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American Journal of Physics

Understanding the Laughlin wave function for the fractional quantum Hall effect

B. L. Johnson

Published Apr 1, 2002• 9 citations
Publisher

Abstract

We study the behavior of an interacting two-dimensional charged particle system in a transverse magnetic field and give a simple picture of the features that lead to the fractional quantum Hall effect. In particular, we discuss how the interparticle interactions give rise to an energy-level substructure in the lowest Landau level and develop a relatively simple model that gives insight into the analytical form of the Laughlin wave function. We analyze the model by transforming to a reference frame that rotates at half the cyclotron angular frequency. In the rotating frame, the vector potential for the magnetic field is replaced by a quadratic scalar potential. It is then evident that the system of interacting particles is a bound state, even in the absence of a charge-neutralizing background. We solve exactly a many-particle system in the rotating frame using a harmonic model interaction. In addition to being a good example for demonstrating basic methods in quantum mechanics, we show explicitly that the Laughlin function is an eigenstate of this model system.

DOI
10.1119/1.1446855

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We study the behavior of an interacting two-dimensional charged particle system in a transverse magnetic field and give a simple picture of the features that lead to the fractional quantum Hall effect. In particular, we discuss how the interparticle interactions give rise to an energy-level substructure in the lowest Landau level and develop a relatively simple model that gives insight into the analytical form of the Laughlin wave function. We analyze the model by transforming to a reference frame that rotates at half the cyclotron angular frequency. In the rotating frame, the vector potential for the magnetic field is replaced by a quadratic scalar potential. It is then evident that the system of interacting particles is a bound state, even in the absence of a charge-neutralizing background. We solve exactly a many-particle system in the rotating frame using a harmonic model interaction. In addition to being a good example for demonstrating basic methods in quantum mechanics, we show explicitly that the Laughlin function is an eigenstate of this model system.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
We study two-dimensional bosons and fermions in a magnetic field with vanishing range interactions. This model was previously proposed for fermions to illustrate the exactness and uniqueness of Laughlin's wave function as a ground state within the lowest Landau level. We show that the restriction to the lowest Landau level is not a valid approximation of these models for arbitrarily high (but finite) cyclotron frequency, since it predicts fractional quantum Hall effect (FQHE) at an incorrect filling factor. In particular, in contrast to the lowest-Landau-level theory, hard-core bosons are shown not to exhibit any FQHE. However, the set of filling factors for the FQHE are identical in the two cases, thus reaffirming the universality of the FQHE.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 13\frac{1}{3}31​ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.
A two-dimensional system of electrons with repulsive two-particle interaction is considered in a homogeneous magnetic field. Assuming the limit of high magnetic field the motion of the electrons is restricted to the lowest Landau level. We present quantum-mechanical wave functions for the many-particle system of N spin-polarized fermions on a parallelogram with periodic boundary conditions and with ${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$ quanta of magnetic flux flowing through its area. These wave functions are generalizations of the Laughlin wave function to more general rational values of the occupation fraction \ensuremath{\nu}=N/${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$. We divide the N electrons into n groups and assume Laughlin-Jastrow-type correlations for electrons within the same group, while electrons in different groups are correlated with a second Jastrow function. The corresponding wave functions exhibit a special structure that depends on \ensuremath{\nu} but is not affected by the interaction between the particles. Furthermore, we consider excited states with quasielectrons and quasiholes. Our theory leads to a unified view of the composite fermion theory of Jain and the hierarchy theory of Haldane, Laughlin, and Halperin. We present a qualitative argument why the fractions of Jain are observed preferably in the experiments.
A simple one-dimensional model is proposed, in which N spinless interacting fermions occupy M>NM>NM>N degenerate states on a circle. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors νN/M=1/(2m+1)\nu N/M=1/(2m+1)νN/M=1/(2m+1). Within this model the complimentary wavefunction for ν=l−I/(2m+1)\nu =l-I/(2m+1)ν=l−I/(2m+1) is found explicitly.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.