$\mathbb{Z}_2$ lattice gauge theories: fermionic gauging, transmutation, and Kramers-Wannier dualities

Abstract: We generalize the gauging of $\mathbb{Z}_2$ symmetries by inserting Majorana fermions, establishing parallel duality correspondences for bosonic and fermionic lattice systems. Using this fermionic gauging, we construct fermionic analogs of $\mathbb{Z}_2$ gauge theories dual to the transverse-field Ising model, interpretable as Majorana stabilizer codes. We demonstrate a unitary equivalence between the $\mathbb{Z}_2$ gauge theory obtained by gauging the fermion parity of a free fermionic system and the conventional $\mathbb{Z}_2$ gauge theory with potentially nonlocal terms on the square lattice with toroidal geometry. This equivalence is implemented by a linear-depth local unitary circuit, connecting the bosonic and fermionic toric codes through a direction-dependent anyonic transmutation. The gauge theory obtained by gauging fermion parity is further shown to be equivalent to a folded Ising chain obtained via the Jordan--Wigner transformation. We clarify the distinction between the recently proposed Kramers--Wannier dualities and those obtained by gauging the $\mathbb{Z}_2$ symmetry along a space-covering path. Our results extend naturally to higher-dimensional $\mathbb{Z}_2$ lattice gauge theories, providing a unified framework for bosonic and fermionic dualities and offering new insights for quantum computation and simulation.