The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the
Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of
. 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
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a funda ...
and possibly also fractional statistics. The 1998
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
was awarded to
Robert Laughlin,
Horst Störmer, and
Daniel Tsui
Daniel Chee Tsui (, born February 28, 1939) is a Chinese-born American physicist, Nobel laureate, and the Arthur Legrand Doty Professor of Electrical Engineering, Emeritus, at Princeton University. Tsui's areas of research include electrical pro ...
"for their discovery of a new form of quantum fluid with fractionally charged excitations"
Laughlin's explanation only applies to fillings
where
is an odd integer. The microscopic origin of the FQHE is a major research topic in condensed matter physics.
Introduction
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 electrons to
magnetic flux quanta
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
)
:
where p and q are integers with no common factors. Here ''q'' turns out to be an odd number with the exception of two filling factors 5/2 and 7/2. The principal series of such fractions are
:
and
:
There were several major steps in the theory of the FQHE.
*Laughlin states and fractionally-charged
quasiparticles: this theory, proposed by
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,
J. R. Schrieffer
John Robert Schrieffer (; May 31, 1931 – July 27, 2019) was an American physicist who, with John Bardeen and Leon Cooper, was a recipient of the 1972 Nobel Prize in Physics for developing the BCS theory, the first successful quantum th ...
, 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
(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.
*Hierarchy states: this theory was proposed by Duncan Haldane, and further clarified by 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 MacDonald and others. Using methods introduced by Moore and Read, based on
conformal field theory explicit wave functions can be constructed for all hierarchy states.
*
Composite fermions: this theory was proposed by
Jain, and further extended by
Halperin Halperin (sometimes spelled as Halparin) is a variation of the Jewish surname Heilprin. Both forms are Southern Yiddish for Heilbrun, that is the German city Heilbronn. The name is sometimes transliterated into the Cyrillic alphabet as Galperin. I ...
, 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.
The FQHE was experimentally discovered in 1982 by
Daniel Tsui
Daniel Chee Tsui (, born February 28, 1939) is a Chinese-born American physicist, Nobel laureate, and the Arthur Legrand Doty Professor of Electrical Engineering, Emeritus, at Princeton University. Tsui's areas of research include electrical pro ...
and
Horst Störmer, in experiments performed on
gallium arsenide heterostructures developed by
Arthur Gossard Arthur C. Gossard was a professor of materials and electrical engineering at the University of California, Santa Barbara. In 1982, he co-discovered the fractional quantum Hall effect. His research is related to molecular beam epitaxy (MBE). He has a ...
. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work.
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.
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
Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public research university in Stony Brook, New York. Along with the University at Buffalo, it is one of the State University of New York syste ...
,
New York. In 1997, two groups of physicists at the
Weizmann Institute of Science in
Rehovot,
Israel
Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
, and at the
Commissariat à l'énergie atomique laboratory near
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. ...
, detected such quasiparticles carrying an
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movin ...
, through measuring quantum
shot noise
Both of these experiments have been confirmed with certainty.
A more recent experiment, which measures the quasiparticle charge extremely directly, appears beyond reproach.
Impact of fractional quantum Hall effect
The FQH effect shows the limits of Landau's
symmetry breaking theory. Previously it was long believed that the symmetry breaking theory could explain all the important concepts and essential properties of all forms of matter. According to this view the only thing to be done is to apply the
symmetry breaking theory to all different kinds of phases and phase transitions.
From this perspective, we can understand the importance of the FQHE discovered by
Tsui, Stormer, and Gossard.
The existence of FQH liquids indicates that there is a whole
new world beyond the paradigm of
symmetry breaking, waiting to be explored.
The FQH effect opened up a new chapter 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 philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergen ...
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
In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases (Phase (matter), phases of matter at absolute zero, zero temperature). Contrary to classical phase transitions, quantum phase transitions can onl ...
.
See also
*
Hall probe
A Hall effect sensor (or simply Hall sensor) is a type of sensor which detects the presence and magnitude of a magnetic field using the Hall effect. The output voltage of a Hall sensor is directly proportional to the strength of the field. I ...
*
Laughlin wavefunction
*
Macroscopic quantum phenomena
Macroscopic quantum phenomena are processes showing quantum behavior at the macroscopic scale, rather than at the atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena are superfluidity and s ...
*
Quantum anomalous Hall effect
*
Quantum Hall Effect
*
Quantum spin Hall effect
*
Topological order
Notes
References
*
*
*
{{DEFAULTSORT:Fractional Quantum Hall Effect
Hall effect
Correlated electrons
Quantum phases
Mesoscopic physics
Unsolved problems in physics
Unexplained phenomena