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

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.

💡 Research Summary

The paper investigates quantum transport on a finite‑generation Bethe lattice (Cayley tree) by introducing non‑Hermitian complex potentials: a source term (+i\gamma_N) on every peripheral site and a drain term (-i\gamma_0) on the central site. The authors formulate a tight‑binding Hamiltonian with uniform hopping (-1) and solve the full eigenvalue problem rather than performing time‑dependent simulations. This approach reveals directly which eigenstates contribute to transport and which remain localized.

A systematic classification of the eigenstates is obtained. The majority of eigenstates are localized on the outermost generation; their number equals the total number of peripheral sites minus those of the previous generation, and they all share the eigenvalue (E=+i\gamma_N). Because their amplitudes vanish on all interior sites, these states cannot carry current; they simply grow exponentially in time as (\exp(\gamma_N t)) due to the source term.

In contrast, only (N+1) eigenstates have non‑zero amplitude at the central site. By projecting the Bethe lattice onto the unique path that connects a peripheral site to the center, the problem reduces to a one‑dimensional PT‑symmetric tight‑binding chain of length (N+1) with gain (+i\gamma_N) at one end and loss (-i\gamma_0) at the other. The effective Hamiltonian for this chain is non‑Hermitian but PT‑symmetric, and its spectrum exhibits an exceptional point (EP) where two eigenvalues coalesce and become a zero‑energy mode. The current flowing from the sources to the drain is carried exclusively by these extended states.

Analytical expressions for the current are derived. The steady‑state current (obtained from the continuity equation for the non‑Hermitian system) is proportional to \