator and complex determinant, leaving its phase diagram poorly understood and inaccessible to standard Monte Carlo simulations.

    Source: §1 · p. 1, §2 · p. 3

  • Method

    The lecture studies determinant-phase effects and the Dirac spectrum using chiral random matrix theory, which is equivalent to QCD in the microscopic domain.

    Source: §3 · p. 5

  • Results

    The determinant phase dramatically changes the phase diagram, and quenching completely fails at nonzero chemical potential because phase-quenched and full QCD undergo different transitions.

    Source: §2 · p. 4, §4.2 · p. 6

  • Takeaways & Limitations

    Random matrix theory contributes analytically solvable explanations for the phase determinant’s physical effects, the failure of quenching, and Dirac-spectrum behavior at nonzero chemical potential.

    Source: §4.9 · p. 11, §5 · p. 12

    The random matrix description beyond the microscopic domain applies to low-energy mean-field results only when m ≪ΛQCD and µ ≪ΛQCD.

    Source: §3 · p. 5

Abstract

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In this lecture we discuss various aspects of QCD at nonzero chemical potential, including its phase diagram and the Dirac spectrum, and summarize what chiral random matrix theory has contributed to this subject. To illustrate the importance of the phase of the fermion determinant, we particularly highlight the differences between QCD and phase quenched QCD.

1. Introduction

QCD at nonzero baryon chemical potential remains difficult because its nonhermitian Dirac operator produces a complex fermion determinant, preventing standard Monte Carlo evaluation. The lecture compares full QCD with phase-quenched QCD and uses chiral random matrix theory to study the determinant phase and Dirac spectrum.

  • A complex fermion determinant from the nonhermitian Dirac operator prevents Monte Carlo evaluation of QCD at nonzero chemical potential.Source: §1 · p. 1
  • Phase-quenched QCD removes the determinant phase, remains lattice-accessible, and has a phase diagram completely different from full QCD.Source: §1 · p. 1
  • Chiral random matrix theory [4] [5] at nonzero chemical potential [6] [7] [8] connects determinant-phase effects with the Dirac spectrum.Source: §1 · p. 2

2. Phase Diagram of QCD

The phase diagram at nonzero baryon chemical potential is poorly constrained by first-principles methods, while phase-quenched QCD is equivalent to QCD at nonzero isospin chemical potential. For µ > mπ/2, the determinant phase removes the would-be pion condensate and changes the phase diagram.

  • First-principles knowledge of the chemical-potential direction is limited because lattice QCD requires a real action, leaving much of the phase diagram model-dependent.The temperature range 150-200 MeV is associated with a crossover from chiral symmetry breaking to restoration.Source: §2 · p. 2
  • The lecture uses the phase-quenched comparison to motivate studying how the fermion determinant phase affects QCD at nonzero chemical potential.Source: §2 · pp. 3–4
  • Phase-quenched QCD corresponds to nonzero isospin chemical potential, where low-temperature pions Bose condense for µ > mπ/2 and simulations agree with theoretical expectations [10] [21] [22] [23] [24] [25] [26].Source: §2 · p. 4
  • The phase diagram of QCD and |QCD| are completely different, with the determinant phase nullifying the would-be pion condensate for µ > mπ/2.Source: §2 · p. 4

3. Random Matrix Model at µ̸ = 0

Chiral random matrix theory replaces Dirac-operator matrix elements with Gaussian random variables while preserving QCD’s global symmetries. In the microscopic domain, it is equivalent to QCD because both reduce to the same static chiral Lagrangian.

  • Chiral random matrix theory replaces Dirac-operator matrix elements with Gaussian random numbers [4] [5], using a complex n×(n+ν) matrix W and chemical-potential structure [6,8].Source: §3 · p. 4
  • The random matrix partition function preserves QCD’s global symmetries and transformation properties.Source: §3 · p. 4
  • In QCD’s microscopic domain, the random matrix model gives the mass and chemical-potential dependence of the partition function because both theories reduce to the same static chiral Lagrangian.Source: §3 · pp. 4–5
  • Beyond the microscopic domain, chiral random matrix theory describes mean-field low-energy results when m ≪ΛQCD and µ ≪ΛQCD.Source: §3 · p. 5

4. Lessons from Random Matrix Theory

Random matrix theory explains key features of the complex Dirac spectrum and shows why the quenched approximation fails at nonzero chemical potential. Homogeneous eigenvalue distributions imply a linear quenched chiral condensate, while phase quenching produces a pion-condensed transition absent from full QCD.

4.1. Lesson 1: Homogeneity of Dirac Eigenvalues

  • Homogeneous, sharply edged complex Dirac spectra follow from eigenvalue repulsion, and imply that the quenched chiral condensate increases linearly with m inside the eigenvalue domain.Source: §4.1 · p. 5

4.2. Lesson 2: Quenched Limit

  • At nonzero chemical potential, the quenched limit is not the n→0 limit of phase-quenched QCD’s determinant ensemble, a distinction demonstrated analytically with random matrix theory.Source: §4.2 · pp. 5–6
  • Phase-quenched QCD undergoes a low-temperature pion-condensation transition at µ = mπ/2, whereas full QCD transitions to nonzero baryon density at µ = mN/3.Source: §4.2 · p. 6
  • Quenching therefore completely fails for µ̸ = 0 because phase-quenched and full QCD undergo different transitions.Source: §4.2 · p. 6

4.3. Lesson 3: Width of the Dirac Spectrum

The phase-quenched transition occurs when the quark mass reaches the Dirac-eigenvalue domain, linking the transition point to the spectrum’s half-width. In the microscopic domain, random matrix theory and lattice evidence further indicate that determinant-related quantities are infrared dominated.

