  Three-dimensional (3D) two-band Hopf insulators are a paradigmatic example of
topological phases beyond the topological classifications based on powerful
methods like $K$-theory and symmetry indicators.Since this class of topological
insulating phases was theoretically proposed in 2008, they have attracted
significant interest owing to their conceptual novelty, connection to knot
theory, and many fascinating physical properties. However, because their
realization requires special forms of long-range spin-orbit coupling (SOC),
they have not been achieved in any 3D system yet. Here we report the first
experimental realization of the long-sought-after Hopf insulator in a 3D
circuit system. To implement the Hopf insulator, we construct basic pseudo-spin
modules and connection modules that can realize $2\times2$-matrix elements and
then design the circuit network according to a tight-binding Hopf insulator
Hamiltonian constructed by the Hopf map. By simulating the band structure of
the designed circuit network and calculating the Hopf invariant, we find that
the circuit realizes a Hopf insulator with Hopf invariant equaling $4$.
Experimentally, we measure the band structure of a printed circuit board and
find the observed properties of the bulk bands and topological surface states
(TSS) are in good agreement with the theoretical predictions, verifying the
bulk-boundary correspondence of the Hopf insulator. Our scheme brings the
experimental study of Hopf insulators to reality and opens the door to the
implementation of more unexplored topological phases beyond the known
topological classifications.
