The fractional quantum Hall effect (FQHE) is a remarkable phenomenon observed in two-dimensional electron systems subjected to strong magnetic fields and low temperatures. It represents a striking departure from the integer quantum Hall effect (IQHE), revealing the emergence of strongly correlated electron states with fractional charge excitations and topological order. The discovery of FQHE in 1982 marked a pivotal moment in condensed matter physics, unveiling new paradigms for understanding quantum phases of matter.

Theoretical frameworks for FQHE are rooted in the concept of topological order, a form of quantum organization that cannot be described by conventional symmetry-breaking theories. The Laughlin wavefunction, proposed by Robert Laughlin in 1983, provides a foundational model for the ν = 1/3 state, where ν denotes the filling factor. This wavefunction captures the essence of electron correlations by introducing a Jastrow factor that enforces antisymmetry while minimizing Coulomb repulsion. The Laughlin state exhibits quasiparticle excitations with fractional charge e/3, a hallmark of FQHE. Subsequent theoretical advances, such as the composite fermion theory, extended this picture to other filling factors like ν = 2/5 or 3/7 by mapping interacting electrons to non-interacting quasiparticles coupled to an effective magnetic field.

Experimental observations of FQHE occur in high-mobility two-dimensional electron gases (2DEGs), typically formed in GaAs/AlGaAs heterostructures. Under perpendicular magnetic fields exceeding several tesla and temperatures below 1 Kelvin, the longitudinal resistivity vanishes while the Hall resistivity plateaus at fractional values of h/e². Precise quantization of these plateaus confirms the topological robustness of FQHE states. Key experiments have identified over 80 distinct fractional states, including those in higher Landau levels and the enigmatic even-denominator fractions like ν = 5/2, which are believed to host non-Abelian anyons.

The statistical properties of quasiparticles in FQHE systems are fundamentally different from those of fermions or bosons. These anyons exhibit fractional statistics, meaning their wavefunction acquires a phase factor other than ±1 upon particle exchange. For example, Laughlin quasiparticles obey Abelian anyonic statistics with an exchange phase of π/3. In contrast, certain non-Abelian states, such as the Moore-Read Pfaffian state proposed for ν = 5/2, support quasiparticles whose braiding operations form a representation of the braid group, enabling topological quantum computation. However, this article focuses solely on their intrinsic properties rather than computational applications.

Topological order in FQHE systems manifests through ground-state degeneracy on multiply connected geometries and edge state phenomenology. The bulk-boundary correspondence implies that gapped bulk states give rise to chiral Luttinger liquid edge modes, whose tunneling characteristics reveal fractional charge and statistics. Measurements of shot noise in quantum point contacts have confirmed fractional charge carriers, while interferometry experiments have provided indirect evidence of anyonic braiding. The topological nature of these states also confers immunity to local perturbations, explaining the extreme precision of Hall plateaus.

Material quality plays a critical role in observing FQHE. High-purity GaAs heterostructures with electron mobilities exceeding 10⁷ cm²/Vs are essential to minimize disorder effects that can destabilize fractional states. Advances in molecular beam epitaxy have enabled the fabrication of such pristine 2DEGs, while dilution refrigerators provide the necessary sub-Kelvin temperatures. Recent work has also explored FQHE in graphene-based systems, though the Dirac spectrum and weaker electron interactions lead to distinct behavior compared to conventional semiconductors.

Theoretical developments continue to refine our understanding of FQHE. The hierarchical construction generalizes Laughlin states to more complex fractions by iteratively forming new incompressible fluids from quasiparticle excitations. Conformal field theory methods have established deep connections between FQHE states and topological quantum field theories, with the Chern-Simons theory providing an effective low-energy description. Numerical techniques like exact diagonalization and density matrix renormalization group calculations have validated many theoretical predictions while uncovering new phases in finite-size systems.

Experimental challenges remain in probing the microscopic details of FQHE states. Scanning tunneling microscopy faces difficulties due to the buried nature of 2DEGs, though progress has been made in surface-sensitive variants. Thermal transport measurements, particularly of the quantized thermal Hall conductance, offer complementary insights into topological order. The quest to unambiguously demonstrate non-Abelian statistics continues to drive innovations in nanofabrication and measurement techniques.

The study of FQHE has profoundly influenced modern condensed matter physics, demonstrating how strong interactions can generate emergent topological phases with exotic excitations. Beyond its intrinsic scientific importance, this field has spurred advances in material synthesis, low-temperature physics, and quantum measurement technologies. Future research may uncover new fractional states in other material systems or reveal deeper connections between topological order and other correlated quantum phenomena. The fractional quantum Hall effect remains a vibrant area of investigation, bridging conceptual breakthroughs with exquisite experimental precision.