A No-Go Theorem for Order-Two Clifford Electric-Magnetic Duality

Abstract: Electromagnetic duality in the toric code has order two at the level of anyon types and acts as a Hadamard-type logical transformation on the encoded quantum information. Here, we ask whether it can likewise be realized microscopically as an order-two Clifford operation. For the $\mathbb{Z}_2$ toric code, we prove that any locality-preserving Clifford realization of electric-magnetic exchange cannot have order two. Our proof is fully general and requires neither translation symmetry with restricted families of system sizes nor a fixed pairing between vertex and plaquette stabilizers. Geometrically, the proof locally emulates the unavoidable crossing between electric and magnetic strings, similar to a cross-cap in a non-orientable manifold. Extending the problem to the $\mathbb{Z}_N$ toric code, we find that an order-two Clifford realization exists for odd $N$, whereas it is impossible for even $N$. Our results show that the group law of an emergent electromagnetic duality need not lift faithfully to its microscopic Clifford realization.