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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 \nu=1/n where n is an odd positive integer. It was constructed to explain the observation of the \nu=1/3
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 \nu = 1/n states as well as quasiparticle excitations with fractional electric charge e/n, 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 \psi_0 is the single particle wavefunction of the LLL state with the lowest orbital angular momenta, then the Laughlin ansatz for the multiparticle wavefunction is : \langle z_1,z_2,z_3,\ldots , z_N \mid n,N\rangle = \psi_(z_1,z_2, z_3, \ldots, z_N ) = D \left \prod_\left( z_i-z_j \right)^n \right\prod^N_\exp\left( - \mid z_k \mid^2 \right) where position is denoted by : z= \left( x + iy\right) 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 ...
) : \mathit l_B = \sqrt and x and y are coordinates in the x–y plane. Here \hbar 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 ...
, e 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 ...
, N is the total number of particles, and B 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 n\hbar .


True ground state in FQHE at ''ν'' = 1/3

Consider n=3 above: resultant \Psi_L(z_1,z_2, z_3, \ldots, z_N)\propto \Pi_ (z_i-z_j)^3 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 \Psi_ 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 m>3 set to zero), Laughlin wavefunction becomes exact, i.e. \langle \Psi_, \Psi_L\rangle=1.


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 : \langle V \rangle = \langle n, N \mid V \mid n, N\rangle, \; \; \; N=2 where the screened potential is (see ') : V\left( r_\right) = \left( \right) \int_0^ \; M \left ( \mathit l + 1, 1, - \right) \;M \left ( \mathit l^ + 1, 1, - \right) \;\mathcal J_0 \left ( k \right) where M 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 \mathcal J_0 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, r_ is the distance between the centers of two current loops, e 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 ...
, r_= \sqrt \mathit l_B 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 L_B is the thickness of the electron gas in the direction of the magnetic field. The angular momenta of the two individual current loops are \mathit l \hbar and \mathit l^ \hbar where \mathit l + \mathit l^ = n. 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 ...
) : k_B^2 = where \omega_c is the cyclotron frequency, and A 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 : u_ = and : v_ = and noted (see Common integrals in quantum field theory) : \int d^2z_1 \; d^2z_2 \; \mid z_1 - z_2 \mid^ \; \exp \left - 2 \left( \mid z_1 \mid^2 + \mid z_2\mid^2 \right) \right\;\mathcal J_0 \left ( \sqrt\; \right) = : \int d^2u_ \; d^2v_ \; \mid u_\mid^ \; \exp \left - 2 \left( \mid u_\mid^2 + \mid v_\mid^2 \right) \right\;\mathcal J_0 \left ( k\mid u_ \mid \right) = : M \left ( n + 1, 1, - \right) . The interaction energy has minima for (Figure 1) : =, , , \mbox and : =, , , \mbox For these values of the ratio of angular momenta, the energy is plotted in Figure 2 as a function of n .


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