Anderson transition and multifractals in the spectrum of the Dirac operator of quantum chromodynamics at high temperature

We investigate the Anderson transition found in the spectrum of the Dirac operator of quantum chromodynamics at high temperature, studying the properties of the critical quark eigenfunctions. Applying multifractal finite-size scaling we determine the critical point and the critical exponent of the transition, finding agreement with previous results, and with available results for the unitary Anderson model. We estimate several multifractal exponents, finding also in this case agreement with a recent determination for the unitary Anderson model. Our results confirm the presence of a true Anderson localization-delocalization transition in the spectrum of the quark Dirac operator at high temperature, and further support that it belongs to the 3D unitary Anderson model class.

Figure 1

Eigenvectors of the Dirac operator in lattice QCD at $T\approx 2.6T_c$ (a) in the insulating regime (with energy $E=0.15$ in lattice units), (b) at criticality ($E=0.3355$), and (c) in the metallic regime ($E=0.365$). Dot sizes are proportional to $\sqrt{|\psi {|}^2}$, with proportionality factors tuned independently for each subfigure to improve visualization. The spatial system size is $L=56$ (in lattice units) for all subfigures. Summation over the temporal direction and over color components has been performed, see the discussion at the end of Sec. 3. Coloring is determined by the value of the $x$ coordinate.

Figure 2

Eigenvectors of the Dirac operator in the critical regime ($E=0.3355$) on different time slices. Dot sizes are proportional to $\sqrt{\sum _c|{\chi }_c(\overrightarrow{x},t){|}^2}$, while coloring is determined by the value of the $x$ coordinate.

Figure 3

Correlations, Eq. (17), between (a) critical eigenfunctions of the unitary Anderson model for $W=18.37$ and system size $L=10$, and (b) eigenfunctions of the QCD Dirac operator at $T\approx 2.6T_c$ in the critical regime, $0.32\le E\le 0.35$, for different system sizes.

Figure 4

GMFEs, (a) ${\tilde{D}}_{0.1}^{\mathrm{ens}}$ and (b) ${\tilde{\alpha }}_{1.0}^{\mathrm{typ}}$, at fixed $\lambda =0.125$. Dots are the raw data, and the solid red line is the best fit obtained by MFSS. Insets show the scaling functions on a log-log scale, after subtracting the irrelevant term. Error bars are not shown in the insets for visual clarity.

Figure 5

Critical parameters obtained via MFFS at fixed $\lambda =0.125$ on the eigenvectors of the QCD Dirac operator. Error bars correspond to the 95% confidence band. Systematic errors are not included. Points corresponding to the same value of $q$ are slightly shifted horizontally for clarity.

Figure 6

Dependence of the fitted (a) critical point and (b) critical exponent, as obtained from $D_{0.1}^{\mathrm{ens}}$ at fixed $\lambda =0.125$, for various energy windows $\mathrm{\Delta }E$. Error bars correspond to the 95% confidence band. Only statistical errors are shown.

Figure 7

Estimated values of the MFEs, (a) ${\alpha }_q$ and (b) $D_q$, in high-temperature QCD, and MFEs of the 3D unitary Anderson model taken from Ref. [28] (slightly shifted horizontally for clarity).