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arXiv:2109.05992 · Progress of Theoretical and Experimental Physics

Non-invertible topological defects in 4-dimensional $\mathbb {Z}_2$ pure lattice gauge theory

Masataka Koide, Yuta Nagoya, Satoshi Yamaguchi B

Published Nov 4, 2021• 112 citations
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Abstract

Abstract We explore topological defects in the 4D pure $\mathbb {Z}_2$ lattice gauge theory. This theory has 1-form $\mathbb {Z}_{2}$ center symmetry as well as Kramers–Wannier–Wegner (KWW) duality. We construct the KWW duality topological defects in a similar way to those constructed by Aasen et al. [J. Phys. A 49, 354001 (2016)] for the 2D Ising model. These duality defects turn out to be non-invertible. We also construct 1-form $\mathbb {Z}_{2}$ symmetry defects as well as the junctions between the KWW duality defects and 1-form $\mathbb {Z}_{2}$ center symmetry defects. The crossing relations between these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.

DOI
10.1093/ptep/ptab145

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Non-invertible topological defects in 4-dimensional Z2\mathbb {Z}_2Z2​ pure lattice gauge theory
Abstract We explore topological defects in the 4D pure Z2\mathbb {Z}_2Z2​ lattice gauge theory. This theory has 1-form Z2\mathbb {Z}_{2}Z2​ center symmetry as well as Kramers–Wannier–Wegner (KWW) duality. We construct the KWW duality topological defects in a similar way to those constructed by Aasen et al. [J. Phys. A 49, 354001 (2016)] for the 2D Ising model. These duality defects turn out to be non-invertible. We also construct 1-form Z2\mathbb {Z}_{2}Z2​ symmetry defects as well as the junctions between the KWW duality defects and 1-form Z2\mathbb {Z}_{2}Z2​ center symmetry defects. The crossing relations between these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.
Defects associated with non-invertible symmetries have attracted significant attention in recent years. Among them, Kramers-Wannier (KW) duality defects have been investigated in both classical statistical systems and quantum Hamiltonian models. Aasen et al. analyzed duality defects in the 2D Ising model and in statistical models built from fusion categories, while Koide et al. later constructed a duality defect in 4D lattice gauge theory. In this work, we extend these developments by providing a systematic construction of KW duality defects/KW defects for a broad class of models formulated within the chain complex framework. Our construction employs the strange correlator, an overlap between a topologically ordered state and a product state, to realize these KW defects.
We study quantum field theories with boundaries by utilizing noninvertible symmetries. We consider three kinds of boundary conditions of the four dimensional Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory at the critical point as examples. The weights of the elements on the boundary are determined so that these boundary conditions are related by the Kramers-Wannier-Wegner (KWW) duality. In other words, it is required that the KWW duality defects ending on the boundary are topological. Moreover, we obtain the ratios of the hemisphere partition functions with these boundary conditions; this result constrains the boundary renormalization group flows.
Wegner duality is essential for Z2 lattice gauge theory, yet the duality on non-trivial topologies has remained implicit. We extend Wegner duality to arbitrary topology and dimension, obtaining a new class of Ising models, in which topology is encoded in non-local domain-wall patterns. Without the overhead of gauge constraints, simulating this model on an L*L torus requires only L*L qubits with two-body couplings, halving the conventional four-body coupled 2L*L qubits, enabling full experimental realization of Z2 lattice gauge theory on near-term devices.
We explore non-invertible symmetries in two-dimensional lattice models with subsystem \mathbb Z_2 ℤ 2 symmetry. We introduce a subsystem \mathbb Z_2 ℤ 2 -gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.
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.