Fractional quantum Hall effect
Encyclopedia
The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised 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
and possibly also fractional statistics. Its discovery and explanation were recognized by the 1998 Nobel Prize in Physics
.
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, a series of plateaus form in the Hall resistance
. Each particular value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta)
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.
The FQHE was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer
, in experiments performed on gallium arsenide heterostructures developed by Arthur Gossard
. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work.
Fractionally charged quasiparticles are neither boson
s nor fermion
s and exhibit anyon
ic 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
.
In 1995, the fractional charge of Laughlin quasiparticles was measured directly in a quantum antidot electrometer at Stony Brook University
, New York
. 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
, detected such quasiparticles carrying an electric current
, through measuring quantum shot noise
.
Both of these experiments are somewhat controversial.
A more recent experiment, which measures the quasiparticle charge extremely directly, appears beyond reproach.
The anyonic statistics, however, have neither been confirmed nor disproved by any of the experiments.
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.
Different FQH states all have the same symmetry
and cannot be described by symmetry breaking theory.
Thus FQH states represent new states of matter that contain a
completely new kind of order—topological order
.
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.
The new type of orders represented by FQH states greatly enrich our
understanding of quantum phases and quantum phase transitions.
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.
The explanation of this effect is open to new relations and interpretations of experiments in light of new theories. It is possible to consider that the FQHE really gives further evidence of the existence of magnetic flux quanta as detected in superconductivity. There are electromagnetic quanta of magnetic flux
and electric elementary charge
. It was shown that these fundamental quanta are determined by a physical geometric unified theory. Each particle carries an integer charge and an integer flux. Classically, if a particle crosses a line in a plane normal to its flux, there is a fractional relation between the charge and flux crossing the line and a corresponding fractional relation between the current and induced voltage if there are no resistive losses. Quantum mechanically, the Hall effect problem may be treated by filling the quantum Landau energy levels
produced by the presence of the magnetic field. The electrons carrying flux quanta may combine to form rotating systems or vortices which act as charge/flux carriers. It has been shown that the Hall electric conductivity is directly proportional to charge/flux ratios of these carriers. Consequently, in accordance with the quantum aspects of the unified theory it is not necessary to assume the existence of fundamental fractional charges.
. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractionalFractionalization
In physics, fractionalization is the phenomenon whereby the quasiparticles of a system cannot be constructed as combinations of its elementary constituents...
elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
and possibly also fractional statistics. Its discovery and explanation were recognized by the 1998 Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...
.
Introduction
The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional system of electrons. In particular magnetic fields, the electron gas2DEG
A two-dimensional electron gas is a gas of electrons free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion in that direction, which can then be ignored for most problems. Thus the electrons appear to be a 2D sheet...
condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low carrier
Charge carrier
In physics, a charge carrier is a free particle carrying an electric charge, especially the particles that carry electric currents in electrical conductors. Examples are electrons and ions...
concentration, and extremely low temperatures. As in the integer quantum Hall effect, a series of plateaus form in the Hall resistance
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...
. Each particular value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta)
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 quasiparticleQuasiparticleIn physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
s: this theory, proposed by LaughlinRobert B. LaughlinRobert Betts Laughlin is a professor of Physics and Applied Physics at Stanford University. Along with Horst L. Störmer of Columbia University and Daniel C. Tsui of Princeton University, he was awarded a share of the 1998 Nobel Prize in physics for their explanation of the fractional quantum Hall...
, 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 quasiparticleQuasiparticleIn physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
s: Bertrand Halperin conjectured, and Daniel Arovas, J. R. 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 
(together with an Aharonov-Bohm phase factor) when identical quasiparticles are exchanged in a counterclockwise sense.
- 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.
- Composite fermionsComposite fermionsA composite fermion is the bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta...
: this theory was proposed by JainJainendra K. JainJainendra K. Jain is an Indian American physicist, who is currently the Erwin W. Mueller Professor of Physics at Pennsylvania State University. Jain is known for his theoretical work on quantum many body systems, most notably for postulating Composite fermions....
, and further extended by HalperinBertrand HalperinBertrand I. Halperin is the Hollis Professor of Mathematics and Natural Philosophy at the physics department of Harvard University.He grew up in Crown Heights, Brooklyn. He attended Harvard University , and did his graduate work at Berkeley with John J. Hopfield .In the 1970s, he, together with...
, 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 partially 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. Composite fermion theory provides a complementary description of the Laughlin and hierarchy states. It gives trial wave functions which, though not identical to those produced from the hierarchy picture (the wave functions for the Laughlin states are identical), are in the same universality class, as shown by Read. There are no experimental tests for fractional quantum Hall states that, even in principle, allow one to confirm the composite fermion description while excluding the hierarchy description.
The FQHE was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer
Horst Ludwig Störmer
Horst Ludwig Störmer is a German physicist who shared the 1998 Nobel Prize in Physics with Daniel Tsui and Robert Laughlin. The three shared the prize "for their discovery of a new form of quantum fluid with fractionally charged excitations"...
