As a departure from axiomatization of quantum mechanics, Non-Hermitian systems have been shown to possess new physics. An example is the capacity to have simultaneous degeneracy of eigenvalues and eigenvectors, qualified as an exceptional point (EP) singularity. The realization of EPs has been limited to parameter- and momentum-space through fine-tuning the gain and loss of driven systems.^[1] The focus of my research at the University of Chicago was to study the realization of EPs in real-space and explore how a non-trivial phase accumulates as a particle moves around an EP.

With known EP-forming models in parameter-space, I ported the analytical model to a real-space description, where the gain and loss vary across a 2D plane. From a representation theory point of view, the thrust of this project was to understand the physical observables present in the mapping from SO(2) to Spin(1), where the former represents real space and the latter the eigenvalue space.

Fig. 1: Riemann surface of complex square root function. Evident is the double cover of SO(2) by Spin(1).^[2]

Open questions to further explore are whether one can design higher-degree EPs that stack atop one another, as opposed to interleaving sheets that have been shown to exist.^[3] An additional line of inquiry is whether one can generate EPs by swapping PT-symmetry for CP-symmetry breaking and move the EP to a Floquet regime.

Fig. 2: Merging of exceptional points of altrenating topological order.^[4]

[1] Dynamically Encircling Exceptional Points: Exact Evolution and Polarization State Conversion
[2] Winding around non-Hermitian singularities
[3] Non-Hermitian topology and exceptional-point geometries
[4] Formation of exceptional points in pseudo-Hermitian systems