Fractional Quantum Hall Matter and Anyons

Fractional quantum Hall matter is an intrinsically topologically ordered fluid: it combines a mobility gap, fractionalized quasiparticles, topology-dependent ground-state structure, quantized response, and an anomalous edge. Anyons are characterized by charge, fusion, topological spin, and braiding—not by fractional charge alone. A reliable phase claim therefore combines projected dynamics with mutually consistent bulk, edge, entanglement, and transport data Nayak et al. 2008, §§ II–IV.

We take electron charge to be $-e$, $e>0$, magnetic field $B>0$ perpendicular to the oriented plane, and counterclockwise exchange as positive. The magnetic length is $\ell_B=\sqrt{\hbar/(eB)}$ and $\nu=2\pi\ell_B^2n$. Reported signs from another field or boundary orientation must be translated before comparison.

Helpful background. Topological order and invertible phases supplies the intrinsic-order distinction; integer quantum Hall matter supplies Landau/Chern response and chiral boundary conventions.

Enter this chapter

The microscopic route begins with Landau-level projection and follows correlations into Laughlin and hierarchical fluids. The data route begins with anyons and the Abelian K matrix, then adds non-Abelian fusion spaces and boundary consistency. The evidence route compares interferometry, noise, thermal transport, and entanglement against device, edge, disorder, and finite-size alternatives.

Reader goal	Suggested route	Capability at the end
Hall foundations	Landau levels → FQH fluids → composite fermions	Connect filling, projected interactions, incompressibility, Jain sequences, and quasiparticle charge
Topological data	Anyons → K matrix → non-Abelian orders	Compute charge, spin, mutual braiding, fusion dimension, torus degeneracy, and edge chirality
Lattice platforms	Berry geometry → FCI/moiré → entanglement diagnostics	Test a fractional Chern phase without assuming Landau-level geometry
Experimental inference	Edges → interferometry → entanglement/evidence	Separate a theory prediction from a device-level or finite-size identification

From projection to topological data

Projection freezes the cyclotron oscillator but leaves noncommuting guiding centers. Their magnetic-translation algebra makes ordinary real-space intuition unreliable and organizes pseudopotentials, structure factors, and many-body momentum sectors. The first figure follows this reorganization into continuum Hall fluids, composite-fermion and parton descriptions, lattice Chern bands, and multicomponent or non-Abelian orders.

The fractional-Hall structure and dictionary. Solid arrows are constructions or response maps; dashed branches require projection, a many-body gap, and a specified gauge constraint. The diagram is schematic and does not infer a phase solely from flatness or filling.

For an Abelian order described by a nondegenerate integer symmetric matrix $K$, charge vector $t$, and integer quasiparticle vector $l$,

$$\nu=t^{\mathsf T}K^{-1}t, \qquad Q_l=e\,t^{\mathsf T}K^{-1}l, \qquad \theta_l=\pi l^{\mathsf T}K^{-1}l,$$ $$\theta_{ll'}^{\rm mutual}=2\pi l^{\mathsf T}K^{-1}l', \qquad \mathcal G_g=|\det K|^g, \qquad c_-=\operatorname{signature}K.$$

$Q_l$ is the positive electric charge of a quasihole in this convention; adding a local electron shifts it by an integer multiple of $e$. Quasiparticles related by $l\sim l+K\Lambda$ are the same topological sector. A basis change $W\in GL(N,\mathbb Z)$ changes $(K,t,l)$ without changing these observables.

Evidence is diagnostic-specific

The second figure separates formal data from observations. Fractional charge can appear in shot noise; Abelian statistical phases can modify interference; fusion channels and quantum dimensions require stronger protocols; thermal conductance constrains $c_-$ but is vulnerable to incomplete equilibration. Entanglement and finite-size spectra are numerical evidence whose interpretation depends on size, aspect ratio, boundary twist, and competing orders.

Validity and evidence map for fractional Hall and anyon claims. Solid branches are candidate predictions, the dotted branch marks required platform-realization checks rather than a prediction, and dashed branches name leading alternatives. Quantized charge or Hall response does not by itself establish braiding, and a finite-size manifold or entanglement pattern does not by itself identify a thermodynamic topological order. The figure is schematic.

