Observation of Topological Phase Transition in Complex Momentum Space

It is well known that topology of a periodic crystal can be characterized by calculating a topological invariant using the real Bloch wave vector. In systems governed by non-Hermitian Hamiltonians, however, this conventional knowledge is challenged due to a phenomenon known as the non-Hermitian skin effect, in which wave functions accumulate at the boundaries. To address the failure of Bloch's theorem in non-Hermitian systems, two representative approaches have been utilized: one involving complex spectral topology in the conventional Brillouin zone and the other revolving wave function topology in the generalized Brillouin zone. The complex spectral topology approach provides the topological information of a system by demonstrating the presence of point gap and line gap in the complex plane. On the other hand, the wave function topology provides a method to calculate non-Hermitian topological invariants using the complex Bloch wave vector, which hosts conventional topological boundary states. Recently, a new approach has been introduced - the intrinsic topology of the generalized Brillouin zone within the complex momentum plane. In this work, we present the first experimental demonstration of topological phase transition within the generalized Brillouin zone, occurring in complex momentum space, through a one-dimensional non-Hermitian electric circuit lattice. More specifically, we designed a one-dimensional non-Hermitian electric circuit lattice inspired by the Hatano-Nelson model with generalized boundary conditions. By adjusting the boundary conditions, specifically the onsite potentials and couplings of boundary nodes, we observed the emergence of topological boundary modes in complex momentum space. Furthermore, we identified that this topological phase transition, known as an exceptional transition, is accompanied by the manifestation of an exceptional point. To build upon and generalize our modified Hatano-Nelson model, we introduced next-nearest-neighbor hopping into the system and observed the emergence of multiple topological boundary modes within the complex momentum space. Our experimental findings hold significant promise for advancing the understanding of bulk-boundary correspondence within non-Hermitian systems, by highlighting the novel topological phase transition that occurs in complex momentum space.