4.3. Lesson 3: Width of the Dirac Spectrum

  • The phase-quenched transition occurs at μ = mπ/2, when the quark mass reaches the domain of Dirac eigenvalues.The coincidence identifies the transition point with the spectral edge and determines the half-width condition.Source: §4.3 · p. 6
  • For sufficiently small chemical potential, the Dirac-spectrum width increases linearly with μ rather than quadratically.The spectrum’s density and width can be used to extract the low-energy constants Σ and Fπ.Source: §4.3 · p. 6

4.4. Lesson 4: Infrared Dominance

  • Although large eigenvalues might contribute because their density grows as ∼V λ^3, finite ultraviolet contributions cannot be excluded at nonzero chemical potential.The concern arises because complex eigenvalues make infrared dominance less immediately clear away from μ = 0.Source: §4.4 · p. 7
  • In the microscopic domain, random matrix theory shows that the average determinant is determined by small Dirac eigenvalues, consistent with lattice results at imaginary chemical potential.A small number of low-lying eigenvalues reproduce the random matrix result [39].Source: §4.4 · p. 7
  • When μ and mπ are of order Fπ or less, agreement with lattice QCD supports chiral perturbation theory and infrared dominance of the quark-determinant magnitude.Because the full QCD partition function is then μ-independent at low temperatures, the μ-dependence of the determinant phase resides in the infrared spectrum.Source: §4.4 · p. 7

4.5. Lesson 5: Failure of Banks-Casher at µ̸ = 0

At nonzero chemical potential, the Banks–Casher relation gives different behavior in phase-quenched and full QCD. Random-matrix analysis explains how the full-QCD condensate can be discontinuous without eigenvalue accumulation, through exponentially large cancellations.

Failure of Banks-Casher at µ̸ = 0

  • For phase-quenched QCD, the Banks–Casher relation correctly gives a vanishing chiral condensate at nonzero chemical potential.The relation uses the density of Dirac eigenvalues and spacetime volume.Source: §4.5 · p. 7
  • The full-QCD chiral condensate is discontinuous when the quark mass crosses the imaginary axis, despite no accumulation of Dirac eigenvalues.Random matrix theory identifies an alternative mechanism for this discontinuity.Source: §4.5 · p. 8

One-dimensional illustration

  • The one-dimensional model has a complex determinant phase, and its condensate stays finite through cancellations among contributions that grow exponentially with the number of lattice points.This is the mechanism behind the Silver Blaze phenomenon in the thermodynamic limit.Source: §4.5 · pp. 8–9
  • In one-dimensional lattice QCD, the condensate remains continuous across the eigenvalue ellipse but is discontinuous at m = 0, where no eigenvalues occur.The left panel shows eigenvalues for one gauge configuration and their thermodynamic-limit support; the right panel shows the condensate versus quark mass.Source: Figure 5 · p. 8, §4.5 · p. 9

4.6. Lesson 6: Quark Mass and Average Phase Factor

The average phase factor distinguishes QCD from phase-quenched QCD, while chiral random matrix theory describes low-lying Dirac spectra at nonzero chemical potential and topology.

4.6. Lesson 6: Quark Mass and Average Phase Factor

  • At nonzero temperature, the average phase factor vanishes in the thermodynamic limit because QCD and phase-quenched QCD have different free energies.Source: §4.6 · p. 9
  • At zero temperature and µ < mπ/2, the average phase factor approaches one in the thermodynamic limit.Source: §4.6 · p. 9
  • The sign problem is most severe when the quark mass lies inside the domain of Dirac eigenvalues.Source: §4.6 · p. 10

4.7. Lesson 7: Distribution of Small Dirac Eigenvalues

  • Chiral random matrix theory results for the radial distribution of the smallest Dirac eigenvalue are compared with lattice histograms across topological charge sectors.Source: §4.7 · p. 10, Figure 7 · p. 10

4.8. Lesson 8: Equality Two Condensates

The two chiral condensates require careful treatment of topology and normalization, especially at nonzero chemical potential, to establish their equality.

4.8. Lesson 8: Equality Two Condensates

  • The chiral condensate can be formulated through two expressions whose equality is connected to the Banks-Casher relation and rigorous normalization results.Source: §4.8 · pp. 10–11
  • In topological-charge sector ν, exact zero modes require a subtle reshuffling of nonzero eigenvalues to satisfy the chiral Ward identity.Source: §4.8 · p. 11
  • For finite Dirac matrices, ν exact zero eigenvalues arise from the rectangular nonzero off-diagonal block, while additional paired zero modes can be ignored as measure-zero contributions.Source: §4.8 · p. 11
  • The two condensates become equal only after the partition functions are correctly normalized, including at nonzero chemical potential.Source: §4.8 · p. 11

4.9. Lesson 9: Test of Algorithms

Because chiral random matrix models are exactly solvable while retaining essential nonhermitian features, they provide testbeds for algorithms used in finite-density lattice QCD.

4.9. Lesson 9: Test of Algorithms

  • Chiral random matrix models have tested the density-of-states method, which was subsequently applied successfully to lattice QCD.Source: §4.9 · pp. 11–12
  • Random matrix models have determined the radius of convergence for algorithms based on Taylor expansion or analytic continuation in µ.Source: §4.9 · p. 12

5. Conclusions

The lecture concludes that the fermion-determinant phase substantially changes finite-density QCD, and that chiral random matrix theory clarifies these effects in an analytically solvable setting.

5. Conclusions

  • The fermion-determinant phase can make the phase diagrams of QCD and phase-quenched QCD completely different.Source: §5 · p. 12
  • In the microscopic domain, chiral random matrix theory captures QCD contributions analytically, including the failure of the quenched approximation and the mechanism behind discontinuities of the chiral condensate.Source: §5 · p. 12
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