Fractional Quantum Hall EffectEdit

The Fractional Quantum Hall Effect (FQHE) is a striking manifestation of strongly correlated quantum physics in two-dimensional electron systems subjected to large perpendicular magnetic fields and low temperatures. Unlike its integer counterpart, the fractional effect reflects the emergence of new quantum fluids whose excitations carry fractional electric charge and obey unusual exchange statistics. The simplest and earliest example occurs at filling factor ν = 1/3, where the many-body ground state can be described by a highly correlated wavefunction proposed by Robert Laughlin and colleagues. Since then a broad family of fractional states has been observed, mapped out in high-murity semiconductor structures and, more recently, in atomically thin materials. The phenomenon sits at the crossroads of condensed matter physics, quantum field theory, and topology, and it has driven the development of theoretical frameworks such as the composite fermion picture and hierarchical constructions, as well as experimental efforts to detect fractional charges and exotic anyonic statistics.

The study of the FQHE has illuminated how strong electron–electron interactions can organize a system into topologically ordered phases that defy a simple local description. In these phases, the quantized Hall conductance is not tied to a single-particle energy spectrum but to a global property of the many-body wavefunction. This has made the FQHE a central example of how topology enters concrete condensed-m matter systems, with implications ranging from edge physics to potential routes for fault-tolerant quantum computation. While the foundational ideas are well established, the field continues to grapple with detailed questions about the precise nature of several fractional states, especially those at higher denominators and the enigmatic 5/2 state, which is a leading candidate for hosting non-Abelian anyons.

Overview

The FQHE arises when electrons confined to a two-dimensional layer are subjected to a strong magnetic field that quantizes their kinetic energy into Landau levels. In the limit where a partial Landau level is sparsely filled and the temperature is well below the interaction energy scale, Coulomb interactions dominate and drive the system into incompressible quantum fluids. The hallmark is a Hall conductance equal to a fractional multiple of e^2/h, accompanied by vanishing longitudinal resistance at low temperatures.

The filling factor ν is the ratio of the electron density to the degeneracy of a Landau level, and fractional plateaus appear at many rational ν values, such as 1/3, 2/5, 3/7, and beyond. The early, prototypical state at ν = 1/3 is well described by Laughlin's wavefunction, a compact analytic expression that captures the essential correlations.
The experimental realization of the FQHE is intimately tied to very clean two-dimensional electron systems, typically in GaAs-based heterostructures (often written as [GaAs/AlGaAs] quantum wells) and at millikelvin temperatures. More recently, graphene and other atomically thin materials have joined the set of platforms enabling fractional states.

The conceptual backbone of the FQHE lies in two complementary theoretical traditions. The Laughlin paradigm provides exact or highly accurate descriptions of a subset of fractions, while composite fermion theory offers a unifying framework that explains a broad sequence of fractions as integer quantum Hall states of emergent quasiparticles. Other hierarchical constructions provide alternative perspectives for certain fractions and for understanding the organization of many experimentally observed states. Across these viewpoints, the common thread is that the physics is controlled by topology and by the way interactions reconfigure the ground state of a highly degenerate single-particle spectrum.

Laughlin state: The ν = 1/m (with m an odd integer) states are described by a single model wavefunction that captures essential correlations and yields quasiparticles with fractional charge e/m. The mathematics of these states ties to holomorphic function theory and to the topology of the many-electron wavefunction.
Composite fermions: Proposed to explain a large range of fractions, this framework maps strongly interacting electrons in a partially filled Landau level to weakly interacting composite fermions at an effective magnetic field, producing a hierarchy of integer-like states at fractional fillings. See Composite fermion for further detail.
Non-Abelian candidates: The ν = 5/2 state stands apart, as discussions center on whether its ground state may host non-Abelian anyons, such as the Pfaffian or related anti-Pfaffian states, with potential implications for fault-tolerant quantum computation.

Physical Principles

Key ingredients behind the FQHE include Landau quantization, strong electron–electron interactions, and finite thickness effects along the confinement direction. In a strong magnetic field, the kinetic energy is quenched and the physics is dominated by Coulomb interactions projected into a partially filled Landau level. The resulting correlated ground state is incompressible and supports excitations with fractional charge.

Landau levels: The discrete energy spectrum created by the magnetic field condenses kinetic energy but leaves a large degeneracy tied to the system size. The separation between Landau levels is large compared to the interaction energy for high-quality samples at low temperatures.
Fractional charge: Quasiparticles in the FQHE carry fractional electric charge, a striking experimental signature inferred from shot-noise measurements and interference experiments in mesoscopic devices.
Topological order: The incompressible states possess global, nonlocal order. Their robust properties, such as quantized Hall conductance and protected edge modes, are insensitive to microscopic details like disorder, so long as the energy gap remains open.
Edge physics: In finite samples, gapless edge excitations accompany the bulk topological order. These edge modes have characteristic chiralities and can be probed via tunneling and interference experiments, providing indirect evidence about the bulk phase.

