Z2 lattice gauge theories: fermionic gauging, transmutation, and Kramers-Wannier dualities
Abstract
We generalize the gauging of Z2 symmetries by inserting Majorana fermions, establishing parallel duality correspondences for bosonic and fermionic lattice systems. Using this fermionic gauging, we construct fermionic analogs of Z2 gauge theories dual to the transverse-field Ising model, interpretable as Majorana stabilizer codes. We demonstrate a unitary equivalence between the Z2 gauge theory obtained by gauging the fermion parity of a free fermionic system and the conventional Z2 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 Z2 symmetry along a space-covering path. Our results extend naturally to higher-dimensional Z2 lattice gauge theories, providing a unified framework for bosonic and fermionic dualities and offering new insights for quantum computation and simulation.
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