Composite Fermions: Fundamentals & Experiments

• Composite fermions are emergent quasiparticles formed by binding an even number of magnetic flux quanta to electrons in a 2D system, creating an effective Fermi sea at specific filling factors.
• Experimental observations through commensurability oscillations and geometric resonance confirm their well-defined Fermi contour and anisotropy transfer from the parent electron band.
• Their topological features, including a π Berry phase and interaction-driven effective mass, are key to understanding the fractional quantum Hall effect and related quantum phases.

Composite fermions (CFs) are emergent quantum particles formed by binding an even number of magnetic flux quanta to electrons in a two-dimensional electron system (2DES) at high perpendicular magnetic field. This mechanism, known as "flux attachment," transforms the strongly correlated electron problem into an effective weakly interacting Fermi system, leading to a fundamental understanding of the fractional quantum Hall effect (FQHE) and related phenomena. CFs inherit many parametric features from their progenitor electrons, establish new forms of quantum order, mediate topological phases, and display nontrivial collective behaviors closely tied to the underlying quantum geometry and symmetry properties of the system.

1. Formation, Fermiology, and Basic Theory

When subjected to a large perpendicular magnetic field, electrons in a 2DES minimize their energy by attaching even numbers (typically two or four) of fictitious flux quanta and thereby transforming into composite fermions. This flux attachment alters the effective magnetic field experienced by the particles—at filling factor ν = 1/(2s), where s is an integer (s = 1 for two-flux, s = 2 for four-flux CFs)—the external field is fully compensated, and CFs move as if in zero average effective field, forming a "Fermi sea".

The canonical CF construction is summarized as follows:

• For electron filling ν and 2s flux quanta attached,

$\nu = \frac{p}{2sp + 1}$

The effective CF filling becomes $p$ (integer), mapping the FQHE of electrons to the integer quantum Hall effect (IQHE) of CFs.

The CF’s Fermi wave vector at ν = 1/2 for the fully spin-polarized case is

$k^* = \sqrt{4\pi n}$

where $n$ is the carrier density. The cyclotron radius of CFs is $R^* = (\hbar k^*)/(eB^*)$ with $B^* = B - B_{1/2}$. This enables the application of Fermi liquid intuition and direct geometric resonance (GR) probes.

2. Experimental Observation and Anisotropy Transfer

Composite fermion Fermi seas have been observed through commensurability oscillation experiments, Shubnikov–de Haas oscillations, and geometric resonance effects. These confirm the existence of a well-defined Fermi contour and the particle’s effective Fermi wave vector in the quantum Hall regime (Kamburov et al., 2014, Jo et al., 2017). GR minima appear when the CF cyclotron orbit becomes commensurate with a periodic external or emergent potential modulation (e.g., superlattice or bilayer-induced periodicity), directly measuring $k^*$.

A crucial question is whether CFs inherit characteristics of the parent electrons’ band structure, such as Fermi surface anisotropy or valley occupancy. Experiments in anisotropic AlAs quantum wells subjected to uniaxial strain confirmed that transport anisotropy survives the CF formation process but is typically reduced: resistance ratios track the square root of the zero-field electronic mass anisotropy, suggesting that the Coulomb interaction mapping modifies, but does not erase, band geometric effects (Gokmen et al., 2010, Jo et al., 2017):

$$\frac{m^*_{CF,\,\|}}{m^*_{CF,\,\perp}} \propto \sqrt{\frac{m_l}{m_t}}$$

This reduced transference, and the resulting geometric deformation of the CF Fermi contour, points to a subtle interplay between the emergent CF band geometry and the microscopic parent structure.

3. Topological Properties, Berry Phase, and Symmetry

It was demonstrated that the effective momentum-space manifold on which CFs reside has nonzero Chern number and Berry curvature, leading to symplectic, rather than Newtonian, equations of motion (Shi, 2017). The Berry curvature is uniformly distributed, and composite fermions at ν = 1/2 (or in generic half-filled Chern bands) acquire a $\pi$ Berry phase when their momentum winds around the Fermi surface. This is directly observed via Landau fan diagrams:

• A linear plot of CF Landau index n vs. inverse effective magnetic field $1/B^*$ shows an intercept of $–\frac{1}{2}$, which corresponds to a $\pi$ Berry phase (Pan et al., 2017).

Fundamentally, the (4+1)-dimensional Chern–Simons description encodes this topology, leading to predictions such as:

• The half-quantized Hall conductance of the CF Fermi sea,

$\sigma_{xy}^{cf} = \frac{1}{2} \frac{e^2}{h},$

• Symmetric treatment of electron and hole filling fractions due to the underlying manifold’s Chern number, with the mapping between n and n+1 Landau levels for filling ν and 1 – ν (Shi, 2017).

4. Interaction, Pairing, and CF Mass

CFs inherit an effective mass $m_{CF}$, set by electron–electron interactions and distinct from the band mass. $m_{CF}$ is typically extracted from the temperature dependence of resistance oscillation amplitudes (Dingle analysis). The effective mass increases with quantum well width (i.e., electron layer thickness), due to the softening of the in-plane Coulomb interaction:

$\hbar \omega_{CF} = \frac{\hbar e B_{eff}}{m_{CF}}$

A key finding is that $m_{CF}$ increases as the quantum well becomes wider, reflecting a decrease in the FQHS energy gap as the charge distribution is spread (Rosales et al., 2022). However, experimentally measured $m_{CF}$ values often exceed theoretical predictions, indicating that existing models may neglect important factors such as Landau level mixing and finite-thickness corrections.

