Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1809.07757

Spin Structures and Exact Dualities in Low Dimensions

Djordje Radicevic

Published Sep 20, 2018• 42 citations
arXivHTMLPDF

Abstract

This paper derives a large web of exact lattice dualities in one and two spatial dimensions. Some of the dualities are well-known, while others, such as two-dimensional boson-parafermion dualities, are new. The procedure is systematic, independent of specific Hamiltonians, and generalizes to higher dimensions. One important result is a demonstration that spin structures in arbitrary lattice fermion theories can always be simply defined as topological gauge fields whose gauge group is the fermion number parity. This definition agrees with other expected properties of spin structures, and it motivates the introduction of "paraspin structures" that serve the same role in parafermion theories.

DOI
10.48550/arxiv.1809.07757

Related papers

Exact bosonization in arbitrary dimensions

Yu-An Chen

We extend the previous results of exact bosonization, mapping from fermionic operators to Pauli matrices, in 2D and 3D to arbitrary dimensions.This bosonization map gives a duality between any fermionic system in arbitrary n spatial dimensions and a class of (n -1)-form Z 2 gauge theories in n dimensions with a modified Gauss's law.This map preserves locality and has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the spatial manifold.A formula for Stiefel-Whitney homology classes on lattices is derived.In the Euclidean path integral, this exact bosonization map is equivalent to introducing a topological Steenrod square term to the space-time action.

arXiv:1911.00017•Physical Review Research•Sep 30, 2020•55 citations

Exact Bosonization in All Dimensions: the Duality Between Fermionic and Bosonic Phases of Matter

Chen, Yu-An

We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.

•Jun 9, 2020•0 citations

Topological dualities in the Ising model

Daniel S. Freed, Constantin Teleman

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

arXiv:1806.00008•Geometry & Topology•Dec 12, 2022•75 citations

Bosonization and Kramers-Wannier dualities in general dimensions

Lei Su, Ivar Martin

It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan–Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers–Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-) dualities in spin systems arising from fermionic minimal translations. The KW (self-) dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.

arXiv:2508.20167•SciPost Physics•Jun 24, 2026•1 citation

Constraints of kinematic bosonization in two and higher dimensions

Arkadiusz Bochniak, Błażej Ruba, J. Wosiek +1 more

Despite being less known, local bosonizations of fermionic systems exist in spatial dimensions higher than 1. Interestingly, the dual bosonic systems are subject to local constraints, as in theories with gauge freedom. These constraints effectively implement long distance exchange interactions. In this work, we study in detail one such system, proposed a long time ago. Properties of the constraints are elaborated for two-dimensional, rectangular lattices of arbitrary sizes. For several small systems, the constraints are solved analytically. It is checked that spectra of reduced spin Hamiltonians agree with the original fermionic ones. The equivalence is extended to fermions in the presence of background Wegner ${\mathbb{Z}}_{2}$ fields coupling to fermionic parity. This is illustrated by an explicit calculation for a particular configuration of Wegner's variables. Finally, a possible connection with the recently proposed web of dualities is discussed.

arXiv:2004.00988•Physical review. D/Physical review. D.•Dec 4, 2020•13 citations

A web of 2d dualities: ${\bf Z}_2$ gauge fields and Arf invariants

Andreas Karch, David Tong, Carl Turner

We describe a web of well-known dualities connecting quantum field theories in d=1+1 dimensions. The web is constructed by gauging Z_2 Z 2 global symmetries and includes a number of perennial favourites such as the Jordan-Wigner transformation, Kramers-Wannier duality, bosonization of a Dirac fermion, and T-duality. There are also less-loved examples, such as non-modular invariant c=1 CFTs that depend on a background spin structure.

arXiv:1902.05550•SciPost Physics•Jul 10, 2019•151 citations
This paper derives a large web of exact lattice dualities in one and two spatial dimensions. Some of the dualities are well-known, while others, such as two-dimensional boson-parafermion dualities, are new. The procedure is systematic, independent of specific Hamiltonians, and generalizes to higher dimensions. One important result is a demonstration that spin structures in arbitrary lattice fermion theories can always be simply defined as topological gauge fields whose gauge group is the fermion number parity. This definition agrees with other expected properties of spin structures, and it motivates the introduction of "paraspin structures" that serve the same role in parafermion theories.
We extend the previous results of exact bosonization, mapping from fermionic operators to Pauli matrices, in 2D and 3D to arbitrary dimensions.This bosonization map gives a duality between any fermionic system in arbitrary n spatial dimensions and a class of (n -1)-form Z 2 gauge theories in n dimensions with a modified Gauss's law.This map preserves locality and has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the spatial manifold.A formula for Stiefel-Whitney homology classes on lattices is derived.In the Euclidean path integral, this exact bosonization map is equivalent to introducing a topological Steenrod square term to the space-time action.
We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.
We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in 222 dimensions, with electromagnetic duality for finite gauge theories in 333 dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.
It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan–Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers–Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-) dualities in spin systems arising from fermionic minimal translations. The KW (self-) dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.
Despite being less known, local bosonizations of fermionic systems exist in spatial dimensions higher than 1. Interestingly, the dual bosonic systems are subject to local constraints, as in theories with gauge freedom. These constraints effectively implement long distance exchange interactions. In this work, we study in detail one such system, proposed a long time ago. Properties of the constraints are elaborated for two-dimensional, rectangular lattices of arbitrary sizes. For several small systems, the constraints are solved analytically. It is checked that spectra of reduced spin Hamiltonians agree with the original fermionic ones. The equivalence is extended to fermions in the presence of background Wegner Z2{\mathbb{Z}}_{2}Z2​ fields coupling to fermionic parity. This is illustrated by an explicit calculation for a particular configuration of Wegner's variables. Finally, a possible connection with the recently proposed web of dualities is discussed.
A web of 2d dualities: Z2{\bf Z}_2Z2​ gauge fields and Arf invariants