QCD Anderson transition at zero and non-zero external magnetic fields

Abstract

The QCD Anderson transition is believed to be connected to both deconfinement and chiral crossovers. These crossovers are substantially affected when external magnetic fields ($B$) are present, most prominently, e.g., via magnetic catalysis and inverse magnetic catalysis. In this work, we use lattice QCD to investigate the Anderson transition in two different setups: (1) at $B=0$ by studying the low-lying eigenmodes of the overlap operator using gauge configurations with $2+1+1$ quark flavors of twisted-mass Wilson fermions. We estimate the mobility edge below which eigenmodes are localized via the inflection point of the so-called relative volume. Previous work has shown that, contrary to expectations, this estimate does not vanish at the temperature of the chiral phase transition. A possible scenario for this apparent contradiction was discussed, and in this work, we present an alternative observable for measuring localization that supports this scenario. And (2) by studying the localization properties of the staggered Dirac operator at $B\neq0$ on configurations with $2+1$ dynamical staggered fermions and 2 stout-smearing steps. Our preliminary results on two lattice spacings ($24^3\times 6$ and $24^3\times 8$) indicate a non-monotonic behavior of the mobility edge with the magnetic field across different temperatures, which hints at a reduction in the Anderson transition temperature in the presence of an external magnetic field.

Figures (2)

• Figure 1: Expectation value of $\tilde{r}$ (bottom), defined in Eq. \ref{['eq:r_tilde']}, compared to the relative volume (top) as functions of $\lambda$ for $T=133(4)\,\mathrm{MeV}$ ($N_\mathrm{t}=24$) and $T=266(8)\,\mathrm{MeV}$ ($N_\mathrm{t}=12$). The data points for $\langle \tilde{r} \rangle$ result from averaging over small bins of size $\Delta\lambda \approx 45\,\mathrm{MeV}$. The vertical red-shaded bar highlights the error range of the mobility edge estimate $\lambda_\mathrm{c}$, which is extracted from the inflection point of $r(\lambda)$, including statistical and systematic (estimated by a second fit) error. The fit window $[\lambda_{\mathrm l}, \lambda_{\mathrm r}]$ is indicated by vertical red lines in the plot of $r(\lambda)$, while the vertical line in the plot of $\langle \tilde{r} \rangle$ displays the position of $\lambda_\mathrm{c}$.
• Figure 2: Relative volume for $N_\mathrm{t}=6$ (left panels) and for $N_\mathrm{t}=8$ (right panels) as a function of the eigenvalues (shown as the dimensionless quantity $\lambda/T$) of the staggered Dirac operator for different magnetic field strengths and several temperatures. The mobility edge estimates are indicated by the red colored circles, which were determined by the displayed fitting curves. For the lowest temperature (bottom plots), the inflection point of $r(\lambda)$ disappears towards the left side. The small error bars at each point represent solely statistical errors. For visibility reasons, we only plot every third data point.