Experimental Observation of Non‐Abelian D 4 Permutations in Non‐Hermitian System

Exceptional points (EPs), as an essential concept in non‐Hermitian systems, give rise to a variety of novel physical phenomena, most notably the exchange of eigenstates induced by encircling in parameter space. Here, we experimentally investigate eigenstate permutations in a parameter space containing three second‐order exceptional points (EP2s), realized on a programmable single‐photon platform capable of synthesizing four‐state non‐Hermitian Hamiltonians. By designing distinct parameter space loops that encircle the EP2s in different sequences, we observe eigenvalue and eigenstate permutations corresponding to the non‐Abelian dihedral group. Furthermore, using a branch‐cut representation of the underlying Riemann‐surface topology, we track the associated eigenstate permutations and show that different encircling paths generate distinct sequences of permutation operations. These results highlight the non‐commutative nature of eigenstate braiding in multi‐EP systems and establish an experimental connection between multi‐EP topology and non‐Abelian group structures in non‐Hermitian physics. More broadly, our work demonstrates the capability of programmable single‐photon platforms for investigating complex non‐Hermitian topological structures and their associated permutation dynamics.