Fractional quantum Hall effect


The fractional quantum Hall effect is the observation of precisely quantized plateaus in the Hall conductance of 2-dimensional electrons at fractional values of, where e is the electron charge and h is the Planck constant.
At the same time, longitudinal resistance drops to zero as for the integer QHE.
It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. The 1998 Nobel Prize in Physics was awarded to Robert Laughlin, Horst Störmer, and Daniel Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations".
The microscopic origin of the FQHE is a major research topic in condensed matter physics.

Descriptions

The fractional quantum Hall effect is a collective behavior in a 2D system of electrons. In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low carrier concentration, and extremely low temperatures. As in the integer quantum Hall effect, the Hall resistance undergoes certain quantum Hall transitions to form a series of plateaus. Each particular value of the magnetic field corresponds to a filling factor
where p and q are integers with no common factors. Here q turns out to be an odd number with the exception of filling factor 5/2
and few others. The principal series of such fractions are
and their particle-hole conjugates
Depending on the fraction, both spin-polarised and zero-spin fractional QHE states may exist.
Fractionally charged quasiparticles are neither bosons nor fermions and exhibit anyonic statistics. The fractional quantum Hall effect continues to be influential in theories about topological order. Certain fractional quantum Hall phases appear to have the right properties for building a topological quantum computer.

History and developments

The FQHE was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer, in experiments performed on heterostructures made out of gallium arsenide developed by Arthur Gossard.
There were several major steps in the theory of the FQHE.
  • Laughlin states and fractionally-charged quasiparticles: this theory, proposed by Robert B. Laughlin, is based on accurate trial wave functions for the ground state at fraction as well as its quasiparticle and quasihole excitations. The excitations have fractional charge of magnitude.
  • Fractional exchange statistics of quasiparticles: Bertrand Halperin conjectured, and Daniel Arovas, John Robert Schrieffer, and Frank Wilczek demonstrated, that the fractionally charged quasiparticle excitations of the Laughlin states are anyons with fractional statistical angle ; the wave function acquires phase factor of when identical quasiparticles are exchanged in a counterclockwise sense. A recent experiment seems to give a clear demonstration of this effect.
  • Hierarchy states: this theory was proposed by Duncan Haldane, and further clarified by Bertrand Halperin, to explain the observed filling fractions not occurring at the Laughlin states'. Starting with the Laughlin states, new states at different fillings can be formed by condensing quasiparticles into their own Laughlin states. The new states and their fillings are constrained by the fractional statistics of the quasiparticles, producing e.g. and states from the Laughlin state. Similarly constructing another set of new states by condensing quasiparticles of the first set of new states, and so on, produces a hierarchy of states covering all the odd-denominator filling fractions. This idea has been validated quantitatively, and brings out the observed fractions in a natural order. Laughlin's original plasma model was extended to the hierarchy states by Allan H. MacDonald and others. Using methods introduced by Greg Moore and Nicholas Read, based on conformal field theory explicit wave functions can be constructed for all hierarchy states.
  • Composite fermions: this theory was proposed by Jainendra K. Jain, and further extended by Halperin, Patrick A. Lee and Read. The basic idea of this theory is that as a result of the repulsive interactions, two vortices are captured by each electron, forming integer-charged quasiparticles called composite fermions. The fractional states of the electrons are understood as the integer QHE of composite fermions. For example, this makes electrons at filling factors 1/3, 2/5, 3/7, etc. behave in the same way as at filling factor 1, 2, 3, etc. Composite fermions have been observed, and the theory has been verified by experiment and computer calculations. Composite fermions are valid even beyond the fractional quantum Hall effect; for example, the filling factor 1/2 corresponds to zero magnetic field for composite fermions, resulting in their Fermi sea.
Tsui, Störmer, and Robert B. Laughlin were awarded the 1998 Nobel Prize in Physics for their work.
Jain, James P. Eisenstein, and Mordehai Heiblum won the 2025 Wolf Prize in Physics "for advancing our understanding of the surprising properties of two-dimensional electron systems in strong magnetic fields".

