A Unified Categorical Description of Quantum Hall Hierarchy and Anyon Superconductivity

TL;DR

This work develops a category-theoretic framework that unifies quantum Hall hierarchy and anyon superconductivity via modular tensor categories over $Rep(U(1))$ or $sRep(U(1)^f)$. Doping is modeled by a generalized stack-and-condense operation that stacks an auxiliary topological order and condenses a composite anyon, with the condensate charge fixed by the symmetry-breaking pattern of $U(1)$ and the chiral central charge adding cumulatively. The approach reproduces known field-theoretic results for both hierarchy transitions and various anyon superconductors, and it predicts new phases such as charge-2e superconductivity from the Laughlin state and charge-$ke$ superconductivity from bosonic Read-Rezayi states. By unifying these phenomena in a single formalism, the work provides a systematic bridge between categorical data, field theory, and experimentally relevant anyonic phases, with potential implications for classifying superconductivity arising from fractional Chern insulators.

Abstract

We present a unified category-theoretic framework for quantum Hall hierarchy constructions and anyon superconductivity based on modular tensor categories over $\mathrm{Rep}(\mathrm{U}(1))$ and $\mathrm{sRep}(\mathrm{U}(1)^f)$. Our approach explicitly incorporates conserved $\mathrm{U}(1)$ charge and formulates doping via a generalized stack-and-condense procedure, in which an auxiliary topological order is stacked onto the parent phase, and the quasiparticles created by doping subsequently condense. Depending on whether this condensation preserves or breaks the $\mathrm{U}(1)$ symmetry, the system undergoes a transition to a quantum Hall hierarchy state or to an anyon superconductor. For anyon superconductors, the condensate charge is determined unambiguously by the charged local bosons contained in the condensable algebra. Our framework reproduces all known anyon superconductors obtained from field-theoretic analyses and further predicts novel phases, including a charge-$2e$ anyon superconductor derived from the Laughlin state and charge-$ke$ anyon superconductors arising from bosonic $\mathbb{Z}_k$ Read-Rezayi states. By placing hierarchy transitions and anyon superconductivity within a single mathematical formalism, our work provides a unified understanding of competing and proximate phases near experimentally realizable fractional quantum Hall states.

Figures (1)

• Figure 1: (a) Schematic description of the stack-and-condense procedure. When a given $\mathrm{U}(1)$-symmetric topological order $\mathcal{C}$ is doped, the emerging anyons form their own topological state $\mathcal{D}$, which is stacked onto $\mathcal{C}$, and then a condensable algebra $\mathcal{A}$ including the doped anyon condenses. The final state is described by $(\mathcal{C} \boxtimes \mathcal{D}) / \mathcal{A}$. (b) Anyon superconductor and hierarchical quantum Hall (QH) state from doping the Laughlin state at filling fraction $\nu = \frac{1}{3}$. The Haldane-Halperin hierarchical state with filling fraction $\nu = \frac{2}{5}$ and chiral central charge $c = 2$ is obtained by choosing $\mathcal{D}$ as the Laughlin state at filling fraction $\nu = \frac{1}{15}$ and condensing a charge-neutral $\mathcal{A}$. A chiral charge-$2e$ anyon superconductor with $c = 1$ is obtained by choosing $\mathcal{D}$ as the charge-neutral $\mathrm{U}(1)_{-2} \times \mathrm{U}(1)_6$ theory and condensing a charged $\mathcal{A}$ including the charge-$2$ local boson $\mathbf{2}$. In the figure, $\Phi_0$ denotes the magnetic flux quantum of the superconductor.