For a rigorous framing, see Topological order and Anyons for the exotic statistics of excitations, including the Abelian and potentially non-Abelian variants.

Theoretical Frameworks

Laughlin state

Laughlin's wavefunction provides an explicit form for the ground state at ν = 1/m and yields charge- e/m quasiparticles with fractional statistics. This milestone anchored the understanding that electron correlations can produce genuinely new collective states.

Composite fermions

The composite fermion picture recasts the problem in terms of emergent particles—electrons bound to an even number of flux quanta—moving in a reduced effective magnetic field. This transformation accounts for a wide array of fractions as integer quantum Hall states of composite fermions, thereby unifying many observed plateaus under a single framework. See Composite fermion for a deeper discussion.

Hierarchical and other constructions

Beyond Laughlin and Jain’s composite fermions, hierarchical models (developed by Haldane, Halperin, and others) organize fractional states through successive condensation of quasiparticles, yielding a broader landscape of possible filling fractions. See Hierarchical quantum Hall effect for a broader view.

Non-Abelian states and the ν = 5/2 enigma

The ν = 5/2 state is a focal point of interest because certain proposed ground states—most notably the Pfaffian and related anti-Pfaffian states—support non-Abelian anyons. Non-Abelian statistics would enable topologically protected qubits and gates, a tempting route for quantum computing. Yet the precise nature of the state, its particle-hole symmetry properties, and the role of Landau level mixing and disorder remain areas of active research and debate. See Pfaffian state and Anti-Pfaffian state.

Experimental Observations

Experiments have mapped a rich set of fractional plateaus in the Hall conductance, with the simplest and most robust observed at ν = 1/3, 2/5, 3/7, and related fractions. Quasiparticle charges consistent with e/3 and other fractions have been inferred from shot-noise measurements, providing compelling evidence for fractionalization. Interferometry experiments in mesoscopic devices have sought to reveal fractional statistics, though a definitive demonstration of non-Abelian braiding remains challenging and is an ongoing objective of the field.

Material platforms: The earliest and most mature realizations come from high-murity GaAs/AlGaAs heterostructures hosting a two-dimensional electron gas. More recently, graphene and other atomically thin materials have yielded additional fractions, including those in regimes with different symmetries or Landau level structure.
Edge channels: Transport along the edges reflects the chiral edge modes predicted by the bulk's topological order. Tunneling experiments and nonlocal measurements help characterize the edge structure and energy gaps.
5/2 state debates: A central experimental question is whether the ν = 5/2 state is described by a Pfaffian-like non-Abelian order or by alternative states (such as anti-Pfaffian), with evidence influenced by disorder, confinement, and Landau level mixing. See Pfaffian state and Anti-Pfaffian state.

Numerous experimental signatures support the broad validity of the FQHE picture, yet precise determination of the topological order for many fractions, and especially the nature of the 5/2 state, remains an area of active investigation.

Materials, Realizations, and Extensions

The FQHE is not limited to a single material platform. GaAs-based two-dimensional electron gases established the canonical regime, but the discovery of FQHE phenomena in graphene and other layered materials has broadened the landscape. The tunability of sample width, carrier density, dielectric environment, and disorder enables systematic tests of theory, including how finite thickness and Landau level mixing modify the phase diagram. Basic transport measurements, charge-sensitive probes, and interferometry all contribute to a more complete understanding of how topology emerges in real, imperfect systems.

Controversies and Debates

While there is broad consensus on the existence and basic structure of many FQHE states, several questions remain debated in the literature:

The precise topological order of the ν = 5/2 state: Multiple candidate states (Pfaffian, anti-Pfaffian, and variants with particle-hole symmetry) fit different subsets of data, and no single experiment has universally settled the issue. The role of edge reconstruction, disorder, and Landau level mixing complicates interpretations. See Pfaffian state and Anti-Pfaffian state.
Particle-hole symmetry in the lowest Landau level: The extent to which particle-hole symmetry constrains possible states, and how this symmetry is broken in real samples, is a focal point of both theory and experiment.
The reach of composite fermion theory: While the composite fermion picture successfully explains a large family of fractions, questions persist about its microscopic underpinnings and its limits for more exotic fractions, especially those in higher Landau levels or under unusual symmetry constraints.
Non-Abelian statistics in practice: Demonstrating non-Abelian braiding in a robust, scalable way remains technically demanding. Some experiments suggest signatures compatible with non-Abelian anyons, but consensus requires unambiguous, reproducible evidence of braiding statistics.
Material-dependent effects: Finite thickness, disorder, and screening influence gap sizes and stability of various states. Discrepancies between different samples or fabrication methods are common and drive ongoing refinements in theory and measurement techniques.

These debates are not about political or ideological disputes; they reflect the natural scientific process of reconciling models with ever-improving data. The FQHE continues to serve as a testing ground for ideas about how topology, symmetry, and interactions shape the quantum world.