Dirac mode localization in QCD near the crossover temperature

TL;DR

This work identifies the localization temperature $T_{ m loc}$ for low Dirac modes in 2+1 flavor QCD using rooted staggered fermions on the lattice. By analyzing the unfolded spectrum and the mobility edge via the integrated level-spacing distribution $I_{s_0}$, the authors locate a localized-delocalized transition in the low-lying Dirac spectrum and determine $T_{ m loc}$ through both bracketing and extrapolation methods. They find $T_{ m loc}$ in the range $155$--$158\,\mathrm{MeV}$, in excellent agreement with the pseudocritical temperatures $T_{ m pc}$ defined from chiral condensates and light-quark susceptibilities, and show that $\lambda_c/m_q$ remains renormalization-group invariant with a nonzero continuum limit above the crossover. The results suggest a close connection between deconfinement, chiral symmetry restoration, and the onset of localization of the low Dirac modes, offering a geometric, gauge-invariant view of the QCD transition that is robust to lattice artifacts.

Abstract

We study the localization properties of the low-lying Dirac eigenmodes in QCD near the crossover temperature, using staggered fermions on the lattice. We find that localized low modes, absent at low temperature, appear at a temperature $T_{\mathrm{loc}}$ in the range $155\,\mathrm{MeV}\le T_{\mathrm{loc}}\le 158\,\mathrm{MeV}$, in excellent agreement with the pseudocritical crossover temperature as determined from the chiral condensate and from the light-quark susceptibility.

Figures (6)

• Figure 1: Upper panels: Integrated unfolded level spacing distribution, $I_{s_0}$ [see Eq. \ref{['eq:iulsd']}], below (left) and above (right) the pseudocritical transition temperature. The values corresponding to Poisson, RMT, and critical statistics are also shown. Lower panels: First moment of the level spacing distribution, $\langle s\rangle =\int_0^\infty ds\, s\,p(s;\lambda;N_{\mathrm{s}} )$, below (left panel) and above (right panel) the pseudocritical transition temperature. Data in all panels correspond to $N_{\mathrm{t}}=8$ simulations.
• Figure 2: Determination of the mobility edge from $I_{s_0}$ [see Eq. \ref{['eq:iulsd']}] at $T=165\,\mathrm{MeV}$. Pairs of lines with the same dashing pattern correspond to the two different choices of the three points used in the linear interpolation, while different patterns correspond to different spectral bin size. Shaded areas denote the corresponding error bands.
• Figure 3: Dependence of our estimate of the mobility edge on the lattice spatial size, for the three lowest temperatures above $T_{\mathrm{pc}}$ for the $N_{\mathrm{t}}=8$ lattices.
• Figure 4: The renormalized mobility edge as a function of the lattice spacing ($1/N_{\mathrm{t}} =aT$) for different aspect ratios at $T=165\,\mathrm{MeV}$. Data points are slightly shifted horizontally to improve readability.
• Figure 5: The integrated unfolded level spacing distribution, $I_{s_0}$ [see Eq. \ref{['eq:iulsd']}], for the four temperatures closest to $T_{\mathrm{pc}}$, for $N_{\mathrm{t}}=8$ and $N_{\mathrm{s}}=80$.