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International Journal of Modern Physics A

ON THE MEAN FIELD ANSATZ IN ANYON THEORIES

D. Boyanovsky

Published Jul 30, 1992• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217751x92002088

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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.
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.
We study the ground state of a large number N of 2D anyons in an external magnetic field. We consider a scaling limit where the statistics parameter α tends to zero when N tends to infinity which allows the statistics to be seen as a "perturbation from the bosonic end". Our model is that of bosons in a magnetic field and interacting through long-range magnetic potential generated by magnetic charges carried by each particle, smeared over discs of radius R. Our method allows to take R tends to not too fast at the same time as N tends to infinity. We use the information theoretic version of the de Finetti theorem of Brandão and Harrow to justify the so-called "average field approximation": the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.
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.
We study the ground state of a large number N of 2D extended anyons in an external magnetic field. We consider a scaling limit where the statistics parameter α is proportional to N−1 when N → ∞, which allows the statistics to be seen as a “perturbation around the bosonic end.” Our model is that of bosons in a magnetic field interacting through long-range magnetic potential generated by magnetic charges carried by each particle, smeared over discs of radius R. Our method allows us to take R → 0 not too fast at the same time as N → ∞ : R = N−1/4+ɛ. We use the information theoretic version of the de Finetti theorem of Brandão and Harrow to justify the so-called “average field approximation”: the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.
We study aspects of long-range order in a model consisting of two interacting layers of anyon gases with opposite statistical charge. We start with bosons with a statistical interaction and second quantize the theory using an approach that uses a local and gauge-invariant order parameter. We then study this model in the Bogoliubov approximation. We explicitly compute the depletion from the condensate and gauge-invariant correlation functions. We show that the ground state is a ``weak'' condensate of charge-flux composites with ``nearly long-range order.'' The condensate becomes statistically neutral in the thermodynamic limit. It is pointed out that the ``nearly long-range order'' at zero temperature has its origins in the logarithmic infrared singularities arising from the two-dimensional Coulomb repulsion induced by the long-range statistical interaction. The quasiparticle spectrum consists of one massless mode, and a plasmon mode with a weak gap. The system is quasiordered at any nonzero temperature, but we present evidence for a finite helicity modulus.
We introduce and analyze a lattice model of anyons in a periodic potential and an external magnetic field, which exhibits a transition from a Mott insulator to a quantum Hall fluid. The transition is characterized by the anyon statistics \ensuremath{\alpha}, which can vary between fermions, \ensuremath{\alpha}=0, and bosons, \ensuremath{\alpha}=1. For bosons the transition is in the universality class of the classical three-dimensional XY model. Near the fermion limit, the transition is described by a massless 2+1 Dirac theory coupled to a Chern-Simons gauge field. Analytic calculations perturbative in \ensuremath{\alpha}, and also a large-N expansion, show that due to gauge fluctuations, the critical properties of the transition are dependent on the anyon statistics. Comparison with previous calculations at and near the boson limit, strongly suggest that our lattice model exhibits a fixed line of critical points, with universal critical properties that vary continuously and monotonically as one passes from fermions to bosons. Possible relevance to experiments on the transitions between plateaus in the fractional quantum Hall effect and the magnetic-field-tuned superconductor-insulator transition are briefly discussed.