New Algorithm Generates Conformal Field Theories from Topological Orders and Anyons.

Researchers developed an algorithm to systematically generate two-dimensional lattice models exhibiting conformal symmetry at critical points. This ‘CFT factory’ utilises topological orders and non-commuting anyon condensates, revealing previously unknown critical points preserving Haagerup symmetries and establishing a route to classifying conformal field theories.

The search for new conformal field theories (CFTs) – mathematical frameworks describing systems at critical points exhibiting scale invariance – remains a central challenge in theoretical physics. These theories underpin diverse phenomena, from condensed matter physics to string theory. Researchers are now detailing a systematic method for generating two-dimensional lattice models that flow to these critical points, effectively creating a ‘factory’ for CFTs. Ling-Yan Hung (Yau Mathematical Sciences Center, Tsinghua University), Kaixin Ji, Yidun Wan and Yu Zhao (State Key Laboratory of Surface Physics, Fudan University) alongside Ce Shen (Beijing Institute of Mathematical Sciences and Applications) present their findings in a paper entitled ‘A 2D-CFT Factory: Critical Lattice Models from Competing Anyon Condensation Processes in SymTO/SymTFT’. Their approach leverages the properties of topological orders and introduces controlled ‘condensation’ of anyons – quasiparticles exhibiting exotic exchange statistics – to engineer critical behaviour and uncover previously unknown critical points, including those preserving Haagerup symmetries.

Unveiling Novel Quantum Phases Through Symmetry and Topological Order

Recent investigations reveal exotic phases of matter arising from non-invertible symmetries, utilising advanced mathematical tools, particularly category theory. Researchers construct two-dimensional lattice models exhibiting conformal behaviour – invariance under scale transformations – at critical points, applying specific boundary conditions informed by three-dimensional topological orders. They employ ‘string-net’ models and controlled condensation of anyons – quasiparticles displaying quantum mechanical statistics distinct from bosons or fermions – establishing a framework for understanding emergent behaviour in condensed matter systems.

A key development lies in a refined ‘condensation tree’ – a systematic method for controlling the non-invertible symmetries preserved at these critical points, generating an infinite family of critical lattice models. This approach encompasses established minimal models and importantly, identifies previously unknown critical points.

Researchers confirm the existence of at least three novel critical points exhibiting Haagerup symmetry – a complex symmetry beyond conventional classifications, indicating a departure from established paradigms and suggesting the potential for entirely new forms of quantum order. Validation extends beyond known examples, such as the 8-vertex model, to these novel instances.

This interdisciplinary work draws heavily from mathematics, physics, and computational methods, highlighting the increasing importance of collaborative approaches in exploring quantum materials. The rapid publication rate, with a significant number of papers appearing in 2024 and 2025, underscores the dynamic nature of this field and its potential to reshape our understanding of fundamental physical phenomena.

The refined condensation tree governs the non-invertible symmetries preserved at the critical point, combined with a generalised Kramers-Wannier duality – a mathematical transformation relating different physical systems – to accurately predict boundary conditions and constrain the global phase diagram, effectively identifying second-order phase transitions.

Researchers establish a direct correspondence between critical couplings – parameters defining the interactions within the model – and categorical data, specifically Frobenius algebras and dimensions within unitary fusion categories, bridging the gap between abstract mathematical structures and physical observables. This connection provides a systematic route for discovering and potentially classifying new conformal field theories, offering a framework for understanding emergent behaviour in condensed matter systems.

Precision numerical simulations, employing a symmetry-preserving tensor-network renormalization group – a computational technique for approximating quantum many-body systems – corroborate these findings and confirm the accuracy of the theoretical predictions. Researchers systematically generate two-dimensional lattice models exhibiting conformal fixed points in the infrared regime, leveraging three-dimensional topological orders realised through string-net models and introducing controlled condensation of non-commuting anyons.

Future work will likely focus on extending this framework to explore a wider range of topological orders and symmetries, and on developing more efficient numerical techniques for simulating these complex systems, promising to accelerate the discovery of new critical phenomena and deepen our understanding of the fundamental principles governing condensed matter physics.