Quantum transport on Bethe lattices with non-Hermitian sources and a drain

Abstract

We consider quantum transport in a tight-binding model on the Bethe lattice of finite generation, which we expect to be the first step toward analyzing electronic transport in a light-harvesting molecule. We seek conditions under which the electronic current from the peripheral sites to the central site reaches its maximum. As a new feature, we add complex potentials for sources at peripheral sites and a drain at the central site, and solve a non-Hermitian eigenvalue problem, instead of simulating an initial-value problem. Solving the eigenvalue problem clearly reveals which electronic channels contribute most to the quantum transport. We find that the number of eigenstates that can penetrate from the peripheral sites to the central site is limited among the total number of eigenstates. All the other eigenstates are localized around the peripheral sites and cannot reach the central site. The former eigenstates can carry current, reducing the problem to quantum transport on a parity-time symmetric tight-binding chain. We find that the current has a maximum with respect to the strengths of the sources and the drain. Counterintuitively, the current decreases as we increase the strengths beyond the maximum and vanishes in the limit of infinite strength. Moreover, we find that the current maximum is given by a zero mode. When the number of links is common to all generations, the current takes the maximum value at the exceptional point where two eigenstates coalesce to a zero mode, which emerges because of the non-Hermiticity due to the PT-symmetric complex potentials. By introducing randomness either into the hopping amplitude or the number of links in each generation of the tree, we obtain a random-hopping tight-binding model, and find that the current reaches its maximum not exactly, but approximately, for a zero mode, although it is no longer located at an exceptional point in general.

Figures (18)

• Figure 1: Schematic view of the tree-like network. The source potentials $+\mathrm{i}\gamma_N$ with $\gamma_N>0$ are added to the peripheral sites, while the drain potential $-\mathrm{i}\gamma_0$ with $\gamma_0>0$ is added to the central site $0$.
• Figure 2: One branch of the tree lattice with the root site $\mu$ in the $(N-1)$th generation and $n_N$ pieces of peripheral sites in the $N$th generation.
• Figure 3: One branch of the tree lattice with the root site $\mu$ in the $(N-2)$th generation and the subsequent sites in the $(N-1)$th generation and the peripheral sites in the $N$th generation.
• Figure 4: Real and imaginary parts of the three eigenvalues of Eq. \ref{['eq320']}. One eigenvalue is pure imaginary with the largest imaginary part, while the other two eigenvalues have a common imaginary part and the real parts of the same magnitude with opposite signs.
• Figure 5: Numerical calculation of the function $f(k)$ in the left-hand side of Eq. \ref{['eq580']} for $1\leq N\leq6$.