Fractional quantum Hall effect

The fractional quantum Hall effect (fractional QHE or FQHE) is the observation of precisely quantized plateaus in the Hall conductance of 2-dimensional (2D) electrons at fractional values of ${\displaystyle e^{2}/h}$, where e is the electron charge and h is the Planck constant. At the same time, longitudinal resistance drops to zero (for low enough temperatures) 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". ^{[1] [2]} The microscopic origin of the FQHE is a major research topic in condensed matter physics.

Descriptions

Unsolved problem in physics

What mechanism explains the existence of the ν=5/2 state in the fractional quantum Hall effect?

More unsolved problems in physics

The fractional quantum Hall effect (FQHE) 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 (the ratio of number of electrons to magnetic flux quanta corresponding to given area)

${\displaystyle \nu =p/q,\ }$

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 ^[3] and few others (7/2 or 2+3/8). The principal series of such fractions are

${\displaystyle {1 \over 3},{2 \over 5},{3 \over 7},{\mbox{etc.,}}}$

and their particle-hole conjugates

${\displaystyle {2 \over 3},{3 \over 5},{4 \over 7},{\mbox{etc.}}}$

Depending on the fraction, both spin-polarised and zero-spin fractional QHE states may exist. ^[4] 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 ${\displaystyle 1/q}$ as well as its quasiparticle and quasihole excitations. The excitations have fractional charge of magnitude ${\displaystyle e^{*}={e \over q}}$.
• 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 ${\displaystyle \theta ={\pi \over q}}$; the wave function acquires phase factor of ${\displaystyle e^{i\theta }}$ (together with an Aharonov-Bohm phase factor) when identical quasiparticles are exchanged in a counterclockwise sense. A recent experiment seems to give a clear demonstration of this effect. ^[5]
• 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' ${\displaystyle \nu =1/q}$. 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. ${\displaystyle \nu =2/5}$ and ${\displaystyle 2/7}$ states from the Laughlin ${\displaystyle \nu =1/3}$ 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, ^[6] 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. ^[7] Using methods introduced by Greg Moore and Nicholas Read, ^[8] based on conformal field theory explicit wave functions can be constructed for all hierarchy states. ^[9]
• 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 (or, in general, an even number of) 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". ^[10]

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 (CF) model, introduced by Jainendra K. Jain in 1989. ^[11] This model successfully reduces the complex problem of strongly interacting electrons in a magnetic field to a simpler problem of weakly interacting quasiparticles, called composite fermions.^{[ citation needed ]}

The central idea of the theory is a conceptual transformation: each electron captures an even number, 2p, of magnetic flux quanta and binds them to itself to form a new quasiparticle, the composite fermion. This transformation has a profound effect:^{[ citation needed ]}

• The Aharonov–Bohm phase acquired by a CF when it encircles another is canceled out by the phase from the attached flux quanta. 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:^{[ citation needed ]}

${\displaystyle \nu ={\frac {\nu ^{*}}{2p\nu ^{*}\pm 1}}}$

where p is a positive integer (typically 1), and the ± sign corresponds to the orientation of the attached flux. This single equation explains the emergence of entire sequences of FQHE states.

The Principal Jain Sequences (ν < 1)

The most prominent FQHE states can be understood as the integer quantum Hall effect (IQHE) 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:^{[ citation needed ]}

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

• Laughlin States and the Main Sequence: For p=1 (electrons bound to two flux quanta) and the + sign, we get the main sequence of fractions:
  • n=1: ν = 1 / (2*1 + 1) = 1/3 (This recovers the primary Laughlin state)
  • n=2: ν = 2 / (2*2 + 1) = 2/5
  • n=3: ν = 3 / (2*3 + 1) = 3/7

...and so on, a sequence of states that has been extensively verified in experiments. ^[12] The state ν=1/5 corresponds to p=2 and n=1.

• Particle-Hole Conjugate States: The second prominent sequence arises from the - sign, which is often conveniently described 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/(2n+1) 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 (from p=2), its conjugate is 1 - 1/5 = 4/5.

These sequences ν = n/(2n-1) and their conjugates account for the vast majority of all observed odd-denominator FQHE states with ν < 1.^{[ citation needed ]}

Higher-Order Hierarchies

The composite fermion model also predicts higher-order hierarchies. The composite fermions themselves are fermions and can, in principle, form their own fractional quantum Hall states. For instance, if the CFs at p=1 form a ν* = 1/3 Laughlin state, the resulting electron filling factor would be:${\displaystyle \nu ={\frac {1/3}{2(1/3)+1}}={\frac {1/3}{5/3}}=1/5}$ This demonstrates how the Laughlin state at ν=1/5 can be viewed as the FQHE of the quasiparticles of the ν=1/3 state, a concept first proposed in the Haldane-Halperin hierarchy theory and elegantly incorporated into the CF framework. ^[12]

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

While the Laughlin wavefunction and the composite fermion model successfully describe the primary odd-denominator FQHE states, a particularly fascinating state was discovered at the even-denominator filling factor ν = 5/2. ^[13] This state cannot be explained by the simple Laughlin theory. 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. ^[14]

