$\mathbb{Z}_2$ lattice gauge theories: fermionic gauging, transmutation, and Kramers-Wannier dualities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912673510195200 |
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| author | Su, Lei |
| author_facet | Su, Lei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20893 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathbb{Z}_2$ lattice gauge theories: fermionic gauging, transmutation, and Kramers-Wannier dualities Su, Lei Strongly Correlated Electrons Statistical Mechanics High Energy Physics - Lattice Quantum Physics 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. |
| title | $\mathbb{Z}_2$ lattice gauge theories: fermionic gauging, transmutation, and Kramers-Wannier dualities |
| topic | Strongly Correlated Electrons Statistical Mechanics High Energy Physics - Lattice Quantum Physics |
| url | https://arxiv.org/abs/2510.20893 |