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Observation of exceptional topology and nonlocal skin effect in Klein bottle electric circuits

Symmetry and its representation play a crucial role in topological phases, including both Hermitian and non-Hermitian paradigms. In the presence of synthetic gauge field, spatial symmetries should be projectively represented, which can modify the Brillouin manifold. However, this is often overlooked in non-Hermitian systems. Here, we present that momentum-space non-symmorphic reflection symmetry, a typical projective symmetry, induce exceptional topology and the nonlocal skin effect in a two-dimensional non-Hermitian electric circuit. We observe the total topological charges 2, rather than 0, for all exceptional points in a Brillouin Klein bottle manifold, and the phase transition when an exceptional point crosses the antiparallel boundary and flips its topological charge. We further observe a novel skin effect that the skin modes at one side are nonlocally connected to those on the opposite side separated by half of the reciprocal lattice. Our results unveil the unique non-Hermitian phenomena enabled by the projective symmetry, and open avenues for exploring the non-Hermitian topology beyond Brillouin torus manifold.