11.6 Fractional quantum Hall effect

Fundamentals of FQHE

The Fractional Quantum Hall Effect (FQHE) occurs when a two-dimensional electron gas, subjected to a strong perpendicular magnetic field at very low temperatures, develops quantized Hall conductance at fractional values of $e^2/h$. Unlike the Integer Quantum Hall Effect, which can be understood from single-particle physics alone, FQHE is fundamentally a many-body phenomenon driven by strong electron-electron interactions.

FQHE states exhibit properties with no classical analog: fractionally charged quasiparticles, topological ground state degeneracy, and anyonic exchange statistics. These features make FQHE one of the richest examples of topological order in condensed matter.

Quantum Hall effect review

The Integer Quantum Hall Effect (IQHE), discovered by von Klitzing in 1980, shows quantized Hall resistance at integer multiples of $h/e^2$. In a 2D electron system under a perpendicular magnetic field, electrons occupy quantized cyclotron orbits called Landau levels, each with a degeneracy proportional to the applied field. When an integer number of Landau levels are completely filled, the Hall conductance locks onto a plateau at $\sigma_{xy} = \nu e^2/h$ with $\nu$ an integer.

Transport in IQHE is carried by chiral edge states that propagate along the sample boundary. Because backscattering is forbidden by topology, these edge channels carry current without dissipation.

FQHE vs IQHE

The critical distinction: IQHE can be fully explained without interactions, while FQHE cannot exist without them. The correlations open an energy gap within a partially filled Landau level, stabilizing an incompressible quantum liquid.

Fractional filling factors

The filling factor is defined as the ratio of electron density $n_e$ to the density of flux quanta threading the sample:

$\nu = \frac{n_e}{B / \Phi_0} = \frac{n_e h}{eB}$

where $\Phi_0 = h/e$ is the magnetic flux quantum.

• The most prominent FQHE states follow the odd-denominator rule: $\nu = 1/3, 2/5, 3/7, 2/3, \ldots$
• A hierarchy of states exists. Primary fractions like $1/3$ and $1/5$ are the strongest. Daughter states like $2/5$ and $3/7$ arise from quasiparticle condensation on top of primary states.
• Rare even-denominator states at $\nu = 5/2$ and $7/2$ break the odd-denominator pattern and are candidates for hosting non-Abelian anyons.

Composite Fermions

Composite fermion (CF) theory, developed by Jain, reframes the strongly interacting electron problem as a weakly interacting problem of new quasiparticles. Each electron binds to an even number of magnetic flux quanta, forming a composite fermion that moves in a reduced effective magnetic field. The FQHE of electrons then maps onto the IQHE of composite fermions.

Composite fermion theory

The construction works as follows:

1. Start with electrons in a strong magnetic field $B$ at a fractional filling $\nu$.
2. Attach $2p$ flux quanta (with $p$ a positive integer) to each electron. This creates a composite fermion.
3. The composite fermions see a reduced effective field and can fill an integer number of their own Landau levels (called $\Lambda$-levels).
4. Integer filling of $\Lambda$-levels for composite fermions corresponds to fractional filling for the original electrons.

This mapping successfully predicts the vast majority of observed FQHE fractions and explains why certain fractions are stronger than others (they correspond to lower CF Landau level indices).

Effective magnetic field

Composite fermions experience an effective magnetic field:

$B_{\text{eff}} = B - 2p \, \Phi_0 \, n_e$

where $p$ is the number of flux pairs attached. The CF filling factor is:

$\nu_{\text{CF}} = \frac{n_e \Phi_0}{|B_{\text{eff}}|}$

The electron filling factor $\nu$ relates to the CF filling factor $n$ (an integer) by:

$\nu = \frac{n}{2pn \pm 1}$

For example, with $p = 1$ and $n = 1$, you get $\nu = 1/3$. With $n = 2$, you get $\nu = 2/5$. The $\pm$ sign accounts for whether $B_{\text{eff}}$ points parallel or antiparallel to $B$.

Chern-Simons transformation

The flux attachment procedure is made rigorous through a Chern-Simons gauge transformation. This introduces a statistical gauge field $a_\mu$ that mediates the flux attachment:

• The original electron Lagrangian is supplemented by a Chern-Simons term that binds $2p$ flux quanta to each particle.
• The transformed theory describes composite fermions coupled to both the external electromagnetic field and the Chern-Simons gauge field.
• At the mean-field level, the Chern-Simons flux cancels part of the external field, recovering the effective field picture above.
• Fluctuations around mean field can be treated perturbatively, allowing systematic calculation of gaps, collective modes, and response functions.

