QCD Anderson transition with overlap valence quarks on a twisted-mass sea

In this work we probe the QCD Anderson transition by studying spectral distributions of the massless overlap operator on gauge configurations created by the twisted mass at finite temperature collaboration (tmfT) with $2+1+1$ flavors of dynamical quarks and the Iwasaki gauge action. We assess finite-size and discretization effects by considering two different lattice spacings and several physical volumes, and mimic the approach to the continuum limit through stereographic projection. Fitting the inflection points of the participation ratios of the overlap Dirac eigenmodes, we obtain estimates of the temperature dependence of the mobility edge, below which quark modes are localized. We observe that it is well described by a quadratic polynomial and systematically vanishes at temperatures below the pseudocritical one of the chiral transition. In fact, our best estimates within errors overlap with that of the chiral phase transition temperature of QCD in the chiral limit.

Figure 1

Top: stereographic projection of the eigenvalues at finite $\tilde{a}=a/\rho$ to the imaginary axis. Bottom: projection of eigenvalues at $T=509.10(5.52)\mathrm{MeV}$ for the D210 dataset.

Figure 2

Histogram of the polar angle distribution $\rho (\phi )$ with bin size $\mathrm{\Delta }\phi =2.5\times {10}^{-3}$ of the overlap eigenvalues for the A370 configurations ($a=0.0936(13)\mathrm{fm}$; $m_{\pi }=364(15)\mathrm{MeV}$).

Figure 3

Same as in Fig. 2 but for the D370 configurations ($a=0.0646(7)\mathrm{fm}$; $m_{\pi }=369(15)\mathrm{MeV}$).

Figure 4

Same as in Fig. 2 but for the D210 configurations ($a=0.0646(7)\mathrm{fm}$; $m_{\pi }=213(9)\mathrm{MeV}$).

Figure 5

Bin averages of the relative eigenmode volume $r(\phi )$ defined in Eq. (6), with $\mathrm{\Delta }\phi =2.5\times {10}^{-3}$, over the polar angle of the overlap eigenvalues, with fits (8), fit windows and extracted inflection points marked in red, for the A370 configurations.

Figure 6

Same as in Fig. 5 but for the D370 configurations.

Figure 7

Same as in Fig. 5 but for the D210 configurations.

Figure 8

Temperature dependence of mobility edge ${\phi }_{\mathrm{c}}(T)$ for the A370 (top), D370 (middle) and D210 (bottom) configurations, as extracted from the bin-averaged angular relative eigenmode volumes in Figs. 5, 6, 7, with fits (9) and $T_0$ extrapolations (red) where ${\phi }_{\mathrm{c}}(T_0)=0$.

Figure 9

Histogram of the distribution $\rho (\lambda )$ of the stereographically projected overlap eigenvalues $\lambda$ for the A370 configurations ($a=0.0936(13)\mathrm{fm}$; $m_{\pi }=364(15)\mathrm{MeV}$).

Figure 10

Same as in Fig. 9, here for the D370 configurations ($a=0.0646(7)\mathrm{fm}$; $m_{\pi }=369(15)\mathrm{MeV}$).

Figure 11

Same as in Fig. 9, here for the D210 configurations ($a=0.0646(7)\mathrm{fm}$; $m_{\pi }=213(9)\mathrm{MeV}$).

Figure 12

Bin averages of the relative eigenmode volume $r(\lambda )$ over the stereographically projected overlap eigenvalues $\lambda$, with fits (11), fit windows and extracted inflection points marked in red, for the A370 configurations.

Figure 13

Same as in Fig. 12, here for the D370 configurations.

Figure 14

Same as in Fig. 12, here for the D210 configurations.

Figure 15

Temperature dependence of mobility edge ${\lambda }_{\mathrm{c}}(T)$ for the A370 (top), D370 (middle) and D210 (bottom) configurations, as extracted from the bin-averaged relative eigenmode volumes of the stereographically projected eigenvalues in Figs. 12, 13, 14, with fits (12) and $T_0$ extrapolations (red) where ${\lambda }_{\mathrm{c}}(T_0)=0$.

Figure 16

Left: Correlation matrices of the stereographically projected eigenvalues (top) and relative volumes (bottom) between all bins within the fit window for $N_{\mathrm{t}}=4$ of the A370 dataset. Right: Same correlation matrices with correlations between neighboring bins removed when using the alternating configurations method.