Quantum transport through hierarchical structures

The transport of quantum electrons through hierarchical lattices is of interest because such lattices have some properties of both regular lattices and random systems. We calculate the electron transmission as a function of energy in the tight-binding approximation for two related Hanoi networks. HN3 is a Hanoi network with every site having three bonds. HN5 has additional bonds added to HN3 to make the average number of bonds per site equal to five. We present a renormalization group approach to solve the matrix equation involved in this quantum transport calculation. We observe band gaps in HN3, while no such band gaps are observed in linear networks or in HN5. We provide a detailed scaling analysis near the edges of these band gaps.

Figure 1

Depiction of the 3-regular network HN3 with a one-dimensional periodic backbone forming a ring, here with $k=5$. The top and bottom sites $n=0$ and $n=2^{k-1}$ require special treatment and are connected to external leads. With these connections, the entire network becomes 3-regular. Note that the graph is planar.

Figure 2

Depiction of the planar network HN5. Black lines demarcate the original HN3 structure; the green-shaded lines are added to make HN5. Note that sites on the lowest level of the hierarchy have degree 3, then degree 5, 7, $\dots$, making up a fraction of 1/2, 1/4, 1/8, $\dots$, of all sites, thereby making for an average degree 5 of this network.

Figure 3

Scattering a quantum electron off a linear version of HN3, here drawn as a branched Koch curve. The incoming electron on the left gets scattered into an outgoing transmitted portion (right) and reflected portion (left) on the attached external leads (blue shaded). In this form of HN3, the one-dimensional backbone is marked by black links while the small-world links are shaded in red. Note, for instance, that the shortest end-to-end path here is the baseline of the Koch curve.

Figure 4

Decimation of a block in the RG. The two sites with on-site energy ${\kappa }_1$ with a connecting bond of strength ${\tau }_1$ are decimated.

Figure 5

Plot of ${\kappa }^{(10)}$ in Eq. (26) as a function of its initial condition ${\kappa }^{(0)}=E$. Even at this small order, the function varies extremely rapidly, such that its (green-shaded) line completely covers the shown domain. Thus, for $-2⩽E⩽2$, ${\kappa }^{(m)}$ for any sufficiently large $m$ is a random function. Correspondingly, the transmission spectrum is dense, with full transmission close to any input energy $E$ such that particles do not localize.

Figure 6

Plot of ${\kappa }_{200}$ in Eq. (29) as a function of its initial condition ${\kappa }_0=E$. Bands of localized and delocalized states intermix. Correspondingly, there are localization-delocalization transitions already before the addition of any additional randomness in HN3, merely as a function of the input energy $E$. On the horizontal axis, we have marked the solutions $E_s^{(i)}$ of Eq. (36) for $s=1$ (red dots), $s=2$ (blue dots), and $s=6$ (small black dots). The accumulation of the latter demonstrates (even for such a small value of $s$) that the band gaps are associated with the absence of such solutions. While the solutions for $s=1$ happen to be interior to the bands, some of those for $s=2$ appear to mark the band edges, in particular the one at $E=E_2^{(2)}=-0.637875$.

Figure 7

Plot of $1/{\kappa }_{\infty }$ as a function of $\sqrt{E-E_s^{(i)}}$ for $E\to E_s^{(i)}$ to test Eq. (49). Here, $E_2^{(2)}=-0.637875$, marked as the second blue dot from the left in Fig. 6.

Figure 8

Plot of the state of Eq. (50) after 1000 iterations for initial energies ${\kappa }_0=E$ and the interpolation parameter $y$ (the resolution is 0.001 in each direction). Shaded points have not, or not yet, converged to a steady state; i.e., those values possess nonzero transmission. At $y=0$, corresponding to HN2, the system transmits for any input energy, $-2⩽E⩽2$. As soon as $y>0$, bands of localized states emerge (especially at $E=0$), and the remaining transmitting states exhibit a chaotic dependence on the parameters. At $y=1$, the case of HN3 marked by a dotted horizontal line, only a few nonlocalized states remain, and the further strengthening of small-world links diminish transmission even more, such that it ceases completely for $y>4$. Any band of transmitting states appears to be accompanied by solutions of Eq. (51), which here are drawn as curves for the lowest orders of the recursion only, $s=0,1,2,$ and 3 corresponding respectively to dot-dot-dashed, dot-dashed, dashed, and solid curves.

Figure 9

Plot of ${\kappa }^{(10)}$ in Eqs. (52) for HN5 as a function of its initial condition ${\kappa }^{(0)}=E$. Bands of localized and delocalized states intermix, but those localized intervals disappear asymptotically.

Figure 10

The last steps in the RG for the linear (left) and the ring (right) geometries. In both cases one is left with only two sites connected to the external leads. Note that for clarity we have dropped the superscripts that denote the RG number on all variables.