Bulk spectra and the non-Hermitian skin effect in systems with long-range couplings

In solid-state physics, band structures are typically calculated using either an infinite lattice model (Method I) or by imposing periodic boundary conditions (Method II). While these methods are equivalent for short-range couplings, their consistency for long-range couplings—common in systems like quantum emitters—is less clear. This paper investigates which method provides a physically and mathematically meaningful description of the bulk energy spectrum. The authors evaluate both methods using one-dimensional quantum emitter chains, both in free space and coupled to a waveguide, the latter of which allows for non-reciprocal couplings and the non-Hermitian skin effect.

The authors demonstrate that Method I, which relies on an infinite chain, leads to severe inconsistencies for long-range couplings. Specifically, in free-space quantum emitter chains with perpendicular polarization, Method I results in divergent band structures. In waveguide-coupled systems, Method I fails to produce the complex-valued spectra necessary to explain the non-Hermitian skin effect, forcing researchers to rely on ad-hoc regularization techniques.

In contrast, Method II, which effectively imposes periodic boundary conditions, remains mathematically stable and physically consistent. It avoids the divergencies found in Method I and correctly captures the spectral features of finite-size systems. Furthermore, the analysis of the waveguide system reveals unique phenomena, such as a non-star-shaped generalized Brillouin zone and strongly localized eigenstates that exhibit zero winding number, which are accurately captured by Method II.

This study highlights a critical pitfall in the theoretical modeling of long-range coupled systems. By showing that the widely used infinite-chain approach (Method I) can produce non-physical artifacts and fail to explain topological phenomena like the non-Hermitian skin effect, the authors provide a necessary correction for researchers working in quantum optics, photonics, and condensed matter physics. The results suggest that periodic boundary conditions are the superior, more robust choice for calculating band structures in systems where coupling strengths do not decay rapidly with distance.