Geometric phase around exceptional points

A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. For general non-Hermitian and complex symmetric Hamiltonians of arbitrary dimension, we study the geometric phase for (double) cycles around exceptional points, at which two energy levels and corresponding eigenvectors coalesce. We show that the geometric phase is exactly $\pi$ 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 $\pi$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence in general multidimensional case, and describe it in terms of interaction of different energy levels