Composite FermionEdit
Composite fermions are a central concept in the study of strongly interacting electrons confined to two dimensions under large perpendicular magnetic fields. The idea, introduced to explain the fractional quantum Hall effect, posits that each electron can bind an even number of magnetic flux quanta to form a new quasiparticle—the composite fermion. In this transformed picture, these composite fermions experience a reduced effective magnetic field and occupy their own set of Landau-like levels, called Λ-levels. When composite fermions fill an integer number of these Λ-levels, the system exhibits the fractional quantum Hall effect at fractions such as 1/3, 2/5, and beyond. The framework also provides a natural account of a compressible state at half filling, where composite fermions form a Fermi sea in the vanishing effective field. The theory connects with Laughlin wavefunction of 1/3 and with broader fractional quantum Hall effect phenomenology, unifying a wide range of observed plateaus under a single construction.
Theoretical framework
Flux attachment and the emergent gauge field: The central maneuver is to attach an even number of flux quanta to each electron, transforming the problem into one of composite fermions moving in an effectively weaker magnetic field. This construction is formalized through Chern-Simons theory, which provides the gauge-field language for flux attachment and the resulting dynamics of CFs.
Reduced field and Λ-levels: The effective magnetic field that composite fermions experience is B* = B − 2mφ0n, where B is the external field, m is the number of attached flux quanta per electron, φ0 = h/e is the flux quantum, and n is the carrier density. In this reduced field, CFs populate Λ-levels, mirroring the way electrons fill Landau levels in the ordinary quantum Hall problem. When CFs completely fill an integer number of Λ-levels, the system displays an integer quantum Hall-like response, which appears experimentally as a fractional quantum Hall plateau in the electron system.
Jain sequences and wavefunctions: The framework explains an extensive series of observed fractions, notably the Jain sequence p/(2mp+1), where p and m are integers. The corresponding many-body wavefunctions can be viewed as a Slater determinant of CFs in Λ-levels multiplied by a Jastrow factor encoding the flux attachment. This construction ties neatly to the broader Laughlin wavefunction and to the idea that many FQHE states arise from a simple integer-quantum-Hall-like picture applied to CFs.
Spin, subbands, and material specifics: Real systems have finite thickness, disorder, and multiple subbands, all of which affect the precise energy scales and the stability of various fractions. The CF description remains robust, but details such as spin polarization, Zeeman energies, and Landau level mixing can shift phase boundaries and influence which fractions are most prominent in a given material, from traditional GaAs quantum wells to newer platforms like graphene.
Particle-hole symmetry and Dirac formulations: At half filling (ν = 1/2), a subtle symmetry emerges between particles and holes. The original CF construction has been complemented by alternative formulations that emphasize particle-hole symmetry, including the idea of Dirac composite fermions, which offer a different way to understand the same phenomenology. These developments illustrate ongoing refinements rather than fundamental disputes about the core idea of flux-attached quasiparticles.
Quasiparticles and statistics: The fractional quantum Hall states supported by the CF framework carry fractional charge and anyonic statistics. The CF picture accounts for the observed charge quantization and braiding properties seen in experiments probing edge channels and interference phenomena, while remaining consistent with the broader topological understanding of FQHE phases.
Historical development and context
Precursor ideas and the Laughlin state: The discovery of the FQHE and the subsequent Laughlin wavefunction provided the essential groundwork for understanding correlated electron states at fractional filling. The Laughlin construction describes the 1/3 state and its close relatives and serves as a natural starting point for more general constructions.
From hierarchical pictures to composite fermions: Early approaches emphasized hierarchical constructions to generate a family of FQHE states. The composite fermion approach, introduced by Jainendra K. Jain, offered a unifying reinterpretation: many observed fractions are simply integer quantum Hall states of CFs. This viewpoint bridged previously separate descriptions and proved highly predictive across a wide range of experiments.
Experimental milestones: Measurements of Hall conductance plateaus at fractional values, activation gaps, and later manifestations such as a CF Fermi surface at ν = 1/2 provided strong empirical support for the CF picture. The development of high-mquality two-dimensional electron systems, including those in GaAs quantum wells and in newer materials, expanded the set of fractions where composite fermions offer a clear description.