, in experiments performed on gallium arsenide heterostructures developed by Arthur Gossard
Arthur Gossard
Arthur C. Gossard is a professor of Materials and Electrical Engineering at the University of California, Santa Barbara. His research is related to molecular beam epitaxy . He has a doctorate from UC Berkeley. He is a member of the United States National Academy of Engineering.-External links:*...
. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work.
Fractionally charged quasiparticles are neither boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
s nor fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s and exhibit anyon
Anyon
In physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
ic statistics. The fractional quantum Hall effect continues to be influential in theories about topological order
Topological order
In physics, topological order is a new kind of order in a quantum state that is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations...
. Certain fractional quantum Hall phases appear to have the right properties for building a topological quantum computer
Topological quantum computer
A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines cross over one another to form braids in a three-dimensional spacetime . These braids form the logic gates that make up the 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
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....
, New York
New York
New York is a state in the Northeastern region of the United States. It is the nation's third most populous state. New York is bordered by New Jersey and Pennsylvania to the south, and by Connecticut, Massachusetts and Vermont to the east...
. In 1997, two groups of physicists at the Weizmann Institute of Science
Weizmann Institute of Science
The Weizmann Institute of Science , known as Machon Weizmann, is a university and research institute in Rehovot, Israel. It differs from other Israeli universities in that it offers only graduate and post-graduate studies in the sciences....
in Rehovot
Rehovot
Rehovot is a city in the Center District of Israel, about south of Tel Aviv. According to the Israel Central Bureau of Statistics , at the end of 2009 the city had a total population of 112,700. Rehovot's official website estimates the population at 114,000.Rehovot was built on the site of Doron,...
, Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...
, and at the Commissariat à l'énergie atomique
Commissariat à l'Énergie Atomique
The Commissariat à l'énergie atomique et aux énergies alternatives or CEA, is a French “public establishment related to industrial and commercial activities” whose mission is to develop all applications of nuclear power, both civilian and military...
laboratory near Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, detected such quasiparticles carrying an electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
, through measuring quantum shot noise
Shot noise
Shot noise is a type of electronic noise that may be dominant when the finite number of particles that carry energy is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are of significance...
.
Both of these experiments are somewhat controversial.
A more recent experiment, which measures the quasiparticle charge extremely directly, appears beyond reproach.
The anyonic statistics, however, have neither been confirmed nor disproved by any of the experiments.
Impact of fractional quantum Hall effect
The FQH effect shows the limits of Landau's symmetry breakingSymmetry breaking
Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations , the choice will appear arbitrary...
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
Symmetry breaking
Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations , the choice will appear arbitrary...
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.
Different FQH states all have the same symmetry
and cannot be described by symmetry breaking theory.
Thus FQH states represent new states of matter that contain a
completely new kind of order—topological order
Topological order
In physics, topological order is a new kind of order in a quantum state that is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations...
.
The existence of FQH liquids indicates that there is a whole
new world beyond the paradigm of symmetry breaking
Symmetry breaking
Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations , the choice will appear arbitrary...
, waiting to be explored.
The FQH effect opened up a new chapter in condensed matter physics.
The new type of orders represented by FQH states greatly enrich our
understanding of quantum phases and quantum phase transitions.
The associated fractional charge, fractional statistics, non-Abelian statistics,
chiral edge states, etc demonstrate the power and the fascination of emergence
Emergence
In philosophy, systems theory, science, and art, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems....
in many-body systems.
The explanation of this effect is open to new relations and interpretations of experiments in light of new theories. It is possible to consider that the FQHE really gives further evidence of the existence of magnetic flux quanta as detected in superconductivity. There are electromagnetic quanta of magnetic flux
Magnetic flux quantum
The magnetic flux quantum Φ0 is the quantum of magnetic flux passing through a superconductor. The phenomenon of flux quantization was discovered B. S. Deaver and W. M. Fairbank and, independently, by R. Doll and M. Nabauer, in 1961...
and electric elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
. It was shown that these fundamental quanta are determined by a physical geometric unified theory. Each particle carries an integer charge and an integer flux. Classically, if a particle crosses a line in a plane normal to its flux, there is a fractional relation between the charge and flux crossing the line and a corresponding fractional relation between the current and induced voltage if there are no resistive losses. Quantum mechanically, the Hall effect problem may be treated by filling the quantum Landau energy levels
Landau quantization
Landau quantization in quantum mechanics is the quantization of the cyclotron orbits of charged particles in magnetic fields. As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels. The Landau levels are degenerate, with the number of electrons...
produced by the presence of the magnetic field. The electrons carrying flux quanta may combine to form rotating systems or vortices which act as charge/flux carriers. It has been shown that the Hall electric conductivity is directly proportional to charge/flux ratios of these carriers. Consequently, in accordance with the quantum aspects of the unified theory it is not necessary to assume the existence of fundamental fractional charges.
See also
- Laughlin wavefunctionLaughlin wavefunctionIn condensed matter physics, the Laughlin wavefunction is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium background when the filling factor of the lowest Landau level is...
- Hall probe
- Quantum Hall Effect
- Topological orderTopological orderIn physics, topological order is a new kind of order in a quantum state that is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations...
- Coulomb potential between two current loops embedded in a magnetic field