Spectral Functions at Small Energies and the Electrical Conductivity in Hot Quenched Lattice QCD

In lattice QCD, the maximum entropy method can be used to reconstruct spectral functions from Euclidean correlators obtained in numerical simulations. We show that at finite temperature the most commonly used algorithm, employing Bryan’s method, is inherently unstable at small energies and gives a modification that avoids this. We demonstrate this approach using the vector current-current correlator obtained in quenched QCD at finite temperature. Our first results indicate a small electrical conductivity above the deconfinement transition.

Figure 1

First four basis functions $u_i(\omega )$ as a function of $a\omega$ for $a{\omega }_{\max }=5$, $N_{\omega }=1000$, $N_{\tau }=24$, using the standard kernel. The inset shows a blowup of the small energy region.

Figure 2

As in Fig. 1, using the redefined kernel $\overline{K}(\omega ,\tau )$.

Figure 3

Vector spectral functions $\rho (\omega )/\omega T$ as a function of $\omega /T$. We used $N_{\omega }=1000$, $a{\omega }_{\max }=5$, $b=1$.

Figure 4

Default model dependence of $\rho (\omega )/\omega T$ for $N_{\tau }=24$ (hot) and 16 (very hot) in the low-energy region. We show results for $N_{\omega }=1000$, 2000 and $b=1.0$, 0.5, 0.1 at fixed $a{\omega }_{\max }=5$.