Filling-Sensitive Spectral Complexity from Hilbert-Space Holonomy in Fragmented Non-Hermitian Systems

TL;DR

This paper reveals how Hilbert-space holonomy influences spectral reality in fragmented non-Hermitian systems, linking geometric phases to the emergence of complex spectra and boundary-condition sensitivity.

Contribution

It introduces a geometric framework using Hilbert-space holonomy to understand spectral properties in non-Hermitian many-body systems, highlighting the role of symmetry sectors and gauge fields.

Findings

Complex spectra occur only in highly symmetric sectors.

Single-particle or spin-flip operations can make spectra real.

Trivial holonomy allows a transformation to Hermitian systems.

Abstract

We show that Hilbert-space holonomy provides a geometric organizing principle for spectral reality in fragmented non-Hermitian many-body systems, complementary to conventional symmetry protection. In two minimal fragmented models, complex spectra can arise only within the most symmetric sectors: half filling in the fermion model and zero magnetization in the spin chain. Adding or removing a single particle, or flipping a single spin, renders the spectra entirely real despite unchanged periodic boundary conditions, reminiscent of boundary-condition sensitivity in systems with a non-Hermitian skin effect. We explain this by viewing nonreciprocal hopping amplitudes as a discrete gauge field on the Krylov graph: trivial holonomy permits a diagonal similarity transformation to the Hermitian limit, whereas nontrivial holonomy obstructs it and allows complex spectra. In certain regimes, trivial holonomy admits an emergent-boundary interpretation, and longer-range models exhibit finite real and complex regions governed by the same criterion.