Geometric phase around exceptional points

Abstract

A wave function picks up, in addition to the dynamic phase, the geometric ͑Berry͒ phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for ͑double͒ cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.

Information Geometry of Complex Hamiltonians and Exceptional Points

“…Incidentally, this expression for the metric is in line with the analysis of geometric phases associated with the eigenstates of complex Hamiltonians [50][51][52][53]. Since the perturbative result of Equation (39) is only valid away from degeneracies, in the next section we shall investigate the generic behaviour close to the exceptional point by employing a more refined perturbative technique.…”

Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution

“…For instance, for a fixed δ, we obtain the exceptional point as a circle of radius |δ|, lying in the plane z = 0. It should be noted that in the recent literature, the term "exceptional point" is applied not only to the particular case when the exceptional "point" is indeed a point but to the general case when the non-Hermitian degeneracy is realized as a submanifold in the parameter space [30,31,32,33]. In the following, we will stick to this more general interpretation of the exceptional point.…”

Observation of bulk Fermi arc and polarization half charge from paired exceptional points

“…One such example is the emergence of a new class of degeneracies, commonly referred to as exceptional points (EPs), where two or more resonances of a system coalesce in both eigenvalues and eigenfunctions [26][27][28]. So far, isolated EPs in parameter space [29][30][31][32][33][34][35] and continuous rings of EPs in momentum space [36][37][38] have been studied across different wave systems due to their intriguing properties, such as unconventional transmission/reflection [39][40][41], relations to parity-time symmetry [42][43][44][45][46][47][48], as well as their unique applications in sensing [49,50] and single-mode lasing [51][52][53].Here, we theoretically design and experimentally realize a new configuration of isolated EP pairs in momentum space, which allows us to reveal the unique topological signatures of EPs in the band structure and far-field polarization, and to extend topological band theory into the realm of non-Hermitian systems. Specifically, we demonstrate that a Dirac point (DP) with nontrivial Berry phase can split into a pair of EPs [54][55][56] when radiation loss-a form of non-Hermiticity-is added to a 2D-periodic photonic crystal (PhC) structure.…”