Generalization of non-Hermitian spectral topology to hyperbolic lattices

Hyperbolic lattices, a compelling platform for exploring matter in non-Euclidean space, challenge conventional band theory by invalidating Bloch’s theorem, necessitating the reexamination of fundamental concepts such as determining spectra in the thermodynamic limit for non-Hermitian systems. Here, we generalize non-Hermitian spectral topology to hyperbolic lattices by developing a reciprocal-space approach to determining such spectra under open boundary conditions, including spectral ranges and gaps. This method introduces supercells that encompass states allowed by non-Abelian translations and performs analytic continuation to leverage point-gap topology. Applying this method to a nonreciprocal model predicts spectral ranges under open boundary conditions that differ from those under periodic ones, revealing higher-dimensional skin effects supported by eigenstate and Green’s function analyses. We further employ a reciprocal semimetal model, uncovering states enabled by non-Abelian translations and topological phase transitions unique to non-Hermitian hyperbolic lattices. Our approach, robust and broadly applicable, offers a valuable framework for investigating spectral topology, non-Hermitian phases, and emergent phenomena in hyperbolic lattices. Hyperbolic lattices in non-Euclidean space challenge conventional band theory due to the breakdown of Bloch’s theorem. The authors develop a reciprocal-space supercell method to determine non-Hermitian spectra under periodic and open boundaries, revealing higher-dimensional skin effects and topological transitions enabled by non-Abelian translations.