Modern refinements and alternatives: The field has evolved to include refinements like the Dirac composite fermion formulation, which emphasizes particle-hole symmetry at half filling, and to acknowledge alternative viewpoints such as the traditional hierarchical picture. The overall consensus is that the CF framework provides a powerful, broadly applicable lens for understanding FQHE phenomena, even as details of specific states invite ongoing discussion.
Experimental evidence and implications
Fractional plateaus and energy gaps: A hallmark of CF theory is the spectrum of fractional Hall plateaus that align with the predicted fractions from the Jain sequence. The corresponding energy gaps grow with sample quality and magnetic field, consistent with the picture of CFs filling Λ-levels and producing robust incompressible states.
Half-filled Landau level and CF Fermi surface: At ν ≈ 1/2, where B* ≈ 0, experiments have detected signatures compatible with a Fermi sea of composite fermions, including commensurability oscillations and specific transport responses that resemble a Fermi liquid but for CFs in a vanishing effective field.
Spin and thickness effects: Real materials show that spin polarization can change with magnetic field and sample width, affecting which CF states are energetically favorable. These factors explain why some fractions are more pronounced in certain quantum wells or in materials with different internal degrees of freedom.
Extensions to other materials: The same physics appears in graphene and other two-dimensional systems, where the interplay of Dirac-like band structure, Landau quantization, and Coulomb interactions yields a CF-like description for observed fractional states, reinforcing the universality of the approach.
Controversies and debates
Nature of flux attachment: A core point of discussion is whether flux attachment represents a literal physical process or a convenient mathematical construction that recasts strong interactions into a weaker effective theory. Proponents emphasize predictive success and close agreement with a broad range of experiments, while critics note that the gauge-field language can obscure underlying microscopic details.
Competing pictures: The Haldane–Halperin hierarchical picture and related models offer alternative routes to the same fractions. Critics of any single framework point to cases where multiple descriptions can fit the data, highlighting that the underlying physics is robust, but the best single language may depend on the regime or observable.
Particle-hole symmetry and Dirac CFs: The recognition of particle-hole symmetry at ν = 1/2 has spurred the exploration of Dirac composite fermions, which reframe the problem in terms of a Dirac-like fermion coupled to a gauge field. This development reflects a healthy theoretical tension rather than a fundamental fracture, illustrating how new ideas refine the understanding without overturning established empirical success.
Non-Abelian candidates and the limits of CF theory: States such as ν = 5/2 are associated with proposals for non-Abelian anyons (e.g., Pfaffian-type states) that may extend beyond the simplest CF picture. While CF theory remains central, these potential non-Abelian states demonstrate that the full taxonomy of quantum Hall phases includes phenomena that test the boundaries of the CF framework.
Widespread acceptance and practical emphasis: In the broad scientific community, CF theory is widely regarded as the most successful and versatile framework for interpreting FQHE phenomena, with its core predictions borne out across a wide array of experimental platforms. Critics tend to focus on edge cases or on foundational interpretations, but the practical utility of the CF approach remains widely endorsed.
Applications and broader impact
Conceptual unification and pedagogy: The composite fermion framework provides a clean, transferable way to understand a large set of FQHE phenomena without resorting to a proliferation of distinct wavefunctions for each fraction. It serves as a teaching tool that connects the integer quantum Hall effect, Laughlin states, and higher-order fractions under a common umbrella.
Topological insights and quantum engineering: The ideas surrounding CFs—emergent gauge fields, topological order, and anyonic statistics—have influenced broader research in condensed matter physics and quantum engineering. The same themes inform discussions about topological qubits and potential directions for fault-tolerant quantum computation, even if non-Abelian states lie beyond the simplest CF description.
Materials science and device platforms: As experimental capabilities expand to different materials, including graphene and other two-dimensional crystals, the composite fermion viewpoint continues to offer a practical framework for predicting and interpreting transport measurements, enabling researchers to design experiments that probe CF dynamics and related topological phenomena.