Additivity, Haag duality, and non-invertible symmetries (2503.20863v1)

Abstract: The algebraic approach to quantum field theory focuses on the properties of local algebras, whereas the study of (possibly non-invertible) global symmetries emphasizes global aspects of the theory and spacetime. We study connections between these two perspectives by examining how either of two core algebraic properties -- "additivity" or "Haag duality" -- is violated in a 1+1D CFT or lattice model restricted to the symmetric sector of a general global symmetry. For the Verlinde symmetry of a bosonic diagonal RCFT, we find that additivity is violated whenever the symmetry algebra contains an invertible element, while Haag duality is violated whenever it contains a non-invertible element. We find similar phenomena for the Kramers-Wannier and Rep(D$_8$) non-invertible symmetries on spin chains.

Analyzing Violations of Additivity and Haag Duality Due to Non-Invertible Symmetries in Quantum Field Theory

This paper presents a comprehensive paper of the interplay between algebraic properties in quantum field theory (QFT) and non-invertible symmetries. Specifically, it examines how additivity and Haag duality—two fundamental algebraic properties—are affected in quantum systems with global symmetries that include non-invertible elements. The research focuses on 1+1D conformal field theories (CFTs) and lattice models, revealing intricate patterns in the symmetric sectors of these systems.

The framework of the paper is rooted in the algebraic approach to QFT, which assigns algebras of operators to subsystems characterized by specific regions. Additivity implies that the algebra associated with a union of regions is generated by the algebras of the individual regions. Haag duality posits that for a given region, the operators in the complement of the region form the commutant of the operators in the region itself. The violations of these properties signal intriguing underlying structures attributable to symmetries.

Key Findings

1. Invertible Symmetries and Additivity Violation: The paper demonstrates that a symmetric sector under an invertible global symmetry can violate additivity. For instance, in the lattice Ising model under $\mathbb{Z}_2$ symmetry, a pair of $\mathbb{Z}_2$-odd operators can exist outside the union of individual local algebras, highlighting an instance of additivity violation.
2. Non-Invertible Symmetries and Haag Duality Violation: The CFTs and lattice models with non-invertible symmetries exhibit potential violations of Haag duality. The Ising CFTs showcase non-invertible Kramers-Wannier symmetries through disorder operators, revealing that Haag duality is violated when these operators are involved.
3. Diagonal Rational Conformal Field Theories: This exploration extends to diagonal RCFTs, where the Verlinde symmetry—consisting of both invertible and non-invertible elements—demonstrates a dual violation: additivity is breached if the symmetry includes an invertible element, while Haag duality experiences violation if it contains non-invertible elements.

Theoretical and Practical Implications

The findings underscore the necessity to further understand symmetric subsectors in quantum field and lattice theories. The peculiar violations of additivity and Haag duality suggest that these sectors might not fully encapsulate the rich global properties and constraints dictated by symmetries. This is illustrative of how non-invertible symmetries, beyond conventional group symmetries, yield rich algebraic structures.

Theoretical advancement lies in exploring generalized symmetry categories, extending beyond conventional gauge theories and spacetime symmetries. Furthermore, these insights offer critical implications for designing lattice models and studying phase transitions in condensed matter physics, where non-invertible symmetries could potentially describe topological orders and duality transformations.

Speculations on AI Implications

The paper subtly hints at broader implications for algorithmic development in artificial intelligence, where these algebraic properties might inform symmetry-based optimization techniques or inspire new approaches in quantum computing frameworks.

Future work can delve into higher-form symmetries and explore additional dimensions to further extend these observed patterns. As AI models evolve, embracing quantum insights may lead to breakthroughs in solving complex problems exhibiting non-trivial symmetries, akin to those discussed in quantum field theories and lattice systems.

Conclusively, this research presents a profound exposition of additivity and Haag duality violations within symmetric sectors of quantum systems. It forms a bridge between algebraic quantum field theory and the emergent understanding of symmetries in physics, offering pathways to novel developments and applications both theoretically and in practical technological domains.