Laughlin Wavefunction

Robert Laughlin proposed a trial wavefunction in 1983 that captures the essential physics of the $\nu = 1/m$ FQHE states. It remains one of the most celebrated exact results in many-body physics, with numerical overlaps exceeding 99% with exact ground states for small systems.

Laughlin's ansatz

The wavefunction for $N$ electrons at filling $\nu = 1/m$ (with $m$ an odd integer) is:

$\Psi_m(\{z_i\}) = \prod_{i < j} (z_i - z_j)^m \, \exp\!\left(-\sum_i \frac{|z_i|^2}{4 l_B^2}\right)$

Here $z_i = x_i + iy_i$ is the complex coordinate of the $i$-th electron in the lowest Landau level, and $l_B = \sqrt{\hbar / eB}$ is the magnetic length.

Key features of this wavefunction:

• The $(z_i - z_j)^m$ factor enforces that electrons avoid each other strongly. The probability of finding two electrons at the same point vanishes as $|z_i - z_j|^{2m}$, much faster than for a simple Slater determinant.
• For $m$ odd, the wavefunction is antisymmetric under particle exchange, as required for fermions.
• Laughlin showed that $|\Psi_m|^2$ maps onto the Boltzmann weight of a classical 2D one-component plasma at a specific temperature. This plasma analogy proves the state is an incompressible liquid with uniform density.

Quasiparticle excitations

Excitations above the Laughlin ground state are created by inserting or removing flux:

• Quasiholes: Inserting one flux quantum at position $z_0$ multiplies the wavefunction by $\prod_i (z_i - z_0)$. This creates a localized deficit of charge $e/m$ (so $e/3$ at $\nu = 1/3$).
• Quasielectrons: Removing a flux quantum creates an excess of fractional charge. Their wavefunctions are more complex and less analytically tractable than quasiholes.
• Both types of excitation are anyons. Exchanging two quasiholes produces a phase factor beyond the usual $\pm 1$ of bosons or fermions.

The energy gap to create a quasihole-quasielectron pair sets the scale for the FQHE plateau width and activation energy measured in transport.

Fractional charge

For the $\nu = 1/m$ Laughlin state, quasiparticles carry charge:

$e^* = \frac{e}{m}$

This fractional charge is not an artifact of the theory. It has been directly confirmed through shot noise measurements at quantum point contacts. When current tunnels between edge states, the noise power scales with the charge of the tunneling quasiparticle. Experiments at $\nu = 1/3$ measured $e^* = e/3$, in excellent agreement with theory.

Charge fractionalization arises because the quasiparticle is a collective excitation of many electrons. No single electron splits; rather, the correlated ground state redistributes charge so that localized excitations carry a fraction of $e$.

Topological Order

FQHE states cannot be characterized by any local order parameter or broken symmetry. Instead, they possess topological order: a type of quantum order defined by long-range entanglement, robust ground state degeneracy on nontrivial manifolds, and anyonic excitations. This places FQHE outside the Landau symmetry-breaking classification of phases.

Ground state degeneracy

On a surface with nontrivial topology (such as a torus), the FQHE ground state becomes degenerate:

• For a Laughlin state at $\nu = 1/m$, the degeneracy on a torus is exactly $m$.
• This degeneracy is topological: it does not depend on system size, Hamiltonian details, or local perturbations. It depends only on the topology of the surface and the topological order of the state.
• The degenerate ground states cannot be distinguished by any local measurement. They differ only in their global quantum numbers (related to the center-of-mass momentum on the torus).
• Ground state degeneracy serves as a diagnostic for identifying and classifying different topological phases.

Edge states in FQHE

The bulk of an FQHE state is gapped, but the boundary supports gapless excitations:

• FQHE edge states are chiral: they propagate in one direction along the sample boundary, determined by the magnetic field orientation.
• Unlike IQHE edges (which are free-electron-like), FQHE edges are described by chiral Luttinger liquid theory. The charge carriers on the edge have fractional charge, and the tunneling density of states shows power-law behavior rather than a step function.
• Tunneling experiments between opposite edges reveal current-voltage characteristics $I \propto V^\alpha$ with a non-integer exponent $\alpha$ that depends on the filling factor.
• Edge states are central to transport measurements and interferometry experiments designed to probe anyonic statistics.

Anyonic statistics

In two dimensions, particle exchange is richer than in 3D. The fundamental group of particle configurations allows for exchange phases beyond $0$ (bosons) and $\pi$ (fermions):

• Exchanging two Laughlin quasiholes at $\nu = 1/m$ produces a phase $e^{i\pi/m}$. These are Abelian anyons because successive exchanges simply accumulate phases.
• For certain states (notably $\nu = 5/2$), the exchange operation is predicted to be a unitary matrix acting on a degenerate subspace of quasiparticle states. These are non-Abelian anyons, where the order of exchanges matters.
• Anyonic statistics are a direct consequence of topological order. They are robust against local perturbations, which is precisely what makes them attractive for quantum computing.

