Many-Body Localization in the Presence of a Central Qudit

We consider a many-body localized system coupled globally to a central $d$-level system. Under an appropriate scaling of $d$ and $L$, we find evidence that the localized phase survives. We argue for two possible thermalizing phases, depending on whether the qudit becomes fully ergodic. This system provides one of the first examples of many-body localization in the presence of long-range (nonconfining) interactions.

Figure 1

Proposed infinite temperature phase diagram for the central qudit model upon taking $L\to \infty$ with $d/\sqrt{L}$ fixed. In addition to MBL and ETH phases of the spin chain, the dotted line indicates the crossover from fully thermal qudit to athermal qudit (“Floquet ETH”). The behavior of the phase boundary near $d/\sqrt{L}=0$ is unclear; a possible ${\mathrm{\Gamma }}_c\sim L^{-1}$ scaling [12] is indicated by the dashed line. Two finite size estimators of the critical $\mathrm{\Gamma }$—defined near Eq. (2) and in Fig. 2—are plotted. The value $r=0.46$ is taken to be halfway between the thermal ($r=0.53$) and nonthermal ($r=0.39$) values [2].

Figure 2

Half chain mutual information variance ${\sigma }_I^2$ for various qudit sizes. The leftward drift of the peaks with system size $L$ is seen to slow down with increasing $d$.

Figure 3

Entanglement entropy $S_{\text{qudit}}\equiv S({\rho }_{\mathcal{Q}})=S({\rho }_{\mathcal{A}\mathcal{B}})$ between the qudit and the spin chain. The dashed line corresponds to the Page value, ${\log }_{2}d-d{(2^{L+1}\log 2)}^{-1}\approx {\log }_{2}d$. The insets show the variance of $S_{\text{qudit}}$.

Figure 4

Variance of the qudit wave function versus size of the spin chain, for $\mathrm{\Gamma }$ in both MBL and ETH phases and near criticality. The red line is given by $0.02L$.