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Journal of Physics C Solid State Physics

Clustering and the ground state of interacting 2D electrons in a magnetic field

P. A. Maksym

Published Apr 30, 1986• 8 citations
Publisher

Abstract

Ground-state energies are computed for interacting 2D electrons subject to periodic boundary conditions in a magnetic field. When the aspect ratio of the periodic cell is small the ground-state energy as a function of magnetic field is found to have cusp-like minima when the number of flux quanta is a multiple of the number of electrons. However, the cusps at even-order fractional filling tend to disappear when the aspect ratio of the periodic cell is increased. This effect is explained in terms of a competition between direct and exchange interactions which favours clustered configurations of electrons when the aspect ratio is large.

DOI
10.1088/0022-3719/19/12/002

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Journal of Physics C Solid State Physics•Jun 20, 1984•5 citations
Ground-state energies are computed for interacting 2D electrons subject to periodic boundary conditions in a magnetic field. When the aspect ratio of the periodic cell is small the ground-state energy as a function of magnetic field is found to have cusp-like minima when the number of flux quanta is a multiple of the number of electrons. However, the cusps at even-order fractional filling tend to disappear when the aspect ratio of the periodic cell is increased. This effect is explained in terms of a competition between direct and exchange interactions which favours clustered configurations of electrons when the aspect ratio is large.
The energy of 2D Bloch electrons in a magnetic field is studied as a function of the filling fraction ν and the magnetic flux Φ. Using a new semi-classical quantization method the total energy E(Φ, ν) is calculated and shown to have an absolute minimum which corresponds to one flux quantum per particle. This optimal flux phenomenon is shown to occur under large conditions and for different lattices. An explicit cusp-like behavior of E νs. Φ at fixed ν is found both for the absolute and for the relative minima of E. Furthermore, the ground state energy is shown to be a smooth function of ν. The implications of our results for the stabilization of Anyons and the flux states are discussed.
The authors calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. They find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. They interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.
We calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. We find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. We interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.
ABSTRACT: We review recent studies on the energetics of fermions confined to a two dimensional square lattice, and the relations of these results to mean-field approaches to the t−J model. Our goal has been to compute the kinetic energy of the Fermi sea of the spinless fermions for any value of the (1) fermion concentration, (2) magnetic flux, and (3) frustration. For the unfrustrated case, we confirm that the ground state energy, χ(Φ), is a minimum for Φ=π(1−δ), which corresponds to one flux quantum per spinless fermion. We then proceed to do a systematic study of frustration effects, coming from longer range couplings, which modify the picture obtained for the unfrustrated case. The frustrating influence of the kinetic energy of the holes (e.g., by breaking magnetic bonds and suppressing the long-range order present in the undoped systems) is the main focus of this work. We find that, in general, E(Φ) always exhibits cusp-like minima which position moves linearly as a function of the fermion density x. Frustration can induce a competition between different local minima. By first considering the local minima for one particle only, we can understand most of the qualitative features of E(Φ). These local minima occur at simple rational fractions of Φ0, and when the flux slightly deviates from these values a one-particle Landau level structure develops. It is precisely such a spectrum that generates a family of cusps that “move away” from the original flux value as x is increased. Every cusp corresponds to an integer number of filled Landau levels, and the minimum energy cusp corresponds to the one level case. Furthermore, we use perturbation theory, valid for low fermion density x, in order to analyze quantitatively the behavior of the cusp-like energy minima; which originate from the Landau level structure when the flux is close to a rational value. If the flux is slightly away from a given rational value [Formula: see text] each of the q subbands generates a secondary Landau level structure. We have derived a t2−t3 phase diagram indicating regions of similar behavior (i.e., adiabatic continuations can be performed with each region, preserving the E(Φ) structure) and the boundaries between them. We have studied several points belonging to those boundaries and found that anomalous behavior, (e.g., cancelation of the k2 term in the dispersion relation) induced by frustration, can occur.
We report finite cluster exact diagonalization studies under periodic boundary conditions of two layers of electrons in a strong magnetic field at a half-filled Landau level. For the system studied by Suen et al., the ground states have more than 97 per cent of its spectral weight in the symmetric single particle levels. The eigenfunctions and eigenvalues up to a constant factor for different aspect ratios are essentially the same as previous numerical results of a single layer of electrons of zero thickness at half-filling.