Non-Hermitian adiabatic transport in the space of exceptional points

An nxn non-Hermitian Hamiltonian matrix H can describe a dissipative system, such as n coupled weakly dissipative classical harmonic oscillators. Under full parametric control over H, the parameter space contains a connected -- but not simply-connected -- subspace of nth order exceptional points, at each of which H is equivalent to an nxn Jordan block. We show that smooth variations of parameters during time T, along a loop within that space, can single out one state that is least dissipative and evolves adiabatically. Its complex adiabatic phase is T times a Puiseux series in powers of T ^-1/n; the coefficient at order T⁰ is the Berry phase, which is a multiple of 2π/n (modulo 2π) and only depends on the homotopy class of the loop within the space of nth order exceptional points.