New Theory Unites Exotic Materials with Potential for Lossless Electricity Transfer

Scientists are increasingly focused on understanding the intricate relationship between the quantum Hall hierarchy and the emergence of anyon superconductivity. Donghae Seo from Pohang University of Science and Technology, Taegon Lee and Gil Young Cho from Korea Advanced Institute of Science and Technology, et al., present a novel category-theoretic framework that unifies descriptions of these phenomena. Their research explicitly incorporates charge conservation and introduces a generalised method for modelling doping processes via stacking and condensation. This work is significant because it not only reproduces previously known anyon superconductors but also predicts entirely new phases, offering a more complete and cohesive understanding of the competing phases observed near fractional quantum Hall states.

Linking Quantum Hall Effects and Anyon Superconductivity via Topological Condensation reveals emergent phenomena in two-dimensional systems

Scientists have developed a unified mathematical framework describing both the formation of quantum Hall hierarchies and the emergence of anyon superconductivity. This work establishes a direct connection between these two phenomena, previously understood as distinct, by utilising category theory and modular tensor categories over Rep(U(1)) and sRep(U(1)f).
The research explicitly incorporates the conservation of electric charge, formulating a process called ‘stacking and condensing’ to model the addition of topological order and subsequent condensation of quasiparticles. Depending on whether this condensation preserves or breaks U(1) symmetry, the system transitions into either a quantum Hall hierarchy state or an anyon superconductor.

Crucially, the framework predicts the charge of the condensate forming in anyon superconductors based on the charged local bosons present within the condensable algebra. This approach successfully reproduces previously known anyon superconductors derived from field-theoretic analyses, while also forecasting novel phases, including a charge-2e anyon superconductor originating from the Laughlin state and charge-ke anyon superconductors arising from bosonic Zk Read-Rezayi states.

By unifying hierarchy transitions and anyon superconductivity within a single formalism, the study offers a comprehensive understanding of competing phases near experimentally accessible fractional quantum Hall states. The core of this achievement lies in extending the ‘stack-and-condense’ operation to account for global U(1) charge conservation, providing a systematic method for determining the condensate charge in anyon superconductors.

This builds upon existing work on quantum Hall hierarchy constructions and applies it to a broader range of scenarios. The framework not only confirms existing theoretical results but also predicts new anyonic phases, expanding the possibilities for designing and understanding novel superconducting materials.

Specifically, the research demonstrates that a charge-2e anyon superconductor can emerge from the Laughlin state at a filling fraction of ν = 1/3, alongside a variety of charge-ke anyon superconductors derived from bosonic Zk Read-Rezayi states. These findings establish a robust link between categorical data, field-theoretic descriptions, and experimentally relevant anyonic phases, paving the way for further exploration of these exotic states of matter and their potential applications in future technologies. The work provides a powerful tool for classifying and predicting the properties of these complex quantum systems.

Categorical modelling of doped topological phases and condensate charge determination are crucial for understanding correlated electron systems

A unified category-theoretic framework underpins this work, explicitly incorporating conserved charge and formulating doping via a generalized stack-and-condense procedure. This procedure involves stacking an auxiliary topological order onto a parent phase, followed by the condensation of quasiparticles created through doping.

Transitions to either fractional quantum Hall hierarchy states or anyon superconductors are then naturally described within this framework. The central methodological contribution lies in extending the stack-and-condense operation to account for global U(1) charge conservation, providing a systematic method for determining the condensate charge in anyon superconductors.

This research begins with a field-theoretic analysis of the Laughlin state at filling fraction ν = 1/3, integrating both hierarchy construction and anyon superconductivity to motivate the categorical approach. The Laughlin state is described using a Lagrangian incorporating terms for a dynamical U(1) gauge field and a background electromagnetic gauge potential, expressed in differential-form notation with wedge products.

Anyons are incorporated by introducing a conserved current minimally coupled to an additional U(1) gauge field, α. To model hierarchy construction, a level-m Chern-Simons term is introduced for α, allowing for the realization of the ν = 2/5 hierarchy state with a specific K-matrix and charge vector. Crucially, setting m = 1/3 within this construction signals the emergence of a superconductor, although fractional Chern-Simons levels are not conventionally allowed.

