In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, the Laughlin wavefunction
[ pp. 210-213] is an
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural ansatzes or, from German, ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be ...
, proposed by
Robert Laughlin for the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
of a
two-dimensional electron gas
A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an Fermi gas, electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels ...
placed in a uniform background
magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
in the presence of a uniform
jellium background when the
filling factor of the
lowest Landau level is
where
is an odd positive integer. It was constructed to explain the observation of the
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 e^2/h, where ''e'' is the electron charge and ''h'' i ...
(FQHE), and predicted the existence of additional
states as well as quasiparticle excitations with fractional electric charge
, both of which were later experimentally observed. Laughlin received one third of the
Nobel Prize in Physics
The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
in 1998 for this discovery.
Context and analytical expression
If we ignore the jellium and mutual
Coulomb repulsion
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the ''electrostatic f ...
between the electrons as a zeroth order approximation, we have an infinitely degenerate lowest Landau level (LLL) and with a filling factor of 1/''n'', we'd expect that all of the electrons would lie in the LLL. Turning on the interactions, we can make the approximation that all of the electrons lie in the LLL. If
is the single particle wavefunction of the LLL state with the lowest
orbital angular momenta, then the Laughlin ansatz for the multiparticle wavefunction is
:
where position is denoted by
:
in (
Gaussian units
Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
)
:
and
and
are coordinates in the x–y plane. Here
is the
reduced Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
,
is the
electron charge
C, or c, is the third letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''cee'' (pronounced ), plural ''cees''.
History
"C ...
,
is the total number of particles, and
is the
magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
, which is perpendicular to the xy plane. The subscripts on z identify the particle. In order for the wavefunction to describe
fermion
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s, n must be an odd integer. This forces the wavefunction to be antisymmetric under particle interchange. The angular momentum for this state is
.
True ground state in FQHE at ''ν'' = 1/3
Consider
above: resultant
is a trial wavefunction; it is not exact, but qualitatively, it reproduces many features of the exact solution and quantitatively, it has very high
overlaps with the exact ground state for small systems. Assuming
Coulomb repulsion
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the ''electrostatic f ...
between any two electrons, that
ground state
can be determined using exact diagonalisation
and the
overlaps have been calculated to be close to one. Moreover, with short-range interaction (Haldane pseudopotentials for
set to zero),
Laughlin wavefunction becomes exact,
i.e.
.
Energy of interaction for two particles
The Laughlin wavefunction is the multiparticle wavefunction for
quasiparticle
In condensed matter physics, a quasiparticle is a concept used to describe a collective behavior of a group of particles that can be treated as if they were a single particle. Formally, quasiparticles and collective excitations are closely relate ...
s. The
expectation value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected va ...
of the interaction energy for a pair of quasiparticles is
:
where the screened potential is (see ')
:
where
is a
confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
and
is a
Bessel function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
of the first kind. Here,
is the distance between the centers of two current loops,
is the magnitude of the
electron charge
C, or c, is the third letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''cee'' (pronounced ), plural ''cees''.
History
"C ...
,
is the quantum version of the
Larmor radius
In physics, cyclotron motion, also known as gyromotion, refers to the circular motion exhibited by charged particles in a uniform magnetic field.
The circular trajectory of a particle in cyclotron motion is characterized by an angular frequency r ...
, and
is the thickness of the electron gas in the direction of the magnetic field. The
angular momenta of the two individual current loops are
and
where
. The inverse screening length is given by (
Gaussian units
Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
)
:
where
is the
cyclotron frequency, and
is the area of the electron gas in the xy plane.
The interaction energy evaluates to:
:
To obtain this result we have made the change of integration variables
:
and
:
and noted (see
Common integrals in quantum field theory)
:
:
:
The interaction energy has minima for (Figure 1)
:
and
:
For these values of the ratio of angular momenta, the energy is plotted in Figure 2 as a function of
.
References
{{reflist
See also
*
Landau level
*
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 e^2/h, where ''e'' is the electron charge and ''h'' i ...
*
Coulomb potential between two current loops embedded in a magnetic field
Hall effect
Condensed matter physics
Quantum phases