Lattice Fusion Category Symmetry Framework Derived From Algebraic Field Theory

Research establishes a formal connection between fusion categories, symmetries and gapless states in one-dimensional systems. It demonstrates that fusion categories define symmetries acting on quasi local algebras and proves a Lieb Schultz Mattis type theorem, revealing gapless symmetric states for categories lacking a fiber functor. Anomalous Kramers-Wannier duality also necessitates gaplessness.

The pursuit of understanding how symmetry manifests in quantum systems continues to drive theoretical development, particularly in condensed matter physics and quantum information theory. Recent work focuses on rigorously defining and exploring symmetries arising from fusion categories, mathematical structures that describe the behaviour of anyons – exotic particles exhibiting fractional statistics. David E. Evans and Corey Jones, alongside their colleagues, present a novel algebraic framework for realising these symmetries on a discrete lattice, a crucial step towards modelling physical systems and potentially harnessing their properties. Their article, ‘An operator algebraic approach to fusion category symmetry on the lattice’, details a formal interpretation of SymTFT decompositions, utilising algebraic field theory to derive expected categorical structures and demonstrating conditions under which these symmetries can be realised in physical systems such as spin chains.

Higher Symmetries Redefine Understanding of Complex Quantum Systems

The study of symmetries underpins much of modern theoretical physics, extending beyond traditional continuous transformations to encompass more abstract, categorical structures. Recent investigations explore symmetries described not by groups, but by higher categorical algebras, offering a richer framework for understanding quantum systems and revealing hidden order and emergent phenomena within them. This current work builds upon this foundation, focusing on the mathematical formalisation of these symmetries on a discrete lattice system and providing a pathway to connect abstract mathematical concepts with concrete physical models.

Topological quantum field theories (TQFTs) provide a powerful toolkit for characterising these categorical symmetries, importing concepts from topology into the realm of quantum physics and enabling the study of symmetries in arbitrary dimensions. SymTFT, or topological holography, represents a significant development, offering a way to decompose a quantum field theory into topological and physical boundaries coupled by a bulk topological theory and allowing for a more tractable mathematical description of complex symmetries.

The application of SymTFT typically focuses on continuum theories; however, recent efforts have begun to translate these ideas into concrete lattice representations, aiming to develop a formulation that allows for the direct observation of the algebraic structures underlying a SymTFT decomposition within the lattice system itself. Identifying the bulk topological order, the gapped boundary, and the symmetry category of defects directly from the mathematical structure of the infinite volume limit system is a key goal, providing a robust and mathematically sound foundation for future research in this field.

This research proposes a framework based on the quasi-local algebra of a one-plus-one dimensional lattice model, leveraging the theory of DHR-bimodules, an extension of superselection theory optimised for spin systems with finite dimensional local algebras. By axiomatising a physical boundary subalgebra within the quasi-local algebra, the framework aims to reconstruct the expected categorical structures and reveal the underlying symmetries of the system, offering a novel way to understand and characterise complex quantum systems on a discrete lattice. The work builds upon the established concepts of algebraic quantum field theory, specifically focusing on the quasi-local algebra and its Hamiltonian operator, representing the observables of the system in the infinite volume limit and describing the system’s dynamics.

Symmetry within Fusion Categories Defines Quasi-Local Observable Algebras

This research establishes a formal framework for understanding symmetry within fusion categories on a one-plus-one dimensional lattice, effectively bridging the gap between abstract mathematical structures and physical observables and centering on axiomatising the physical boundary subalgebra of quasi-local observables. Quasi-local algebras represent physical quantities defined within a limited spatial region, and by focusing on their boundaries, researchers can rigorously define how symmetries act on the system, leveraging algebraic field theory, a mathematical language designed to describe quantum fields and their interactions. The core innovation lies in demonstrating that a fusion category, acting on bimodules – structures describing how different parts of a system interact – can be canonically associated with a quasi-local algebra, effectively encoding the symmetries within the algebra’s properties.

The researchers establish a direct link between the mathematical properties of fusion categories and the physical behaviour of systems, demonstrating that the fusion ring – a set of rules governing how different symmetry transformations combine – acts as locality-preserving channels on the quasi-local algebra and maintaining the physical realism of the model. Crucially, the invariant operators, those remaining unchanged under symmetry transformations, are recovered from this framework, providing a concrete connection between the abstract mathematical description and the observable physical quantities. The team then investigates the conditions under which these fusion categories can be realised as symmetries of a tensor product spin chain, a simplified model of a quantum system, demonstrating that integer dimensions of the category’s objects are essential for this realisation and that the existence of a fiber functor – a mapping that connects the category to a simpler, more manageable structure – is required for an on-site action, meaning the symmetry acts locally on each element of the spin chain.

Further extending this framework, the researchers provide a formal definition of a topological symmetric state, a state exhibiting symmetry under topological transformations, and prove a Lieb-Schultz-Mattis type theorem, a result predicting the existence of gapless excitations – low-energy states – in certain quantum systems. This theorem, applied to their framework, demonstrates that any fusion category lacking a fiber functor necessarily leads to gapless, pure symmetric states on an anyon chain, a one-dimensional system exhibiting exotic quantum statistics, highlighting a deep connection between the mathematical properties of the symmetry category and the physical behaviour of the system.

Furthermore, the study applies this framework to analyse states covariant under anomalous Kramers-Wannier type duality, a symmetry relating different physical configurations, demonstrating that any state exhibiting this symmetry must be gapless, reinforcing the connection between symmetry, topology, and the emergence of gapless excitations.

Fusion Category Symmetry Emerges from Quasi-Local Algebras via Bimodule Reconstruction

This research establishes a formal framework for understanding fusion category symmetry on a (1+1)-dimensional lattice, interpreting SymTFT decompositions through the axioms of algebraic field theory and centering on the physical boundary subalgebra of quasi-local observables. It demonstrates how a canonical fusion category emerges from these algebras via bimodules, with the fusion ring acting via locality-preserving channels, effectively reconstructing the original quasi-local algebra from its invariant operators, providing a powerful tool for understanding how symmetries manifest in physical systems and how they constrain observable phenomena. The work rigorously connects abstract mathematical structures – fusion categories – with physical systems exhibiting symmetry.

The authors prove a crucial condition for realising a fusion category as symmetries of a tensor product spin chain: all objects within the category must possess integer dimensions, while the existence of a fiber functor enables on-site action, establishing precise mathematical criteria for translating categorical symmetries into physically observable phenomena within spin chain models. The research introduces a formal definition of a topological symmetric state and subsequently proves a Lieb-Schultz-Mattis type theorem, demonstrating that any fusion category lacking a fiber functor necessarily gives rise to gapless, pure symmetric states on an anyon chain, highlighting a fundamental constraint on the behaviour of anyon chains.

Mathematical Rigour Increasingly Defines Understanding of Topological Matter’s Complex Behaviour

This comprehensive body of work demonstrates a robust and interconnected research landscape at the intersection of condensed matter physics and mathematics, revealing a clear trend towards increasingly rigorous mathematical formalisation of concepts central to topological phases of matter, notably through the extensive use of category theory, specifically fusion and braided tensor categories. These mathematical structures provide a powerful language for describing the anyonic excitations that characterise topological order and underpin potential applications in quantum computation, actively exploring the implications of non-invertible symmetries, an emerging area that challenges conventional understandings of symmetry in quantum systems.

This investigation extends beyond purely theoretical considerations, with a significant body of work employing computational and numerical methods to complement analytical approaches, suggesting a concerted effort to bridge the gap between abstract mathematical frameworks and concrete physical realisations.