Laughlin wavefunction

In 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 \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 (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 in 1998 for this discovery.

In 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 \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 (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 in 1998 for this discovery. = \psi_{n,N}(z_1,z_2, z_3, \ldots, z_N ) = D \left[ \prod_{1 \leqslant i where position is denoted by z={1 \over 2 \mathit l_B} \left( x + iy\right) in (Gaussian units) \mathit l_B = \sqrt{\hbar c\over e B} and x and y are coordinates in the x–y plane. Here \hbar is the reduced Planck constant, e is the electron charge, N is the total number of particles, and B is the magnetic field, which is perpendicular to the xy plane. The subscripts on z identify the particle. In order for the wavefunction to describe fermions, 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_{i 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 between any two electrons, that ground state \Psi_{ED} 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_{ED}|\Psi_L\rangle=1 . H_{\text{parent}} = \sum_{i where: The sum is over all pairs of particles (i,j). P_{ij}(l) is the operator that projects the pair (i,j) onto a state with relative angular momentum l. V_l are positive coefficients ( V_l > 0 ) representing the energy cost for a pair to be found in the l-th relative angular momentum channel. For the parent Hamiltonian, only the V_l for l need to be non-zero. This Hamiltonian is a sum of positive semi-definite operators, so its energy eigenvalues are always non-negative. A ground state with zero energy can only exist if it is annihilated by every term in the sum. This means the ground state wavefunction |Ψ 0 ⟩ must satisfy: P_{ij}(l) |\Psi_0\rangle = 0 \quad \text{for all pairs } (i,j) \text{ and all } l The Laughlin ν = 1/m state, by its very construction, perfectly satisfies this condition. It contains no components where any pair of particles has relative angular momentum less than m. Therefore, the Laughlin state is an exact zero-energy eigenstate of this parent Hamiltonian. Furthermore, for a given number of particles, it can be shown that the Laughlin state is the unique, densest state (i.e., the state with the most particles per unit of magnetic flux) that satisfies this set of conditions. This makes it the unique ground state of this idealized Hamiltonian. This formalism is extremely powerful. It proves that there exists a local Hamiltonian for which the Laughlin state is the exact ground state, solidifying its physical relevance. It also provides a crucial tool for exact diagonalization studies. The ground state of a more realistic interaction, like the Coulomb potential, can be computed numerically and its overlap with the ideal Laughlin state can be calculated. A large overlap indicates that the Laughlin state is an excellent approximation to the true ground state of the system. = \langle n, N \mid V \mid n, N\rangle, \; \; \; N=2 where the screened potential is (see ) V\left( r_{12}\right) = \left( { 2 e^2 \over L_B}\right) \int_0^{\infty} {{k\;dk \;} \over k^2 + k_B^2 r_{B}^2 } \; M \left ( \mathit l + 1, 1, -{k^2 \over 4} \right) \;M \left ( \mathit l^{\prime} + 1, 1, -{k^2 \over 4} \right) \;\mathcal J_0 \left ( k{r_{12}\over r_{B}} \right) where M is a confluent hypergeometric function and \mathcal J_0 is a Bessel function of the first kind. Here, r_{12} is the distance between the centers of two current loops, e is the magnitude of the electron charge, r_{B}= \sqrt{2} \mathit l_B is the quantum version of the Larmor radius, 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^{\prime} \hbar where \mathit l + \mathit l^{\prime} = n . The inverse screening length is given by (Gaussian units) k_B^2 = {4 \pi e^2 \over \hbar \omega_c A L_B} 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_{12} = {z_1 - z_2 \over \sqrt{2} } and v_{12} = {z_1 + z_2 \over \sqrt{2} } and noted (see Common integrals in quantum field theory) {1 \over \left( 2 \pi\right)^2\; 2^{2n} \; n! } \int d^2z_1 \; d^2z_2 \; \mid z_1 - z_2 \mid^{2n} \; \exp \left[ - 2 \left( \mid z_1 \mid^2 + \mid z_2\mid^2 \right) \right] \;\mathcal J_0 \left ( \sqrt{2}\; { k\mid z_1 - z_2 \mid } \right) = {1 \over \left( 2 \pi\right)^2\; 2^{n} \; n! } \int d^2u_{12} \; d^2v_{12} \; \mid u_{12}\mid^{2n} \; \exp \left[ - 2 \left( \mid u_{12}\mid^2 + \mid v_{12}\mid^2 \right) \right] \;\mathcal J_0 \left ( {2} k\mid u_{12} \mid \right) {\mathit l \over n} ={2\over3}, {3\over 5}, {4\over 7}, \mbox{etc.} For these values of the ratio of angular momenta, the energy is plotted in Figure 2 as a function of n .

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