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arXiv:cond-mat/9810328 · Physical review. B, Condensed matter

Interacting electrons in magnetic fields: Tracking potentials and Jastrow-product wave functions

Gábor Fáth, Stephen B. Haley

Published Jul 15, 1998• 4 citations
arXivHTMLPDF

Abstract

The Schr\"odinger equation for an interacting spinless electron gas in a nonuniform magnetic field admits an exact solution in Jastrow product form when the fluctuations in the magnetic field track the fluctuations in the scalar potential. For tracking realizations in a two-dimensional electron gas, the degeneracy of the lowest Landau level persists and the ``tracking'' solutions span the ground state subspace. In the context of the fractional quantum Hall problem, the Laughlin wave function is shown to be a tracking solution. Tracking solutions for screened Coulomb interactions are also constructed. The resulting wave functions are proposed as variational wave functions with potentially lower energy in the case of non-negligible Landau level mixing than the Laughlin function.

DOI
10.1103/physrevb.58.1405

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The Schr"odinger equation for an interacting spinless electron gas in a nonuniform magnetic field admits an exact solution in Jastrow product form when the fluctuations in the magnetic field track the fluctuations in the scalar potential. For tracking realizations in a two-dimensional electron gas, the degeneracy of the lowest Landau level persists and the ``tracking'' solutions span the ground state subspace. In the context of the fractional quantum Hall problem, the Laughlin wave function is shown to be a tracking solution. Tracking solutions for screened Coulomb interactions are also constructed. The resulting wave functions are proposed as variational wave functions with potentially lower energy in the case of non-negligible Landau level mixing than the Laughlin function.
We study the two-dimensional electron gas in a quantizing magnetic field for the cases of Coulomb and harmonic interactions among the electrons. Numerical solutions for the quantum states of clusters of up to five electrons show that the strength of the excitation gap is a strongly oscillating function of the density not unlike what is observed in the anomalous quantum Hall effect. We present analytic results for the case of harmonic interactions and show that the variational wave function recently proposed by Laughlin for the Coulomb problem is in fact an exact eigenstate of the harmonic problem.
The topological morphology--order of zeros at the positions of electrons with respect to a specific electron--of Laughlin state at filling fractions 1/m1/m1/m (mmm odd) is homogeneous as every electron feels zeros of order mmm at the positions of other electrons. Although fairly accurate ground state wave functions for most of the other quantum Hall states in the lowest Landau level are quite well-known, it had been an open problem in expressing the ground state wave functions in terms of flux-attachment to particles, {\em a la}, this morphology of Laughlin state. With a very general consideration of flux-particle relations only, in spherical geometry, we here report a novel method for determining morphologies of these states. Based on these, we construct almost exact ground state wave-functions for the Coulomb interaction. Although the form of interaction may change the ground state wave-function, the same morphology constructs the latter irrespective of the nature of the interaction between electrons.
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.
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.
We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power mmm of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.