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Progress of Theoretical Physics Supplement

Where Has the Conformal Symmetry of Anyons Gone in a Magnetic Field?

Tadashi Awaji, Masahiro Hotta

Published Jan 1, 1992• 0 citations
Publisher

Abstract

It is known that a many-body system of free anyons has the conformal symmetry and that the symmetry is broken explicitly when an external magnetic field is applied to the system. However, we can show that a modified conformai symmetry still exists in the presence of the magnetic field and plays a crucial role in anyon energy spectrum.
DOI
10.1143/ptps.110.375

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We study the properties of particles with fractional statistics (anyons) in a uniform magnetic field, to investigate the interplay between the statistical and physical flux. By exact solution of the two-body case, both free and with a harmonic attraction, we find a very clear example of the way in which an external field can offset the frustration caused by the statistics. These solutions also give the second virial coefficient, and hence a thermodynamic signature of the same effect. Finally, the same approach is used to study the problem of many anyons in a magnetic field, in particular the ground state and excitation spectrum.
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 study the physics of the mean field ansatz in a simplified anyon model. The model is solved exactly in a finite geometry with particular attention to boundary conditions compatible with a mean statistical magnetic field. The spectrum consists of bosonic plasma-like density fluctuations with a gap. We find that the repulsive Coulomb interactions introduced by the fluctuations around the mean field force the statistical charge and magnetic field to become localized near the boundaries. Translational invariance, broken by the mean field, is restored in the infinite volume limit as the statistical charge and magnetic field escape to infinity. The polarization effects of an external magnetic field and the role of “large” gauge transformations are also studied.
Abstract The contribution of anyonic degrees of freedom emerging in the non-Abelian spin sector of a one-dimensional system of interacting fermions carrying both spin and SU ( N f ) orbital degrees of freedom to the thermodynamic properties of the latter is studied based on the exact solution of the model. For sufficiently small temperatures and magnetic fields the anyons appear as zero energy modes localized at the massive kink excitations (Tsvelik 2014 Phys. Rev. Lett . 113 066401). From their quantum dimension they are identified as spin- anyons. The density of kinks (and anyons) can be controlled by an external magnetic field leading to the formation of a collective state of these anyons described by a parafermion conformal field theory for large fields. Based on the numerical analysis of the thermodynamic Bethe ansatz equations we propose a phase diagram for the anyonic modes.