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Modern Physics Letters B

ON THE VALIDITY OF MEAN-FIELD THEORY FOR ANYONS ON A LATTICE

А. А. Овчинников, An. A. OVCHINNIKOV

Published Dec 30, 1992• 2 citations
Publisher

Abstract

We examine the validity of the mean-field approximation for anyons on a lattice at high density. The phase fluctuations for a large deviation from the Fermi statistics, in particular for the hard core bosons, are shown to be large. The importance of the phase fluctuations in different fermionic mean-field approaches for the antiferromagnetic Heisenberg model is stressed.

DOI
10.1142/s0217984992001666

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We examine the validity of the mean-field approximation for anyons on a lattice at high density. The phase fluctuations for a large deviation from the Fermi statistics, in particular for the hard core bosons, are shown to be large. The importance of the phase fluctuations in different fermionic mean-field approaches for the antiferromagnetic Heisenberg model is stressed.
We study a tight-binding model for anyons on a square lattice using a self-consistent mean-field theory, where the anyon gauge field is approximated by its expectation value. For any anyon density per site ${\mathrm{\ensuremath{\rho}}}_{0}$=p/q and statistics \ensuremath{\nu}=1-1/n, the anyon system has a linear collective mode if and only if q is greater than twice the largest common factor of p and n. The collective mode dispersion, \ensuremath{\omega}(k), is calculated numerically for n=1 (hard-core bosons) and n=2 (semions) at certain densities.
Some genreal features of the scattering of boson-based anyons with an added nonstatistical interaction are discussed. Periodicity requirements of the phase shifts are derived, and used to illustrate the danger inherent in separating these phase shifts into the well-known pure Aharanov-Bohm phase shifts, and an additional set which arises due to the interaction. It is proven that the added phase shifts, although due to the nonstatistical interaction, necessarily change as the statistical parameter is varied, keeping the interaction fixed. A hard-disk interaction provides a concrete illustration of these general ideas. In the latter part of the paper, scattering with an additional hard-disk interaction is studied in detail, with an eye towards providing a criterion for the validity of the mean-field approximation for anyons, which is the first step in virtually any treatment of this system. We find, consistent with previous work, that the approximation is justified if the statistical interaction is weak, and that it must be more weak for boson-based than for fermion-based anyons.
The mean field equations for anyons in an external homogeneous magnetic field are studied and two classes of mean field solutions are explicitly constructed. The first class of solutions, corresponding to one filled Landau level in the combined external and smeared out statistical magnetic fields, is compared to the corresponding exact groundstate solutions. The energies are found to agree well at low densities and when the statistical field is not too strong compared to the background field. At high densities the fermi pressure increases the mean field energy indefinitely. The mean field solutions also miss the lower part of the energy spectrum. To the second class of mean field solutions no corresponding anyon state is known. These solutions represent the anyonic continuation of the fermi state with two filled Landau levels.
We examine the validity of vector-mean-field theory (VMFT) for statistical transmutation on large lattices with a high density of particles per site (1/2 and 1/4). We take as a difficult test case the representation of hard-core bosons as fermions plus attached flux tubes. We use a variational Monte Carlo method to test the variational properties of the mean-field ground-state wave function against the predictions of the VMFT. We find a discrepancy of order 1 in the thermodynamic limit. This leads us to postulate that a better formulation on a lattice may be that of a renormalized vector-mean-field theory. We show that the renormalization coefficients can be understood by an analysis of the phase fluctuations (whose magnitude we estimate) of the long-range gauge interaction. These phase fluctuations are of order \ensuremath{\pi} on the lattice (thus leading to a breakdown of VMFT on the lattice) while they vanish in a continuum formulation. We give a detailed discussion of the qualitative differences of VMFT on the lattice versus the continuum. In particular, we examine the effect of having lines of zeros (lattice) versus points of zeros (continuum) for the nodes of the many-body wave function. In addition, a remarkable variational theorem is discovered for the ground-state wave function of the VMFT.