Quantum $k$-core conduction on the Bethe lattice

Classical and quantum conduction on a bond-diluted Bethe lattice is considered. The bond dilution is subject to the constraint that every occupied bond must have at least $k-1$ neighboring occupied bonds, i.e., $k$-core diluted. In the classical case, we find the onset of conduction for $k=2$ is continuous while for $k=3$, the onset of conduction is discontinuous with the geometric random first-order phase transition driving the conduction transition. In the quantum case, treating each occupied bond as a random scatterer, we find for $k=3$ that the random first-order phase transition in the geometry also drives the onset of quantum conduction giving rise to a new universality class of Anderson localization transitions.

Figure 1

Here $k=3$ and $z=4$. The shaded circles denote branches that are $k-1$ connected to infinity. The removal of bonds 1 and 2 eventually triggers removal of bond 3 and bonds emanating from vertex $a$, including the shaded circles. The remaining three branches emanating from the center site survive the removal process.

Figure 2

$\mathcal{H}\left(s\right)$ for ${\sigma }_0=1/2$.

Figure 3

(Color online) Left: plot of $H_c\left(\sigma \right)$ with bin size 0.0025 for $k=3$, $z=4$ classical bond percolation. Right: plot of $K\left(\sigma \right)$ for the same conditions. $H_c\left(\sigma \right)$ converges after 14 iterations and $K\left(\sigma \right)$ after six iterations.

Figure 4

(Color online) Left: plot of $H_c\left(\sigma \right)$ with bin size 0.0025 for $k=3$, $z=4$ classical site percolation. Right: plot of $K\left(\sigma \right)$ for the same conditions.

Figure 5

Two quantum scatterers in series where $t_1$ and $r_1$ denote the transmission amplitude and reflection amplitude, respectively, for the first scatterer with the incident wave coming from the left, and $t_2$ and $r_2$ the transmission and reflection amplitudes for the second. The prime denotes the incident wave coming from the right.

Figure 6

(Color online) Left: plot of $H_c^q\left(g\right)$ with bin size 0.005 for $k=3$, $z=4$ quantum bond percolation. Right: plot of $K^q\left(g\right)$ for the same conditions.