Imprints of $U_A(1)$ chiral anomaly and disorder in the Dirac eigenspectrum of QCD at finite temperature

Abstract

We perform a comprehensive study of the properties of Dirac eigenvalue spectrum in QCD as a function of temperature on the lattice. In addition to effects due to interplay between interactions and disorder inherently present in a many-body system, the Dirac spectrum also contains crucial information about the effective restoration of different subgroups of almost exact two flavor chiral symmetry in QCD. We calculate the infrared eigenvalues of the overlap Dirac operator on 2+1 flavor QCD ensembles generated using domain wall fermion discretization, on a large volume lattice. From the normalized level spacing ratios we identify those eigenvalues which have intermediate level statistics, distinctly different from the majority in the bulk spectrum that follow universal level fluctuations similar to a random matrix of Gaussian unitary type. We provide an explanation of these intermediate level ratios in terms of a specific random matrix model and quantify correlation between these eigenstates and disorder in the gauge fields manifested in the renormalized Polyakov loop values. Whereas existence of these intermediate eigenmodes are intimately connected to the effective restoration of different subgroups of chiral symmetry close to chiral crossover transition, these arise due to effects of random uncorrelated disorder at higher temperatures when the $U_A(1)$ is effectively restored. We also, for the first time, calculate the Thouless conductance for the Dirac spectrum that quantifies the structural rigidity of the eigenvectors, and use it as a diagnostic tool to understand the restoration of the anomalous $U_A(1)$ subgroup of chiral symmetry and localization driven due to disorder.

Figures (14)

• Figure 1: The variation of $\langle \tilde{r} \rangle$ in the bins of eigenvalues in the temperature range where the non-singlet part of chiral symmetry is effectively restored but not its $U_A(1)$ subgroup. The orange and black solid lines represent the expected $\langle \tilde{r} \rangle$ for the GUE and Poisson statistics respectively.
• Figure 2: The variation of $\langle \tilde{r} \rangle$ in small bins in $\lambda/T$ at high temperatures $T\gtrsim 1.5~T_{pc}$ MeV when the $U_A(1)$ is effectively restored compared to the data at $195$ MeV.
• Figure 3: Probability distribution of $\tilde{r}$ for the intermediate (left) and bulk (right) eigenvalues at different temperatures compared with its values for random matrices belonging to GUE (solid line) and those belonging to uncorrelated eigenvalues following Poissonian level statistics (dotted line).
• Figure 4: The probability distribution of the normalized level ratios $P(\tilde{r})$ for our first matrix model where a GUE block is additionally mixed with a block of uncorrelated eigenvalues following level statistics similar to a Poisson distribution selected with a probability distribution weight for $\tilde{r}$ determined from a GUE. The $P(\tilde{r})$ for the newly added uncorrelated block, and the entire mixed matrix are denoted as triangles and diamonds respectively with $3\%$ (blue) and $15\%$ (orange) admixture.
• Figure 5: The $P(\tilde{r})$ for our second matrix model where a GUE block is additionally mixed with a block of uncorrelated eigenvalues selected with a probability distribution weight for $\tilde{r}$ determined from a Poissonian distribution. The $P(\tilde{r})$ for the newly added uncorrelated block, and the entire mixed matrix are denoted as triangles and diamonds respectively with $3\%$ (blue) and $15\%$ (orange) admixture.