The Moore–Read state represents a fundamentally new type of quantum fluid. Its key physical idea is that the composite fermions (at an effective filling factor of 1/2) 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 (where N must be even) and, like the Laughlin state, is built in the lowest Landau level. It has two essential components:

${\displaystyle \Psi _{\text{MR}}={\text{Pf}}\left({\frac {1}{z_{i}-z_{j}}}\right)\prod _{i<j}(z_{i}-z_{j})^{m}}$

1. A standard Laughlin–Jastrow factor, ${\displaystyle \prod (z_{i}-z_{j})^{m}}$, where m is an even integer (typically m=2 for the ν=5/2 state). This factor ensures that the wavefunction is antisymmetric under electron exchange (when combined with the Pfaffian's properties) and keeps the electrons apart.

2. A Pfaffian term, ${\displaystyle {\text{Pf}}(1/(z_{i}-z_{j}))}$. 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 ${\displaystyle 1/(z_{i}-z_{j})}$ is the wavefunction for a pair of particles with relative angular momentum l=1 (a p-wave pair).

The original construction by Moore and Read was highly innovative, using techniques from conformal field theory (CFT). 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.

Physical Properties and Significance

The Moore–Read state is not just another FQHE state; its properties are profoundly different from the Laughlin states.

• Pairing of Composite Fermions: It is the first realistic model of a topological p-wave paired state of fermions in two dimensions. This links the FQHE to concepts from superconductivity and superfluidity (like Helium-3).
• Non-Abelian Anyons: This is its most significant feature. The quasiparticle excitations of the Moore–Read state are predicted to obey non-Abelian statistics . When two Laughlin quasiparticles are exchanged, the system's wavefunction acquires a simple phase factor (Abelian statistics). In contrast, when two Moore–Read quasiparticles are exchanged, the final state of the system depends on the order in which the exchanges are performed. Braiding these non-Abelian anyons performs a rotation within a degenerate space of quantum states. The elementary excitations are often called Ising anyons due to the CFT connection.
• Candidate for Topological Quantum Computation: The existence of non-Abelian anyons makes the ν=5/2 state a leading physical candidate for realizing fault-tolerant topological quantum computation. A quantum bit (qubit) can be encoded in the degenerate ground state of several well-separated anyons, and quantum gates can be performed by physically braiding them around each other. Because the information is stored non-locally, it is intrinsically robust against local noise and errors. ^[15]

While the Moore–Read state is the leading theoretical candidate for the ν = 5/2 plateau, conclusively demonstrating its non-Abelian nature through experiment remains a major goal of condensed matter physics. Experiments measuring thermal Hall transport have provided strong evidence in favor of the Pfaffian state, but a definitive braiding experiment has not yet been achieved. ^[16]

Evidence for fractionally-charged quasiparticles

Experiments have reported results that specifically support the understanding that there are fractionally-charged quasiparticles in an electron gas under FQHE conditions.

In 1995, the fractional charge of Laughlin quasiparticles was measured directly in a quantum antidot electrometer at Stony Brook University, New York. ^[17] In 1997, two groups of physicists at the Weizmann Institute of Science in Rehovot, Israel, and at the Commissariat à l'énergie atomique laboratory near Paris, ^[18] detected such quasiparticles carrying an electric current, through measuring quantum shot noise ^{[19] [20]} Both of these experiments have been confirmed with certainty.^{[ citation needed ]}

A more recent experiment, ^[21] measures the quasiparticle charge. In 2020, interferometry experiments conducted by two different groups, at Paris ^[22] and Purdue, ^[23] were both able to probe and confirm the braiding statistics of anyons.

Impact

The FQH effect shows the limits of Landau's symmetry breaking theory. Previously it was held that the symmetry breaking theory could explain all the important concepts and properties of forms of matter. According to this view, the only thing to be done was to apply the symmetry breaking theory to all different kinds of phases and phase transitions. ^[24] From this perspective, the importance of the FQHE discovered by Tsui, Stormer, and Gossard is notable for contesting old perspectives.

The existence of FQH liquids suggests that there is much more to discover beyond the present symmetry breaking paradigm in condensed matter physics. Different FQH states all have the same symmetry and cannot be described by symmetry breaking theory. The associated fractional charge, fractional statistics, non-Abelian statistics, chiral edge states, etc. demonstrate the power and the fascination of emergence in many-body systems. Thus FQH states represent new states of matter that contain a completely new kind of order—topological order. For example, properties once deemed isotropic for all materials may be anisotropic in 2D planes. The new type of orders represented by FQH states greatly enrich our understanding of quantum phases and quantum phase transitions.

See also

• Hall probe
• Laughlin wavefunction
• Macroscopic quantum phenomena
• Quantum anomalous Hall effect
• Quantum Hall Effect
• Quantum spin Hall effect
• Topological order
• Fractional Chern insulator

Notes

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References

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