The Composite Fermion Hierarchies

While the Laughlin wavefunction provides an exceptionally accurate description for states at filling factors ν = 1/3, 1/5,..., a vast number of other fractional states have been experimentally observed. The vast majority of these can be understood within a single, unified framework known as the composite fermion theory, introduced by Jainendra K. Jain in 1989. This theory postulates that the complex problem of strongly interacting electrons in a magnetic field reduces to a simpler problem of weakly interacting quasiparticles, called composite fermions.
The central idea of the theory is a conceptual transformation: each electron captures an even number, 2p, of magnetic flux quanta to form a new quasiparticle, the composite fermion. This transformation has a profound effect:
  • The Aharonov–Bohm phase acquired by the electron for a closed loop is in part canceled by the Berry phase from the attached flux quanta or vortices. This effectively "screens" the external magnetic field from the perspective of the CFs.
  • The composite fermions then move in a much weaker effective magnetic field, B*.
  • The problem of strongly interacting electrons at a filling factor ν is mapped onto a problem of weakly interacting composite fermions at a new, effective filling factor ν*.
The relationship between the electron filling factor ν and the composite fermion filling factor ν* is given by the master equation:
where p is a positive integer, and the ± sign corresponds to the orientation of the attached flux. This single equation explains the emergence of entire sequences of FQHE states.

The Jain Sequences (ν < 1)

The most prominent FQHE states can be understood as the integer quantum Hall effect of composite fermions. In this scenario, the weakly interacting CFs completely fill n of their own emergent "composite-fermion Landau levels," leading to an integer filling factor for them: ν* = n. Substituting this into the master equation generates the Jain sequences of FQHE states:
  • The Main Sequence: For p=1 and the + sign, we get the main sequence of fractions:
  • * n=1: ν = 1 / = 1/3
  • * n=2: ν = 2 / = 2/5
  • * n=3: ν = 3 / = 3/7

...and so on, a sequence of states that has been extensively verified in experiments.
  • Particle-Hole Conjugate States: The second prominent sequence arises from the - sign, or, alternatively, via particle-hole symmetry. A state at filling factor ν is closely related to a state at 1-ν. The particle-hole conjugates of the main sequence n/ are:
  • * For ν=1/3, its conjugate is 1 - 1/3 = 2/3.
  • * For ν=2/5, its conjugate is 1 - 2/5 = 3/5.
  • * For ν=1/5, its conjugate is 1 - 1/5 = 4/5.
These sequences ν = n/ and their conjugates account for the vast majority of all observed odd-denominator FQHE states with ν < 1.

Higher-Order States

The composite fermion theory also admits higher-order states. The composite fermions themselves are fermions and can, in principle, form their own fractional quantum Hall states. The states of composite fermions carrying 4 flux quanta may be viewed as the fractional quantum Hall states of composite fermions carrying two flux quanta: for example, the states at n/ can be obtained from with. Similarly, fractional quantum Hall effect of composite fermions at produces, through, states at 4/11 and 5/13.

The Moore–Read Pfaffian State (ν = 5/2)

While the Laughlin wavefunction and the composite fermion theory successfully describe the primary odd-denominator FQHE states, a state was discovered at the even-denominator filling factor ν = 5/2. In 1991, Gregory Moore and Nicholas Read proposed a groundbreaking trial wavefunction, now known as the Moore–Read state or Pfaffian state, which has become the leading theoretical description for this enigmatic phase.
The Moore–Read state represents a fundamentally new type of quantum fluid. Its key physical idea is that the composite fermions do not form a simple Fermi sea, but instead form a p-wave paired state, analogous to the Cooper pairs in a p-wave superconductor. This pairing is the source of its unique and remarkable properties.

Wavefunction and Construction

The Moore–Read wavefunction is constructed for a system of N electrons and, like the Laughlin state, is built in the lowest Landau level. It has two essential components:
1. A standard Laughlin–Jastrow factor,, where m is an even integer. This factor ensures that the wavefunction is antisymmetric under electron exchange and keeps the electrons apart.
2. A Pfaffian term,. The Pfaffian is a polynomial that can be thought of as the "square root" of the determinant of a N x N anti-symmetric matrix. This mathematical object naturally encodes the pairing of particles. The term is the wavefunction for a pair of particles with relative angular momentum l=1.
The original construction by Moore and Read used techniques from conformal field theory. They showed that this wavefunction could be formally represented as a correlation function of operators in the Ising model CFT. This CFT connection provides a deep theoretical structure and allows for the properties of the quasiparticle excitations to be calculated rigorously.