Exact ${\mathbb{Z}}_2$ electromagnetic duality of ${\mathbb{Z}}_2$ toric code is non-Clifford

Abstract

The 2D ${\mathbb{Z}}_2$ toric code admits a global symmetry exchanging electric and magnetic quasiparticles, known as electromagnetic duality. Known realizations include lattice translation symmetry, an exact ${\mathbb{Z}}_4$ symmetry generated by a Clifford circuit, and an exact ${\mathbb{Z}}_2$ symmetry generated by a non-Clifford circuit. We show that a Clifford electromagnetic duality cannot realize an exact internal ${\mathbb{Z}}_2$ symmetry. This is proved rigorously for symmetries with coarse translation invariance by $l$ lattice units for generic odd $l$. Therefore an exact internal ${\mathbb{Z}}_2$ electromagnetic duality must be non-Clifford, whereas generic internal Clifford realization necessarily has ${\mathbb{Z}}_{2^m}$ algebra with $m\ge 2$. Our result suggests an unexpected connection between the algebra of exact electromagnetic duality and Clifford hierarchy of circuits.

7 pages, 2 figures. Added a ref, minor edit