[Paper Review] Quark localization in QCD above $T_c$

Matteo Giordano, Tamás G. Kovács+1 more|Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)|Nov 7, 2013

Quantum Chromodynamics and Particle Interactions5 references3 citations

TL;DR

This study uses large-scale dynamical QCD simulations with physical quark masses to show that in the deconfined phase above $T_c$, the lowest Dirac eigenmodes remain localized and follow Poisson statistics, while higher modes are delocalized and follow random matrix theory. The mobility edge—the boundary between localized and delocalized modes—scales properly in the continuum limit and increases linearly with temperature, acting as an effective gap for long-range hadronic correlators despite the absence of a true spectral gap.

ABSTRACT

It was previously found that at high temperature the lowest part of the QCD Dirac spectrum consists of localized modes obeying Poisson statistics. Higher up in the spectrum, modes become delocalized and their statistics can be described by random matrix theory. The transition from localized to delocalized modes is analogous to the Anderson metal-insulator transition. Here we use dynamical QCD simulations with staggered quarks to study this localization phenomenon. We show that the "mobility edge", separating localized and delocalized modes, scales properly in the continuum limit and rises steeply with the temperature. Using very high statistics simulations in large volumes we find that the density of localized modes scales precisely with the spatial volume and even at $T=2.6T_{c}$ the lowest part of the spectrum extends all the way down to zero with no evidence of a spectral gap.

Motivation & Objective

• To investigate the localization properties of Dirac eigenmodes in QCD above the chiral phase transition temperature $T_c$.
• To determine whether a true spectral gap exists in the Dirac spectrum at high temperatures.
• To examine the scaling behavior of the mobility edge—the transition point between localized and delocalized modes—in the continuum limit.
• To assess the impact of localized modes on long-range hadronic correlators, particularly their effective gap behavior.

Proposed method

• Simulations are performed using dynamical staggered fermions in SU(3) gauge theory with physical quark masses across a range of temperatures from 260 to 800 MeV.
• Large-volume ensembles ($N_s = 24, 36, 48$) with high statistics ($N_{\text{conf}} \sim 10^4$) are generated to ensure accurate spectral analysis.
• The participation ratio $PR = \left( \sum_x |\psi_i^\dagger(x)\psi_i(x)|^2 \right)^{-1} / V$ is used to distinguish localized ($PR \to 0$) from delocalized ($PR \sim \text{const}$) modes.
• Spectral density is computed and normalized by spatial volume to test scaling behavior; power-law fits are used to analyze the behavior near $\lambda \to 0$.
• The localization length $l = a (V \cdot PR)^{1/4}$ is computed to quantify the spatial extent of localized modes.
• The rescaled mobility edge $\lambda^{rs}_c = \lambda_c / m_{ud}$ is used to test continuum scaling and to study its temperature dependence.

Experimental results

Research questions

• RQ1 Does the Dirac spectrum in QCD above $T_c$ exhibit a true spectral gap, or do localized modes persist down to zero eigenvalue?
• RQ2 How does the mobility edge—the boundary between localized and delocalized modes—scale with temperature and lattice spacing in the continuum limit?
• RQ3 To what extent do localized low-lying eigenmodes act as an effective gap in long-range hadronic correlators?
• RQ4 What is the spatial extent (localization length) of the lowest Dirac eigenmodes, and how does it scale with the inverse temperature?

Key findings

• The spectral density near $\lambda = 0$ vanishes as $\lambda^{4.047 \pm 0.001}$, indicating a smooth, non-analytic behavior without a true spectral gap.
• The participation ratio of low-lying modes decreases with increasing spatial volume, confirming their localization, while bulk modes maintain a constant $PR$, indicating delocalization.
• The mobility edge $\lambda_c$ scales properly in the continuum limit and increases linearly with temperature, with $\lambda^{rs}_c = \lambda_c / m_{ud}$ showing a nearly linear dependence on $T/T_c$.
• The localization length of low-lying modes is approximately one inverse temperature in units of the lattice spacing, indicating that localized modes are squeezed in all spatial and temporal directions.
• The mobility edge acts as an effective gap for long-range hadronic correlators, as localized modes do not contribute to correlations over distances larger than the localization length.
• Extrapolation of the mobility edge to zero yields a critical temperature of $T \approx 170$ MeV, consistent with the absence of localized modes below the chiral crossover.