Spin Structures and Exact Dualities in Low Dimensions

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.

45 pages. v2: typos corrected, references added, v3: more references added, improved presentation, 2 figures

References in corpus (5)

• Generalized Global Symmetries
• Fermionic SPT phases in higher dimensions and bosonization
• Duality between $(2+1)d$ Quantum Critical Points
• Bosonization in three spatial dimensions and a 2-form gauge theory
• Comments on Defining Entanglement Entropy

Cited by in corpus (12)

• A Web of 2d Dualities: ${\mathbf{Z}}_2$ Gauge Fields and Arf Invariants
• Topological Transition on the Conformal Manifold
• Gauging the Kitaev chain
• Constraints of kinematic bosonization in two and higher dimensions
• Systematic Constructions of Fracton Theories
• Scaling of the Formation Probabilities and Universal Boundary Entropies in the Quantum XY Spin Chain
• Abelian Bosonization, OPEs, and the "String Scale" of Fermion Fields
• The Ultraviolet Structure of Quantum Field Theories. Part 1: Quantum Mechanics
• The Ultraviolet Structure of Quantum Field Theories. Part 2: What is Quantum Field Theory?
• The Ultraviolet Structure of Quantum Field Theories. Part 3: Gauge Theories
• Spin chains, defects, and quantum wires for the quantum-double edge
• Confinement and Flux Attachment