Fractional Quantum Hall Matter and Anyons

Fractional Hall fluids support excitations with fractional charge and anyonic statistics. Their topological description connects microscopic projection to fusion, braiding, ground-state sectors, and edge response.

Enter this chapter

Begin with Topological Order, Invertible Phases, and Matter Diagnostics to distinguish intrinsic topological order from invertible response phases.

Helpful background. Topological Order, Invertible Phases, and Matter Diagnostics and Integer Quantum Hall Matter and Chern Insulators develop the definitions used by the chapter’s opening pages.

Topics

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 limit on the conclusion.
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.

How the ideas fit together

Landau-level or Chern-band projection constrains the microscopic dynamics, while an effective topological theory summarizes the resulting gapped phase. In Abelian phases, charge response, mutual statistics, and topology-dependent degeneracy follow from common integer data; non-Abelian phases additionally carry fusion spaces Nayak et al. 2008, §§ II–IV.

These relationships explain why fractional charge alone does not determine braiding. Edge spectra and interference signals also depend on reconstruction, equilibration, and device effects. Compatible bulk, boundary, and quasiparticle measurements are needed to distinguish candidate orders.

The comparison below separates the assumptions and tests needed for these different conclusions.

Claim-validity table

Scroll horizontally to compare all columns. Use the Left and Right arrow keys when the table region is focused.

Candidate fractional Hall orders and tests of their topological predictions.

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

Table data (JSON, version 1.0.0) · JSON schema

Where to go next

Continue to Fractionalization and Emergent Gauge Fields to identify the redundancy and physical-state constraint in a parton representation.

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.