Knot Topology of Exceptional Point and NonHermitian NoGo Theorem
Abstract
Exceptional points (EPs) are peculiar band singularities and play a vital role in a rich array of unusual optical phenomena and nonHermitian band theory. In this paper, we provide a topological classification of isolated EPs based on homotopy theory. In particular, the classification indicates that an $n$th order EP in two dimensions is fully characterized by the braid group B$_n$, with its eigenenergies tied up into a geometric knot along a closed path enclosing the EP. The quantized discriminant invariant of the EP is the writhe of the knot. The knot crossing number gives the number of bulk Fermi arcs emanating from each EP. Furthermore, we put forward a nonHermitian nogo theorem, which governs the possible configurations of EPs and their splitting rules on a twodimensional lattice and goes beyond the previous fermion doubling theorem. We present a simple algorithm generating the nonHermitian Hamiltonian with a prescribed knot. Our framework constitutes a systematic topological classification of the EPs and paves the way towards exploring the intriguing phenomena related to the enigmatic nonHermitian band degeneracy.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.11346
 Bibcode:
 2021arXiv211111346H
 Keywords:

 Condensed Matter  Mesoscale and Nanoscale Physics;
 Condensed Matter  Quantum Gases;
 Mathematical Physics;
 Physics  Optics;
 Quantum Physics
 EPrint:
 6+3 pages, 2+1 figures