Claim-validity table

Proposed order	Filling and shift	Topological input	Quasiparticles	Edge prediction	Closed-manifold and entanglement test	Experimental discriminator	Principal ambiguity
Laughlin $1/m$	$\nu=1/m$, sphere shift $m$	$K=(m)$, $t=(1)$	Minimal charge $e/m$, exchange $\pi/m$	One chiral boson, $c_-=1$	$m^g$ genus-$g$ sectors; $\gamma=\tfrac12\ln m$	Hall plateau plus charge and statistical-phase tests	Edge reconstruction, Coulomb-dominated interference, or finite-size charge-density order
Jain Abelian state	$\nu=p/(2sp\pm1)$; shift sequence-dependent	Multicomponent $K,t$ or projected composite-fermion state	Charges and braiding from $K^{-1}$	Several modes; upstream content depends on signature and equilibration	Expected torus manifold and counting after momentum folding	Charge, thermal, and interface equilibration compared jointly	Spin/valley polarization, Landau-level mixing, and alternative hierarchy state
Candidate non-Abelian state	Filling and shift candidate-specific	Fusion category or CFT data plus charge sector	Quantum dimension greater than one; matrix braid action	Charged plus neutral sectors; candidate-specific $c_-$	Fusion-space growth, modular data, entanglement counting, size convergence	Fusion-channel-sensitive interferometry and equilibrated thermal response	Abelian competitor, edge nonequilibration, disorder, or insufficient size
Fractional Chern insulator	Fractional lattice-band filling; no universal shift on a torus	Many-body Chern response and topological data	Lattice analogues of continuum anyons	Depends on boundary and lattice symmetry	Spectral flow, manifold counting, entanglement, absence of density order	Zero-field fractional Hall response with independent incompressibility and magnetism tests	Competing charge/magnetic order, inhomogeneity, band mixing, or device-dependent phase diagram
Interferometric anyon claim	Filling and localized quasiparticle number fixed	Predicted Aharonov–Bohm plus statistical phase	Charge and braid phase or fusion channel stated	Coherent edge path with calibrated area and velocity	Not sufficient alone	Phase slips/periods tied to controlled quasiparticle addition and reproduced across regimes	Coulomb charging, area breathing, telegraph traps, neutral-mode dephasing, and selection
Entanglement identification	Geometry, cut, and conserved sectors stated	Candidate total quantum dimension and edge counting	Sector-dependent quantum dimensions if resolved	Low-lying counting may mirror an edge CFT	Multi-size extrapolation of entropy and entanglement gap	Indirect; must agree with response or quasiparticle tests	Nonuniversal area term, cut dependence, aliasing, and small-system mimicry

The table complements the two figures by requiring filling, shift, quasiparticle data, edge content, finite-size evidence, and experimental alternatives to be checked independently. Together, the adjacent prose, relationship-centered alt text, and table give the complete nonvisual account of the diagrams.

Guide to the pages

1. Topological Order, Invertible Phases, and Matter Diagnostics separates intrinsic order from short-range-entangled response phases.
2. Landau Levels, Projection, and Magnetic Translations derives guiding-center algebra and projected density kinematics.
3. Fractional Quantum Hall Fluids constructs the Laughlin state, fractional charge, and low-energy response.
4. Composite Fermions and Hierarchical Hall States develops Jain sequences while retaining projection and gauge constraints.
5. Fractional Chern Insulators and Moiré Hall Matter states geometric flat-band criteria and the current platform evidence ceiling.
6. Anyons as Quasiparticles in Quantum Matter defines exchange, braiding, fusion, spin, and superselection sectors.
7. Abelian Topological Orders and K-Matrix Data derives charge, statistics, degeneracy, Hall response, and signature.
8. Non-Abelian Topological Orders adds fusion spaces, $F$ and $R$ consistency, and quantum dimensions.
9. Chiral Edges, Interfaces, and Bulk–Boundary Tests distinguishes anomaly-fixed edge data from reconstruction-dependent spectra.
10. Anyon Interferometry and Evidence Standards compares braiding-sensitive signals with charging and edge alternatives.
11. Topological Entanglement and Spectrum Diagnostics gives finite-size tests and their evidentiary limits.

Review the chapter

Normalization. For $K=(3)$, $t=(1)$, the vector $l=(1)$ has $Q=e/3$, exchange angle $\pi/3$, and three torus sectors. Replacing $l$ by $l+3$ attaches a local electron and leaves its topological sector unchanged.

Boundary. Two edges with the same bulk $K$ can have different reconstructed mode content. Their net anomaly, Hall response, and equilibrated chiral central charge remain bulk constrained; individual velocities and extra counterpropagating pairs do not.

Evidence. A nearly threefold low-energy manifold on one torus size is compatible with a $\nu=1/3$ phase but also with translation breaking. Flux spectral flow, momentum sectors, density correlations, entanglement, and size/aspect-ratio stability are needed before the stronger conclusion.

References

• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, “Non-Abelian Anyons and Topological Quantum Computation,” Reviews of Modern Physics 80 (2008) 1083–1159, doi:10.1103/RevModPhys.80.1083.