We consider the space of n×n non-Hermitian Hamiltonians (n=2, 3, ...) that are equivalent to a single n×n Jordan block. We focus on adiabatic transport around a closed path (i.e., a loop) within this space, in the limit as the time scale T=1/ɛ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is ɛ−1 times an expansion in powers of ɛ1/n. The exponential of the term of nth order (which is equivalent to the “geometric” or Berry phase modulo 2π) is thus independent of ɛ as ɛ→0; it depends only on the homotopy class of the loop and is an integer power of e2πi/n. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.

• Received 10 August 2020
• Accepted 12 August 2020

DOI:https://doi.org/10.1103/PhysRevA.102.032216