Experimental Observations

Observing FQHE requires extreme conditions: very high sample purity, millikelvin temperatures, and strong magnetic fields. The precision of the quantized Hall plateaus at fractional values rivals that of the integer effect, and detailed measurements have confirmed fractional charge, energy gaps, and edge state structure.

High-mobility samples

FQHE is only visible when disorder is weak enough that the interaction-driven gap is not washed out. This demands very high electron mobility:

• Mobilities exceeding $10^6 \, \text{cm}^2/\text{Vs}$ are typically required; the highest-quality GaAs/AlGaAs heterostructures reach $> 10^7 \, \text{cm}^2/\text{Vs}$.
• Modulation doping spatially separates the dopant ions from the 2D electron gas, reducing ionized impurity scattering.
• Samples are grown by molecular beam epitaxy (MBE), which provides atomic-layer precision and ultra-low defect densities.
• Sample quality directly determines which FQHE fractions are observable. Weaker states (higher-order daughters, even-denominator fractions) appear only in the cleanest samples.

Measurement techniques

• Temperatures in the millikelvin range (dilution refrigerator) and magnetic fields above ~10 T are standard.
• Four-terminal measurements separate longitudinal ($R_{xx}$) and Hall ($R_{xy}$) resistances. FQHE states show $R_{xx} \to 0$ and $R_{xy}$ quantized at $h/(\nu e^2)$.
• Tilted field experiments add an in-plane magnetic field component. Since FQHE depends only on the perpendicular component, tilting helps distinguish orbital from spin effects.
• Local probes such as single-electron transistors (SETs) can image the charge distribution and detect individual quasiparticles near the edges.

Fractional plateaus

The Hall resistance shows plateaus at:

$R_{xy} = \frac{h}{\nu \, e^2}$

for fractional $\nu$.

• The strongest plateaus in the lowest Landau level appear at $\nu = 1/3, 2/5, 3/7, 2/3, 3/5, \ldots$
• In the second Landau level, the $\nu = 5/2$ and $7/2$ states stand out as even-denominator exceptions, potentially hosting non-Abelian anyons.
• Activation gap measurements ($R_{xx} \propto e^{-\Delta / 2k_BT}$) extract the energy gap $\Delta$ for each state. Gaps are typically fractions of a kelvin in the strongest states and decrease for higher-order fractions.

Theoretical Models

Beyond the Laughlin wavefunction and composite fermion picture, several theoretical frameworks extend our understanding to the full zoo of observed FQHE states.

Hierarchy states

The Haldane-Halperin hierarchy builds daughter states on top of primary Laughlin states:

1. Start with a Laughlin state at $\nu = 1/m$.
2. Create quasiparticles (quasiholes or quasielectrons) above this ground state.
3. These quasiparticles themselves form a correlated liquid at a new fractional filling, producing a daughter FQHE state.
4. The process can be iterated, generating a hierarchy of fractions.

This construction predicts FQHE at $\nu = p/(2np \pm 1)$ and explains why odd-denominator fractions dominate: even-denominator states would require bosonic quasiparticle condensation, which is generically less stable.

Jain series

Jain's composite fermion approach produces the same set of fractions as the hierarchy but from a more physically transparent starting point:

$\nu = \frac{n}{2pn \pm 1}$

where $n$ is the number of filled CF Landau levels and $p$ counts flux pairs. For $p = 1$:

• $n = 1 \Rightarrow \nu = 1/3$
• $n = 2 \Rightarrow \nu = 2/5$
• $n = 3 \Rightarrow \nu = 3/7$

The CF picture also predicts the relative strength of different fractions: lower $n$ states have larger gaps and are more robust. Numerical studies confirm that Jain's CF wavefunctions have extremely high overlap with exact diagonalization results.

Moore-Read state

The $\nu = 5/2$ state does not fit into the standard Laughlin or Jain frameworks. Moore and Read proposed a Pfaffian wavefunction that pairs composite fermions in a manner analogous to BCS superconductivity:

$\Psi_{\text{MR}} = \text{Pf}\!\left(\frac{1}{z_i - z_j}\right) \prod_{i<j}(z_i - z_j)^2 \, e^{-\sum_i |z_i|^2 / 4l_B^2}$

where Pf denotes the Pfaffian (square root of the determinant of an antisymmetric matrix).