Furthermore, pairing instabilities of CFs can drive transitions from a compressible CF Fermi sea to incompressible paired quantum Hall phases, as observed at even denominator filling factors (notably ν = 5/2 and in wide quantum wells at ν = 1/2 and ν = 1/4). The onset and robustness of these paired states (e.g., Pfaffian states) depend sensitively on parameters such as quantum well width and the symmetry of the charge distribution (Sharma et al., 2023, Nešković et al., 13 Jun 2024).

5. Quantum Geometry, Band Structure, and Generalized Theories

Contemporary CF theory recognizes the importance of the underlying band’s quantum geometry. In Chern bands or for FCIs, CFs are formed by binding vortices to Bloch electrons, resulting in an enlarged Hilbert space. Key geometric quantities such as the Berry curvature ($\Omega_\mathbf{k}$) and quantum metric ($g_\mathbf{k}$) enter the CF single-particle Hamiltonian, which reduces to a generalized Hofstadter problem at small momentum (Hu et al., 5 Aug 2025):

$$H_{\text{sing}}(\mathbf{k}, z_{v}) = \epsilon_\mathbf{k} + \left(\frac{\hbar^2}{2 m^* \ell_e^2 \ell_v^2}\right) [\mathcal{G}_\mathbf{k} + \Omega_\mathbf{k}] + \left(\frac{\hbar^2}{2 m^* \ell_e^2 \ell_v^2}\right) (2\mathcal{D} - z_v)(2\bar{\mathcal{D}} - \bar{z}_v)$$

Satisfying the "trace condition" $\mathcal{G}_\mathbf{k} + \Omega_\mathbf{k} = 0$ results in exactly flat CF bands, optimizing FCI stability. This band-geometric approach has enabled a close match between theory and exact diagonalization for twisted MoTe$_2$ (Hu et al., 5 Aug 2025).

6. Particle-Hole Asymmetry, Minority Carrier Models, and Layer/Spin Degrees of Freedom

Breaking of particle–hole symmetry in real systems is revealed in commensurability experiments and transport: resistance minima positions corresponding to CF cyclotron resonance appear asymmetric about ν = 1/2, explained quantitatively by assigning the CF density to the minority carriers in the lowest Landau level (Kamburov et al., 2014, Kamburov et al., 2014). The minority-carrier model is essential for accurately mapping the Fermi contour and for interpreting phenomena where spin or valley degrees of freedom play a role.

Relatedly, in bilayer quantum wells or systems with multiple valleys, the internal structure of CFs can encode two-component correlations, leading to new ground states, such as the two-component Ψ$_{111}$ state near ν = 1 and to paired states with interlayer excitonic correlations in graphene bilayers (Li et al., 2019, Mueed et al., 2016).

7. Quantum Hall Phases in Non-Landau-Level Systems and Future Directions

The CF paradigm has been extended to lattice systems, notably fractional Chern insulators (FCIs), where Bloch electrons form CFs through vortex binding. In such systems, the interplay between band topology, quantum geometry, and interactions is critical. The dipole picture of CFs and generalized preferred charge substitution formalism facilitate computationally efficient identification of stable FCIs with microscopic accuracy (Hu et al., 5 Aug 2025). The projected many-body CF wavefunctions show exceptional overlap with ground states found by exact diagonalization.

Moreover, the nature of CF dispersions and higher LL physics can drive qualitative differences in quantum Hall states: for example, in the 1LL, non-quasiconvex CF dispersions destabilize Laughlin-like states in favor of alternative topological orders, such as the fermionic Haffnian (Jin et al., 15 Jul 2024).

Table: Representative Quantities and Formulas

Property/Feature	Formula or Rule	Reference Section
CF cyclotron energy	$\hbar\omega_{CF} = \hbar e B_{eff} / m_{CF}$	Section 4
CF Fermi wave vector (ν=1/2)	$k^* = \sqrt{4 \pi n}$	Section 1, 2
Anisotropy transfer	$\alpha_{CF} = \sqrt{\alpha_{F}}$	Section 2
Fermi contour anisotropy model	$$V(x, y) \propto 1/\sqrt{\gamma^2 x^2 + y^2 / \gamma^2}$$	Section 2
Berry phase in CF Fermi sea	$\pi$ (signaled by Landau fan intercept $-1/2$)	Section 3
Hall conductance, half-filling	$\sigma_{xy}^{cf} = \frac{1}{2}(e^2/h)$	Section 3
Trace condition, FCI stability	$\mathcal{G}_\mathbf{k} + \Omega_\mathbf{k} = 0$	Section 5

Conclusion

Composite fermions constitute the central organizing concept for understanding the FQHE, quantum Hall ferromagnetism, and fractional Chern insulators. Their properties—emergent band structure, effective mass, geometric response, and topological pairing potential—are controlled by both microscopic material parameters and emergent quantum geometry. Modern research frames CFs as nontrivial, topologically decorated quasiparticles whose phenomenology unifies quantum Hall physics across Landau levels, lattice systems, and multi-component settings, while experimental and theoretical investigations continue to refine and challenge the boundaries of this correspondence.