Anyon Fusion Rules Classification Achieves General Algebraic Formula for Quantum Phase Transitions

The behaviour of anyons, exotic particles appearing in topologically ordered systems, presents a fundamental challenge in condensed matter physics, and Yoshiki Fukusumi from National Center for Theoretical Sciences, National Taiwan University, along with colleagues, now offer a powerful new method for classifying these elusive entities. The researchers develop a general algebraic formula that describes how anyons transform across boundaries, known as domain walls, and relate to quantum phase transitions, building upon the established Verlinde formula for commutative fusion rings. This achievement provides a framework for understanding the connections between different topological phases of matter, and importantly, allows scientists to predict how symmetries emerge when systems undergo changes, offering insights into both fundamental physics and potential applications in quantum computing. The team’s approach extends to complex models created through techniques like orbifolding, ultimately providing a comprehensive system for classifying symmetry-enriched topological orders and criticalities.

The research investigates the ring homomorphism between C-linear commutative fusion rings, also known as symmetry topological field theories, or SymTFTs. A core principle of this work is the assumption that the Verlinde formula holds true for commutative fusion rings, and its combination with specific data from these settings allows for the classification of anyons, particles with exotic quantum properties, suitable for developing categorical formulations. Furthermore, the formalism provides descriptions of massless renormalization group flows between conformal field theories, or CFTs, and also represents a series of measurement-induced quantum phase transitions, all within the language of SymTFTs. This is achieved through the well-established correspondence linking CFTs and topological quantum field theories, or TQFTs, and extends understanding of relationships between these theoretical frameworks.

Ising Model, Chiral Fermions, and Topological Phases

This appendix details connections between conformal field theories (CFTs), topological phases, and symmetry-protected topological (SPT) phases, specifically focusing on the Ising model and its extensions. The work explains how to construct a chiral Majorana fermion model, a type of topological phase, from a Z2-extended Ising model, detailing the fusion rules and homomorphisms involved, and discussing the implications for symmetry-protected topological order. Key concepts include CFTs, which describe critical phenomena in 2D systems, and the Ising model, a fundamental model in statistical mechanics exhibiting a phase transition. The appendix describes how to combine the Ising CFT with a Z2 symmetry and then apply a procedure called bulk semionization to create a new topological phase, the chiral Majorana fermion model.

The resulting operators satisfy the fusion rules of a double semion model, a topological phase with two types of anyons. A homomorphism, a structure-preserving map, is established between the fusion ring of two copies of the Ising model and the fusion ring of the chiral Majorana fermion model, revealing how operators in the two Ising models can be mapped to operators in the chiral Majorana fermion model. To complete the mapping, the author introduces an additional object, leading to an emergent object related to the difference between bosonic and fermionic descriptions. This highlights that the properties of topological phases can differ from the properties of the underlying microscopic models. The research provides a concrete way to construct a chiral Majorana fermion model from a simpler Ising model, offering insights into the emergence of topological order and the role of symmetry protection, where breaking the symmetry can destroy the Majorana fermions.

Anyon Transformations and Topological State Transitions

Scientists have established a formula to describe how anyons transform during transitions between different topological states of matter. This work centers on symmetry topological field theories, and the team’s approach utilizes ring homomorphisms to precisely define these transformations. The core of their method relies on the Verlinde formula, a well-established principle governing the behavior of these systems, combined with specific data characterizing the initial and final states, allowing for the classification of anyons. Experiments revealed that the method accurately predicts the number of possible transitions between conformal field theories, demonstrating a correspondence between massless renormalization group flows and measurement-induced changes in these systems.

Specifically, when examining a flow from one minimal model to another, the team’s calculations matched existing results, correctly predicting a specific number of homomorphisms. The research introduces a powerful technique for representing anyons using idempotents, algebraic elements that simplify the process of determining the ring homomorphisms governing their behavior. This idempotent representation directly yields the necessary algebraic data, streamlining the analysis of domain wall transformations. The method can be extended using mathematical techniques like orbifolding and ring isomorphism, enabling classifications of more complex symmetry-enriched topological orders. The team developed a formula to calculate the coefficients defining these homomorphisms, analogous in form to the Verlinde formula itself, allowing researchers to identify the ideal structure within a massless renormalization group and relate it to the module structure of a corresponding massive system, drawing parallels to Nambu-Goldstone bosons.

Anyon Classification via Boundary Transformation Formula

This research establishes a classification of anyons, fundamental particles appearing in certain quantum systems, based on the properties of their fusion rings. Scientists developed a formula to describe how these anyons transform when encountering a boundary, or domain wall, within a topologically ordered system. This formula relies on the Verlinde formula and allows for the categorization of anyons in a way that supports the development of more comprehensive mathematical descriptions of these systems. The work extends beyond simply classifying anyons, demonstrating connections to broader concepts in theoretical physics.

The team showed how their approach applies to fractional supersymmetric and nonlocal models, and provides insights into generalized symmetry through analogies with Nambu-Goldstone bosons, which emerge when symmetries are broken. By combining their formula with mathematical techniques like orbifolding and extension, researchers can classify more complex, symmetry-enriched topological orders and criticalities. The authors acknowledge that while their method provides a systematic way to describe domain walls, the analytical properties of these walls remain largely unexplored, even in simpler models. Future research will focus on understanding these analytical properties and exploring related phenomena such as conformal interfaces and symmetry-breaking branes, alongside applications to higher-dimensional models.