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Journal de physique

Ground state of the Fermi gas on 2D lattices with a magnetic field

R. Rammal, Jean Bellissard

Published Jan 1, 1990• 10 citations
Publisher

Abstract

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.

DOI
10.1051/jphys:0199000510190215300

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Fractional quantum Hall states on a square lattice

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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.

Journal of Physics Condensed Matter•Jun 10, 1991•5 citations
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.
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.
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.
By using a mean-field Hartree-Fock treatment, models of strongly correlated fermions on a square lattice are reduced to the problem of fermions propagating in a magnetic field. We compute the kinetic energy of the Fermi sea for any value of the fermion density, magnetic flux, and frustration. From these we obtain, among other things, the optimal flux associated with the global energy minimum and the location of the local energy minima. Every cusp corresponds to an integer number of filled Landau levels, and the minimum-energy cusp corresponds to the one-level case. The breaking of time-reversal symmetry, when the diagonal coupling is turned on, produces a family of asymmetric cusps around \ensuremath{\Phi}=${\mathrm{\ensuremath{\Phi}}}_{0}$/2, in the one-particle energy. We use perturbation theory, valid for low fermion density, in order to analyze quantitatively the behavior of the cusplike energy minima; these minima are due to the Landau-level structure when the flux is close to a rational value. We have derived a phase diagram indicating regions of similar behavior and the points that exhibit anomalous dispersion induced by frustration.
We present analytic formulas for the ground state energy of two-dimensional (2D) anyon gas in strong magnetic field (Landau level filling factor νL ≤ 1). The formulas are obtained by applying harmonic potential regularization for the vanishing confinement to harmonically confined Coulomb anyon gas. In a case of absence of the Coulomb interaction our analytic result provides an exact solution. It contains a contribution of the anyon gauge field characterized by the anyon parameters ν and νL. In a case of presence of the Coulomb interaction we introduce a function depending on the parameters ν, νL and the density parameter rs. The function is determined by fitting the Fano-Ortolani interpolation equation in the fractional quantum Hall effect regime for spin-polarized electrons and by consistence requirement with known results for the ground state energy of the 2D Coulomb Bose gas in strong magnetic field. We show that our formulas are valid not only for fermions (ν=1) but quite generally for anyons (0 ≤ ν ≤ 1).
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.