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

TL;DR

The paper develops a unified framework for bosonic and fermionic $ obreak{\mathbb{Z}_2}$ gauging on lattices, inserting Majorana fermions to define fermionic gauging and constructing dualities to Ising-like gauge theories. It demonstrates explicit unitary circuits that map between bosonic and fermionic toric codes, revealing direction-dependent anyon transmutation and connecting these mappings to folded Majorana chains and Jordan–Wigner transformations. A key insight is that Kramers–Wannier dualities in Majorana-dual theories are related to, yet distinct from, KW dualities obtained by gauging a space-covering symmetry, with higher-dimensional generalizations to 3D provided. The results deliver a coherent picture unifying bosonic and fermionic dualities, with potential implications for quantum computation, simulation of fermionic models, and exploration of symmetry-topological field theory in lattice systems. The framework naturally extends to higher dimensions and offers new perspectives on state preparation, error correction, and quantum simulators for fermionic gauge theories.

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.

Figures (10)

• Figure 1: Gauging and dualities of bosonic and fermionic systems. Green (red) arrows indicate bosonic (fermionic) gauging. Gray (dotted) arrows indicate fermionic gauging via a nonlocal disentangling unitary to and from a fermionic gauge theory with nonlocal terms.
• Figure 2: (a) Operators in the bosonic and fermionic Ising-dual gauge theories on the square lattice. In the bosonic (fermionic) theory, blue and red lines represent $X, Z$ [$(-1)^{F}, \gamma$], respectively. The first operator corresponds to the Gauss law. (b) Operators in the bosonic Majorana-dual theories. Green lines represent $Y$. The overall minus sign in the Gauss-law operator is omitted. (c) Fermionic gauging of the TFIM. Two Majorana fermions (red lines) are assigned to an edge, and four Majorana fermions associated with a plaquette form a plaquette-like operator.
• Figure 3: Quantum circuits used to map the gauge theories to ancilla-decorated Ising models. The rightmost dotted edges are identified with the leftmost edges. (a, d) The sequential application of the CNOT cascades is indicated by the arrows. (e) The CZ gate layer is required for the Majorana-dual theory. (f) The SWAP gate layer is applied last to the Ising-dual gauge theory across the boundary, while additional layers for the Majorana-dual gauge theory are shown in Fig. \ref{['sec_additional']}SM.
• Figure 4: Exchange statistics and direction-dependent transmutation of flux excitations. (a) Two pairs of fluxes in the bosonic toric code are created by horizontal string operators. One string (cyan) is deformed to cross the other (red). At the intersection, they split along the dashed diagonal into two new strings, which are then deformed back to the horizontal direction. (b) Similar process in the fermionic toric code, except that deformation along the vertical direction is accompanied by Wilson loops extending to the boundary. The anticommutation between $X$ and $Z$ on the shaded edge introduces a minus sign, accounting for the fermionic statistics of the flux excitations.
• Figure 5: Folded path used in the JW transformation on a torus. The periodic boundary (dotted line) is shifted to the center for clarity.