Constructing Non-Hermitian Theories with Tunable Exceptional Points and Controlled State Purification

Research Question and Approach

This paper investigates how to systematically engineer and control exceptional points (EPs)—spectral singularities where both eigenvalues and eigenvectors coalesce—in many-body quantum systems. The authors propose a framework based on momentum-space deformation, applying it to quadratic Bogoliubov-de Gennes (BdG) Hamiltonians. By decomposing non-Hermitian Hamiltonians into Hermitian and anti-Hermitian components, they derive universal criteria for the existence of EPs and demonstrate how these points can be tuned by adjusting deformation strengths in specific momentum sectors.

The researchers identify that EPs can be engineered in any momentum block by tuning the deformation strength, with the critical momentum values determined solely by the intrinsic structure of the Hermitian parent Hamiltonian. A key result is that an EP in a single momentum block induces an exponential proliferation of pairwise eigenvector coalescences across the entire many-body spectrum. Furthermore, the authors establish EPs as a mechanism for state purification, identifying three distinct purification regimes. They uncover a striking odd-even system-size dichotomy: while even-sized systems can achieve complete purification of arbitrary mixed states, odd-sized systems are fundamentally limited by the presence of unpaired momentum modes.

This work provides a unified, systematic protocol for reverse-engineering non-Hermitian quantum matter, including both short-range and long-range, reciprocal and nonreciprocal models. By linking momentum-space engineering to many-body dynamics and state purification, the framework offers a practical design tool for quantum technologies, such as error correction, cooling, and resource distillation, where controlled dissipation and spectral singularities are highly desirable.