Abstract
The non-Hermitian skin effect dramatically reshapes the energy bands of non-Hermitian systems, meaning that the usual Bloch band theory is fundamentally inadequate as their characterization. The non-Bloch band theory, in which the concept of Brillouin zone is generalized, has been widely applied to investigate non-Hermitian systems in one spatial dimension. However, its generalization to higher dimensions has been challenging. Here, we develop a formulation of the non-Hermitian skin effect and non-Bloch band theory in arbitrary spatial dimensions, which is based on a natural geometrical object known as the amoeba. Our theory provides a general framework for studying non-Hermitian bands beyond one dimension. Key quantities of non-Hermitian bands, including the energy spectrum, eigenstates profiles, and the generalized Brillouin zone, can be efficiently obtained from this approach.
- Received 16 July 2023
- Accepted 5 February 2024
DOI:https://doi.org/10.1103/PhysRevX.14.021011
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
In quantum mechanics, the evolution of an isolated system is described by a mathematical construct called the Hamiltonian. Specifically, the Hamiltonian must be of a type known as Hermitian, which guarantees the energy spectrum is real and that probabilities are preserved. But for open systems, a non-Hermitian Hamiltonian is required. One of the most interesting consequences of this is the non-Hermitian skin effect, in which the energy eigenstates of the system are squeezed toward its boundaries. This indicates that the usual energy-band theory should be fundamentally revised for non-Hermitian systems. However, the existing formulation is restricted to 1D systems. Here, we formulate a non-Hermitian band theory applicable to any number of spatial dimensions.
The key mathematical tool in our work is an elegant geometrical object called an amoeba. The amoeba is essentially a shadow or a projection of a polynomial of complex variables onto real space. This name is suggested by its typical appearance, featuring elongated “tentacles” and sometimes “vacuoles” in its body. We show that a surprisingly rich amount of information about a non-Hermitian system is encoded in an associated amoeba. For instance, the energy spectrum can be straightforwardly obtained by looking at whether a certain vacuole appears inside the amoeba. A systematic treatment further enables calculations of many key quantities of non-Hermitian systems.
Our formulation provides a quantitative theory of the non-Hermitian skin effect and the associated energy bands beyond one dimension. It paves the way for a complete non-Hermitian band-theory framework and its concomitant applications.
