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International Journal of Modern Physics B

Anyons: a possible mechanism for binding charged particles

D. Schmeltzer, A. R. Bishop

Published Jun 1, 1991• 2 citations
Publisher

Abstract

The hard core boson gas in (2+1)-dimensions in the presence of a Chern-Simons term is considered, we find that the long wave description is given in terms of Coulomb (longitudinal) and transversal electrodynamics. The longitudinal part is described in terms of electric charges which carry vorticity. When binding of electrical charges occurs the transversal part is described by a photon field and corresponds to a superfluid. Using the same language we find that a spinless fermion is described by a massive photon. We use these concepts to understand the relation between the negative-U Hubbard model in the presence of a magnetic field, and vortices which carry electric charge induced by doping in the repulsive Hubbard model.

DOI
10.1142/s0217979291001656

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•Jan 1, 1997•1 citation
The hard core boson gas in (2+1)-dimensions in the presence of a Chern-Simons term is considered, we find that the long wave description is given in terms of Coulomb (longitudinal) and transversal electrodynamics. The longitudinal part is described in terms of electric charges which carry vorticity. When binding of electrical charges occurs the transversal part is described by a photon field and corresponds to a superfluid. Using the same language we find that a spinless fermion is described by a massive photon. We use these concepts to understand the relation between the negative-U Hubbard model in the presence of a magnetic field, and vortices which carry electric charge induced by doping in the repulsive Hubbard model.
I consider a gas of ``free'' anyons with statistical parameter \ensuremath{\delta}, with hard cores, on a two-dimensional square lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on the same lattice coupled to a Chern-Simons gauge theory with coupling \ensuremath{\theta}=1/2\ensuremath{\delta}. At the semiclassical level, the system is found to be equivalent to a gas of fermions, with the same density, in an average effective magnetic field \ensuremath{\rho}/\ensuremath{\theta}. I consider the case in which an integer number of the Landau bands of the saddle-point problem are completely filled. If \ensuremath{\delta}=\ensuremath{\pi}/m and the density \ensuremath{\rho}=r/q, with m, r, and q integers, the system is a superfluid, provided that q is larger than twice the largest common factor of m and r. If q is even and the system is half filled, the state may be either a superfluid or a quantum Hall phase. For all other values of \ensuremath{\rho} and \ensuremath{\delta}, compatible with integer filling of the Landau bands, the system is in a quantum Hall phase. The dynamical stability of the superfluid state is ensured by the topological invariance of the quantized Hall conductance of the fermion problem. I find a close analogy between anyon superconductivity and the Schwinger mechanism. The effective Lagrangian for the low-energy modes coupled to the electromagnetic field is derived. The energies of fermion and flux states are logarithmically divergent, but finite for the anyon state. The system has flux quantization, a zero-temperature Hall effect with a quantized Hall conductance, Meissner effect, charged vortices, screening with induced magnetic fields for static charges, and different masses for the longitudinal and transverse components of the electromagnetic field.
We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions ’ charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the-translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. This lecture is based on ref.[1] and on previous work done on the problem of a translational invariant formulation of the Anyon mean field theory and its properties with respect to superfluidity and possibly superconductivity (for a general review of the previous work see ref.[2]). Actually, to discuss those issues in a proper way it is necessary to introduce a suitable mathematical formalism:
We investigate fermion liquids interacting with longitudinal and transverse Abelian gauge fields via bosonization. In two spatial dimensions we obtain the fermion quasiparticle propagator for the specific case of a Coulomb plus Chern-Simons gauge action. In the Halperin-Lee-Read theory of the half-filled Landau level this action describes composite fermions moving in zero net magnetic field. We identify the novel non-Fermi-liquid fixed point which controls the low-energy \ensuremath{\nu}=1/2 physics and show how Kohn's theorem is satisfied.
Several simple models of strongly correlated bosons on three-dimensional lattices have been shown to possess exotic fractionalized Mott insulating phases with a gapless ``photon'' excitation. In this paper we show how to view the physics of this ``Coulomb'' state in terms of the excitations of proximate superfluid. We argue for the presence of ordered vortex cores with a broken discrete symmetry in the nearby superfluid phase and that proliferating these degenerate but distinct vortices with equal amplitudes produces the Coulomb phase. This provides a simple physical description of the origin of the exotic excitations of the Coulomb state. The physical picture is formalized by means of a dual description of three-dimensional bosonic systems in terms of fluctuating quantum mechanical vortex loops. Such a dual formulation is extensively developed. It is shown how the Coulomb phase (as well as various other familiar phases) of three-dimensional bosonic systems may be described in this vortex loop theory. For bosons at half-filling and the closely related system of spin-1∕21∕21∕2 quantum magnets on a cubic lattice, fractionalized phases as well as bond- or ``box''-ordered states are possible. The latter are analyzed by an extension of techniques previously developed in two spatial dimensions. The relation between these ``confining'' phases with broken translational symmetry and the fractionalized Coulomb phase is exposed.
Anyon gas with interparticle (retarded) Coulomb interaction has been studied. The resulting system is shown to be a collection of dressed anyons, with a screening factor introduced in their spin. Close structural similarity with the Chern-Simons construction of anyons has helped considerably in computing the screening effect. Finally the present model is compared with the conventional Chern-Simons construction. The possible existence of particles having arbitrary spin and statistics has been proved quite some time back [1, 2]. However, dynamical model building, on the other hand, has proved to be rather controversial, with the dispute still continuing. Basically there are two broad lines along which the models are conceived: (i) The Chern-Simons (CS) construction [3], where a point charge is coupled to CS eletrodynamics. Removal of the auxiliary (or statistical) CS gauge field renders the particle anyonic. (ii) The construction of minimal anyon field equations, where one starts from very general physical postulates, such as the mass shell and Pauli-Lubanski condition for the particle [4]. A variant of the latter scheme is the spinning particle model [5, 6, 7], with which we are concerned in the present Letter. The connection between the latter two is elaborated in [8].