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International Journal of Modern Physics B

EXACT MANY BODY FERMI-BOSE TRANSMUTATION IN 2+1 DIMENSIONS

D. M. Gaitonde, Sumathi Rao

Published Nov 20, 1992• 0 citations
Publisher

Abstract

We show that the low energy limit of relativistic fermions interacting with a statistical gauge field also includes a scalar interaction. When the Chern-Simons (CS) parameter µ=e2/2π and the scalar interaction is precisely that which is obtained through relativistic reduction, the many-body Hamiltonian can be solved exactly, directly in the fermion gauge, for the ground state energy which is zero and the ground state wavefunction which is gauge equivalent to one, characteristic of free bosons. Conversely, for N bosons interacting with a CS gauge field with µ=e2/2π, the mean-field ground state energy is πN2/m, which is characteristic of N free fermions.

DOI
10.1142/s0217979292001626

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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:
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
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The low-energy spin excitations in the flux phase of the (2+1)-dimensional large-nnn SU(n)\mathrm{SU}(n)SU(n)Heisenberg antiferromagnet are relativistic fermions coupled to a U(1) gauge field. These excitations are studied in the physical n=2n=2n=2 case with the use of a 1n\frac{1}{n}n1​ expansion. A Chern-Simons term is included to remove gauge configurations forbidden in the n=2n=2n=2 theory.