Coupled-wire constructions of non-Abelian quantum Hall states: generality, microscopic quasiparticle braiding, and realization beyond the anisotropic limit

The coupled-wire construction (CWC) builds two-dimensional fractional quantum Hall phases from an array of interacting one-dimensional wires analyzed by bosonization, yielding solvable microscopic electronic Hamiltonians. Teo and Kane (Phys. Rev. B 89, 085101 (2014)) extended the CWC to the non-Abelian Moore–Read and Read–Rezayi states and posed an open research program: whether essentially every quantum Hall state admits a coupled-wire realization; how the quasiparticle ground-state degeneracy and non-Abelian braiding follow microscopically from the coupled-wire model rather than only from inherited conformal-field-theory data; and whether concrete interacting lattice Hamiltonians (Chern-insulator analogs) can realize these non-Abelian phases away from the extreme anisotropic limit, together with the related questions of the global phase diagram off the solvable gapping line and of intrinsically three-dimensional non-Abelian phases built from wires. Many specific families (hierarchies, non-Abelian spin singlets, parton states, orbifold states, chiral spin liquids) now have coupled-wire constructions, and the ground-state degeneracy of the bosonic Moore–Read state has been derived from wire models, but no general existence construction, no microscopic derivation of full non-Abelian braiding matrices from wire dynamics, no phase diagram off the solvable line, and no intrinsically 3D non-Abelian wire construction have been established: the core of the problem remains open.

Background

A fractional quantum Hall (FQH) state is a two-dimensional incompressible electron fluid at a rational filling factor $\nu$ whose low-energy physics is topologically ordered: it supports fractionally charged quasiparticles that are anyons, i.e. particles whose exchange can act by a phase (Abelian statistics) or by a non-commuting matrix on a degenerate ground-state space (non-Abelian statistics). The Moore–Read state (Moore and Read, 1991) at $\nu=1/2$ is the simplest non-Abelian FQH state; its quasiparticles realize Ising anyons with a Majorana neutral sector. The Read–Rezayi sequence (Read and Rezayi, 1999) generalizes it, with neutral sectors described by $Z_k$ parafermion conformal field theories (CFTs) — chiral CFTs whose primary fields obey a non-Abelian fusion algebra. Non-Abelian spin-singlet states (Ardonne and Schoutens, 1999) are multicomponent non-Abelian states based on $SU(n)$ coset CFTs, and the parton construction describes an even wider class of FQH states by fractionally decomposing the electron into partons. The coupled-wire construction (Kane, Mukhopadhyay and Lubensky, 2002; Teo and Kane, 2014) provides a solvable microscopic route to such phases. An array of parallel one-dimensional quantum wires in a magnetic field is described at low energy by Luttinger liquids, i.e. by bosonization, which rewrites each wire's low-energy fermions in terms of chiral boson fields $\varphi_{j,p}$ (wire index $j$, chirality $p=R/L$) with canonical commutators. At special filling factors, correlated multi-electron tunneling processes between neighboring wires become momentum-conserving sine-Gordon terms $\propto\cos(\ldots)$; when these interactions are relevant and their arguments commute, the array is driven into a gapped, exactly solvable strong-coupling phase that carries the Hall conductance, fractional charges, and chiral edge modes of the target FQH state. For non-Abelian states, Teo and Kane showed how the original chiral modes fractionalize into coset sectors of the edge CFT (the coset construction builds a new CFT from the degrees of freedom of a chiral algebra $G$ after quotienting out a subalgebra $H$, denoted $G/H$), with different sectors gapped 'in different directions'; the Read–Rezayi neutral sector arises from a $[SU(2)_k]_k/SU(2)_k$ coset. On the solvable line $u_{ab}=v_{ab}$ of interaction couplings the gapping terms decouple, but the operator $\Xi_{ab}$ (their Eq. (4.22)) is a nonprimary field (not a primary field, i.e. not one of the CFT fields that transform simplest under conformal transformations) acting simultaneously in two coset sectors, and it is not clear how $\Xi^R\Xi^L$ competes with the solvable terms away from this line. The construction is naturally formulated at an extreme anisotropic limit (strong interactions, weak inter-wire tunneling), and its connection to Chern insulators — lattice analogs of quantum Hall states that need no external magnetic field — and to the abstract parton and wave-function descriptions of FQH states was left as an open question.

Problem Statement

Complete the coupled-wire construction as a general microscopic theory of non-Abelian quantum Hall phases, as posed by Teo and Kane (2014). The requested results, which share the single objective of making the wire array a universal, solvable, electronic description of non-Abelian topological order, are: (1) Generality — determine whether essentially every quantum Hall state admits a coupled-wire realization, by general proof or systematic construction or by exhibiting an obstruction; in particular the hierarchical generalizations of the Moore–Read and Read–Rezayi states, the non-Abelian spin-singlet states based on $SU(3)_2$ coset CFT, and states accessible through the parton construction. (2) Topological order from the microscopic model — demonstrate the quasiparticle ground-state degeneracy and the non-Abelian statistics (braiding matrices) of these phases directly from the coupled-wire Hamiltonian's operators and their dynamics, beyond identifying the neutral parafermion content and beyond reading the braiding data off the associated conformal field theory. (3) New phases and higher dimensions — determine whether the construction gives concrete microscopic models for phases not already well understood, including fractional and non-Abelian Chern insulators and fractional topological insulators in two and possibly three spatial dimensions. In the Read–Rezayi coupled-wire model this includes the unresolved global phase diagram off the solvable line $u_{ab}=v_{ab}$, where the nonprimary operator $\Xi_{ab}$ (Eq. (4.22) of the source) competes with the solvable gapping terms; and, for physical realizations, the construction of concrete interacting lattice Hamiltonians that realize the non-Abelian phases away from the extreme anisotropic limit together with their stability under more isotropic perturbations.

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