• The paired structure gives rise to non-Abelian anyonic excitations, specifically Majorana-like zero modes bound to quasiholes.
• Braiding these quasiholes performs unitary transformations on the degenerate ground state manifold, which is the basis for proposals of topological quantum computation.
• Experimental confirmation of the non-Abelian nature of the $5/2$ state remains an active area of research. Thermal Hall conductance measurements have provided partial evidence, but the question is not yet fully settled.

FQHE in Other Systems

While GaAs heterostructures remain the workhorse for FQHE studies, the phenomenon has been observed and predicted in several other platforms, each offering unique advantages.

Graphene and FQHE

Graphene's Dirac spectrum and additional internal degrees of freedom produce a distinctive FQHE landscape:

• FQHE has been observed in both suspended graphene and graphene encapsulated in hexagonal boron nitride (hBN), where mobilities can exceed $10^6 \, \text{cm}^2/\text{Vs}$.
• Graphene has a four-fold degeneracy from spin and valley, leading to an approximate SU(4) symmetry. This produces FQHE sequences not seen in GaAs, including states within the zeroth Landau level.
• Fractional quantum Hall ferromagnetism occurs when interactions spontaneously break the SU(4) symmetry, polarizing the system in spin, valley, or both.
• Tilted field experiments help disentangle orbital and Zeeman contributions, revealing the symmetry-breaking pattern.

Fractional Chern insulators

Fractional Chern insulators (FCIs) are lattice analogs of FQHE states that arise in topological flat bands without any external magnetic field:

• The role of the magnetic field is played by the Berry curvature of the band. When a nearly flat band has nonzero Chern number and is partially filled, interactions can stabilize FCI states.
• Theoretical predictions include both Abelian and non-Abelian FCI phases, as well as states with higher Chern numbers that have no direct FQHE counterpart.
• Experimental realizations are being pursued in moiré superlattices (twisted bilayer graphene, twisted MoTe$_2$), optical lattices with cold atoms, and engineered photonic systems. Recent experiments in twisted MoTe$_2$ have reported signatures of FCIs at zero magnetic field.

Non-Abelian states

Several FQHE filling factors are predicted to host non-Abelian anyons:

• The $\nu = 5/2$ Moore-Read state supports Majorana zero modes bound to its quasiholes. Four quasiholes share a two-dimensional degenerate Hilbert space, and braiding them performs non-commuting unitary operations.
• The Read-Rezayi states at $\nu = 12/5$ are predicted to support Fibonacci anyons, which are computationally universal: any quantum gate can be approximated to arbitrary accuracy by braiding Fibonacci anyons.
• Experimental signatures are sought through edge-state interferometry (Fabry-Pérot and Mach-Zehnder geometries), thermal Hall conductance quantization, and tunneling spectroscopy.

Applications and Future Directions

Topological quantum computing

Non-Abelian anyons offer a fundamentally different approach to quantum computation:

• Information is stored in the degenerate ground state of a collection of non-Abelian anyons. Because this degeneracy is topological, the qubit is inherently protected from local noise and decoherence.
• Quantum gates are performed by braiding anyons around each other. The gate depends only on the topology of the braid, not on the speed or exact path, providing built-in fault tolerance.
• Majorana zero modes at $\nu = 5/2$ could realize a subset of topological gates. Fibonacci anyons (if realized) would provide a computationally universal gate set.
• The main experimental challenges are reliably creating well-separated non-Abelian anyons and performing controlled braiding operations at the required millikelvin temperatures.

Fractional quantum Hall interferometry

Interferometry is the most direct route to detecting anyonic statistics:

• In a Fabry-Pérot interferometer, quasiparticles travel along edge states that encircle a confined region. The interference pattern depends on both the Aharonov-Bohm phase and the anyonic phase from quasiparticles trapped inside.
• Mach-Zehnder interferometers offer a complementary geometry but are more sensitive to dephasing from Coulomb interactions.
• Recent experiments have reported oscillation patterns consistent with anyonic braiding phases at $\nu = 1/3$, providing some of the first direct evidence for anyonic statistics.
• Coulomb blockade spectroscopy in quantum dots coupled to FQHE edges has independently measured fractional charge.

Novel fractional states

Active frontiers in FQHE research include:

• Bilayer and multilayer systems, where interlayer correlations can stabilize new states, including the $\nu = 1$ bilayer exciton condensate and paired states with interlayer coherence.
• Fractional quantum spin Hall effect in time-reversal-invariant topological insulators, where counter-propagating edge states with opposite spin could each carry fractional charge.
• Bosonic FQHE in systems of cold atoms or exciton-polariton condensates, where the statistics of the underlying particles differ but the topological physics can be analogous.
• Exploration of FQHE in new moiré materials continues to expand the catalog of platforms where correlated topological phases can be studied and potentially controlled.