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arXiv:1704.01640

Transition of the uniform statistical field anyon state to the nonuniform one ar low particle densities

Matej Hudák, O. Hudák

Published Apr 6, 2017• 1 citation
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Abstract

Using results of our exact description of the spinless fermion motion in a nonhomogeneous magnetic field \( {\bf B} = B( 0, 0, 1/cosh^{2}( \frac{x-x_{0}}{ δ})) \) we study a gas of these particles moving in this field. For lower densities \( ν< ν_{c}(B, δ) \) the corresponding total energy is lower than that of the uniform field state. Thus when the density of anyons decreases a transition from the uniform statistical field state to the nonhomogeneous field state is predicted.

DOI
10.48550/arxiv.1704.01640

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Using results of our exact description of the spinless fermion motion in a nonhomogeneous magnetic field ( {\bf B} = B( 0, 0, 1/cosh^{2}( \frac{x-x_{0}}{ δ})) ) we study a gas of these particles moving in this field. For lower densities ( ν< ν_{c}(B, δ) ) the corresponding total energy is lower than that of the uniform field state. Thus when the density of anyons decreases a transition from the uniform statistical field state to the nonhomogeneous field state is predicted.
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 develop a density functional treatment of noninteracting Abelian anyons, which is capable, in principle, of dealing with a system of a large number of anyons in an external potential. Comparison with exact results for a few particles shows that the model captures the behavior qualitatively and semiquantitatively, especially in the vicinity of the fermionic statistics. We then study anyons with statistics parameter 1+1/n1+1/n1+1/n, which are thought to condense into a superconducting state. An indication of the superconducting behavior is the mean-field result that, for uniform density systems, the ground-state energy increases under the application of an external magnetic field independent of its direction. Our density functional theory based analysis does not find that to be the case for finite systems of anyons, which can accommodate a weak external magnetic field through density transfer between the bulk and the boundary rather than through transitions across effective Landau levels, but the ``Meissner repulsion'' of the external magnetic field is recovered in the thermodynamic limit as the effect of the boundary becomes negligible. We also consider the quantum Hall effect of anyons, and show that its topological properties, such as the charge and statistics of the excitations and the quantized Hall conductance, arise in a self-consistent fashion.
Diamagnetism of 2D-Fermions in the Strong Nonhomogeneous Static Magnetic Field (B=B(0,0,1/cosh2(x−x0δ)))( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))(B=B(0,0,1/cosh2(δx−x0​​))) : gas magnetization, ... and gas compressibility
We study diamagnetism of a gas of fermions moving in a nonhomogeneous magnetic field (B=B(0,0,1/cosh2(x−x0δ)))( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))(B=B(0,0,1/cosh2(δx−x0​​))) The gas magnetization, the static magnetic susceptibility, the chemical potential and the gas compressibility are discussed and compared with the uniform field case. General need to study dynamics of electrons in different types of magnetic fields follows from a large number of experimental situations in which its understanding enables physicists to obtain new information.
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A ground state energy variational calculation of anyon gas with Hamiltonian included the interaction of spins of particles with anyon vector potential induced, i.e. statistical, magnetic field exhibits exact cancelation of terms connected with fractional statistics. This leads to bosonization of anyons due to coupling of their spins with statistical magnetic field. We presume that at the dense gas fluctuations of effective spins destroy the coupling and bosons become anyons. At the assumption that pseudogap (PG) boundary is temperature independent and when anyons are fermions we use this model to interpret experimental phase diagrams of Tallon and Loram hole and electron doped High-T_c superconductors below PG energy E_g and find the qualitative and quantitative agreement. We do the hypothesis that phase transition (PT) of bosons into Bose-Einstein condensate is not of second order, but of first order, close to second one, PG regime is meta stable phase of bosons, and E_g=0 is the critical point of this PT. Bosons undergo PT into fermions on PG boundary. Described in the literature non-Fermi quasi-particles might be related to bosons with effective spins.
The energy density is computed for a U(2) Chern-Simons theory coupled to a non-relativistic fermion field (a theory of ``non-Abelian anyons'') under the assumptions of uniform charge and matter density. When the matter field is a spinless fermion, we find that this energy is independent of the two Chern-Simons coupling constants and is minimized when the non-Abelian charge density is zero. This suggests that there is no spontaneous breaking of the SU(2) subgroup of the symmetry, at least in this mean-field approximation. For spin-1/2 fermions, we find self-consistent mean-field states with a small non-Abelian charge density, which vanishes as the theory of free fermions is approached.