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Anyon Mean Field as an Exact Limit of a Gauge Theory

Roberto Iengo A

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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:

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Mean Field Theory for Fermion-based U(2) Anyons

Patrick N. McGraw

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.

arXiv:hep-th/9606038•Jun 10, 1996•0 citations

GAUGE INVARIANCE AND BROKEN SYMMETRIES IN ANYON SUPERFLUIDS

D. Boyanovsky

We review aspects of broken symmetry and the nature of long range order in theories of anyons starting with bosons with a statistical interaction. We introduce a novel gauge invariant quantization scheme that allows the identification of local and gauge invariant order parameters. The connection between spin and statistics is reviewed and the consequences of broken symmetries in the anyon representation are discussed. An anyon gas is studied in the Bogoliubov approximation, it is determined that the ground state is a condensate of charge-flux composites with “quasi-long-range order” at zero temperature, a “weak” gap in the spectrum and finite helicity modulus. The system is disordered at nonzero temperatures. The disorder is not caused by Goldstone bosons but by the strong infrared behavior arising from the Coulomb interaction induced by the long-range statistical interaction. The properties of topological vortices in nonrelativistic and in relativistic Landau-Ginzburg theories are studied in detail. We study the physics of the mean-field ansatz and quasi-long range order in a simple exactly soluble relativistic model. This model exhibits a novel phenomenon of charge redistribution to the boundaries and restoration of translational invariance in the infinite volume limit. It also illuminates the physics of quasi-long-range order with a gap in the spectrum, statistical charge polarization by external magnetic fields and the role of “large” gauge transformations.

International Journal of Modern Physics A•Sep 30, 1992•12 citations

Toward Hamiltonian simulations of Maxwell-Chern-Simons theory: constant modes and gauge field truncation

Andrea Bulgarelli, Maria Cristina Diamantini, Nico Dichter +6 more

Maxwell-Chern-Simons (MCS) theory in $2+1$ dimensions provides a paradigmatic example of a topological gauge theory with both dynamical and topological degrees of freedom. Its Euclidean formulation suffers from a sign problem, making Hamiltonian numerical approaches particularly attractive. As a first step toward the non-perturbative Hamiltonian study of MCS theory, we investigate the constant mode sector on a spatial torus. Being analytically solvable in the continuum, it provides an ideal benchmark for understanding how the topological properties of the theory are encoded in a finite-dimensional lattice Hilbert space. We construct a finite-dimensional discretization of the torus of flat connections and show that the resulting lattice problem maps onto a generalized Harper-Hofstadter model with twisted boundary conditions. We identify the commensurability conditions under which the finite lattice exactly reproduces the magnetic translation algebra and the topological degeneracy of the continuum theory. A systematic analysis of gauge field truncation and its convergence toward the continuum limit is then presented.

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

D. M. Gaitonde, Sumathi Rao

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.

International Journal of Modern Physics B•Nov 20, 1992•0 citations

Bosonisation in any dimension

H. S. Sharatchandra

The mechanism underlying any bosonisation or fermionisation is exposed.It is shown that any local theory of fermions on a lattice in any spatial dimension greater than one is equivalent to a local theory of Ising spins coupled to a $Z_{2}$ gauge field.There is a close relation to the description of anyons using a Chern-Simons term.

arXiv:hep-th/9301130•Jan 31, 1993•1 citation

1D anyons in relativistic field theory

Arata Yamamoto

Abstract We study relativistic anyon field theory in 1+1 dimensions. While (2+1)-dimensional anyon fields are equivalent to boson or fermion fields coupled with the Chern–Simons gauge fields, (1+1)-dimensional anyon fields are equivalent to boson or fermion fields with many-body interaction. We derive the path integral representation and perform a lattice Monte Carlo simulation.

arXiv:1709.06323•Progress of Theoretical and Experimental Physics•Feb 26, 2018•0 citations
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:
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.
We review aspects of broken symmetry and the nature of long range order in theories of anyons starting with bosons with a statistical interaction. We introduce a novel gauge invariant quantization scheme that allows the identification of local and gauge invariant order parameters. The connection between spin and statistics is reviewed and the consequences of broken symmetries in the anyon representation are discussed. An anyon gas is studied in the Bogoliubov approximation, it is determined that the ground state is a condensate of charge-flux composites with “quasi-long-range order” at zero temperature, a “weak” gap in the spectrum and finite helicity modulus. The system is disordered at nonzero temperatures. The disorder is not caused by Goldstone bosons but by the strong infrared behavior arising from the Coulomb interaction induced by the long-range statistical interaction. The properties of topological vortices in nonrelativistic and in relativistic Landau-Ginzburg theories are studied in detail. We study the physics of the mean-field ansatz and quasi-long range order in a simple exactly soluble relativistic model. This model exhibits a novel phenomenon of charge redistribution to the boundaries and restoration of translational invariance in the infinite volume limit. It also illuminates the physics of quasi-long-range order with a gap in the spectrum, statistical charge polarization by external magnetic fields and the role of “large” gauge transformations.
Maxwell-Chern-Simons (MCS) theory in 2+12+12+1 dimensions provides a paradigmatic example of a topological gauge theory with both dynamical and topological degrees of freedom. Its Euclidean formulation suffers from a sign problem, making Hamiltonian numerical approaches particularly attractive. As a first step toward the non-perturbative Hamiltonian study of MCS theory, we investigate the constant mode sector on a spatial torus. Being analytically solvable in the continuum, it provides an ideal benchmark for understanding how the topological properties of the theory are encoded in a finite-dimensional lattice Hilbert space. We construct a finite-dimensional discretization of the torus of flat connections and show that the resulting lattice problem maps onto a generalized Harper-Hofstadter model with twisted boundary conditions. We identify the commensurability conditions under which the finite lattice exactly reproduces the magnetic translation algebra and the topological degeneracy of the continuum theory. A systematic analysis of gauge field truncation and its convergence toward the continuum limit is then presented.
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.
The mechanism underlying any bosonisation or fermionisation is exposed.It is shown that any local theory of fermions on a lattice in any spatial dimension greater than one is equivalent to a local theory of Ising spins coupled to a Z2Z_{2}Z2​ gauge field.There is a close relation to the description of anyons using a Chern-Simons term.