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, or the Cayley tree, 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 light-harvesting sites to the central site reaches its maximum.
As a new feature for analyzing quantum transport, 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.
Indeed, we find that the number of eigenstates that can penetrate from the peripheral sites to the central site is quite 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 (¥PT)-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 (a zero-energy eigenstate).
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.