This necessitates a U(1)−2×U(1)6 Chern-Simons theory, described by a Lagrangian with dynamical gauge fields b and c, and characterized by a specific 6×6 K-matrix. Performing an SL(4, Z) basis transformation and integrating out massive gauge fields associated with the trivial topological sector yields a Lagrangian describing a charge-2e superconductor with chiral central charge c = 1. This demonstrates that the same framework can describe both hierarchy states and superconductivity, establishing a unified approach to understanding competing phases near fractional Hall states.

Doping, condensation and symmetry breaking in category-theoretic constructions of Hall hierarchies and anyon superconductivity reveal deep connections to topological phases of matter

A unified category-theoretic framework has been established for Hall hierarchy constructions and anyon superconductivity based on tensor categories over and . This work explicitly incorporates conserved charge and formulates doping via a generalized stack-and-condense procedure, wherein an auxiliary topological order is stacked onto a parent phase, followed by the condensation of quasiparticles created by doping.

The resulting state is either a Hall hierarchy state or an anyon superconductor, dependent on whether the symmetry is preserved or broken during condensation. Specifically, the condensate charge within anyon superconductors is uniquely determined by the charged local bosons contained within the condensable algebra.

This framework successfully reproduces all previously known anyon superconductors derived from field-theoretic analyses, while also predicting novel phases. A charge-anyon superconductor is derived from the Laughlin state, and additional charge-anyon superconductors arise from bosonic Read-Rezayi states.

The research demonstrates that doping charge-e/4 anyons in a Pfaffian state yields a conventional charge-2e superconductor. Within the categorical formulation, the condensate charge of the superconducting phase is uniquely fixed by the symmetry-breaking pattern of the global U(1) symmetry, and the chiral central charge follows directly from the same framework.

Notably, the framework reproduces field-theoretical results on anyon superconductors, including those from non-Abelian parent states, and predicts new classes of anyonic phases. Charge-2e anyon superconductivity emerges from the Laughlin state at a filling fraction of ν = 1/3, and a variety of charge-ke anyon superconductors are derived from bosonic Zk Read-Rezayi states.

The K-matrix formulation reveals a singular matrix with zero determinant when m = 1/3, signaling the emergence of a superconductor. An effective level of m = 1/3 is realized through a U(1)−2×U(1)6 Chern-Simons theory, resulting in a charge-2e superconductor with a chiral central charge of c = 1. This correspondence between anyon superconductivity and quantum Hall hierarchy constructions is elucidated by the categorical approach, establishing a systematic bridge between categorical data, field-theoretic descriptions, and experimentally relevant anyonic phases.

Topological phase transitions via stack-and-condense procedures in category theory offer a novel framework for understanding emergent phenomena

A unified mathematical framework has been developed to understand both the construction of quantum Hall hierarchies and the emergence of anyon superconductivity. This approach utilises category theory, specifically tensor categories, to explicitly incorporate conserved charge and model the process of doping through a stack-and-condense procedure.

This procedure involves combining a primary topological phase with an auxiliary one, followed by the condensation of specific quasiparticles. The resulting phase depends on whether the condensation process preserves or breaks the system’s symmetry. Preservation of symmetry leads to a quantum Hall hierarchy state, while symmetry breaking results in an anyon superconducting phase.

Importantly, the charge of the condensate in anyon superconductors is uniquely determined by the charged local bosons present within the condensing algebra. This framework successfully reproduces previously known anyon superconductors and predicts novel phases, including charge-anyon superconductors originating from the Laughlin state and bosonic Read-Rezayi states.

The authors acknowledge a limitation in that their current framework primarily focuses on Abelian topological orders, leaving non-Abelian cases for future exploration. Further research should investigate the classification of anyon superconductivity within fractional Chern insulators, offering a promising avenue for extending these findings. These results establish a systematic route towards understanding competing phases near fractional Hall states and provide a unified perspective on their underlying principles.