The thesis is organized as follows. In Chapter one we recall how the notions of fractional spin and statistics appeared in two-dimensional physics and expose briefly the role played by anyons in the explanation of the Fractional Quantum Hall Effect and the mechanism of anyonic superconductivity. In chapter two we treat in some detail the canonical quantization of the pure abelian Chern-Simons theory at arbitrary rational Chern-Simons coupling constant k: our formulation of the theory of anyons on the torus in the next chapter is based on this theory. We use an algebraic approach to investigate the main properties of this system concentrating on the construction of the Hilbert space and on its modular properties. We explain in detail how to manage consistently the global anomaly mentioned earlier. The structure of the Hilbert space which results, in the coherent state representation, constitutes the starting frame for the coupling to matter in chapter three. In chapter three we introduce a (non relativistic) matter field and develop its coupling to the Chern-Simons gauge field, which induces the statistics flip, in first quantization. Due to the presence of an integer number of anyons and to the Dirac quantization condition on a torus k is forced to be rational. We construct the Hamiltonian and the total momentum operator, evidencing the crucial role played by the topological components of the Chern-Simons field with respect to their commutativity, and determine the conditions which define the Hilbert space. These conditions are then completely solved to obtain an explicit basis for the whole Hilbert space. We define then a gauge-fixed Hilbert space proving that it carries a representation of the modular group and that, a necessary consistency check, the physics is independent of the gauge-fixing. In section 4.6 we find the exact ground state solutions of a "self-dual" Hamiltonian [22 • 231. Throughout this thesis we work in a "gauge" where the wave function obeys ordinary statistics and the fractional statistics is presented by a (non trivial) Hamiltonian which contains the Chern-Simons field. On the plane there exists a singular gauge transformation which transforms the Hamiltonian in the free one and the wave function in a "function" which picks up a phase if two particles are interchanged. In section 4.7 we determine the singular gauge transformation on the torus which transforms the Hamiltonian in an "essentially free" Hamiltonian and the (bosonic or fermionic) wave function in an anyonic one. We find that the "essentially free" Hamiltonian can be further reduced to the free one at the expense of introducing a multi-component wave function, in agreement with general braid group analysis results on non simply connected surfaces . In chapter five we consider a system of anyons at integer coupling constant k with the purpose of investigating its superconducting properties. We consider in particular its Mean Field Approximation which turns out to be translation invariant. In this approximation the Hamiltonian problem of the many-body system can be completely solved; the many-body energy eigenstates at fixed total momentum turn out to constitute a kind of translation invariant Landau-levels, with a collective degeneracy which turns out to be somewhat smaller than the one obtained on the plane by taking direct products of single-particle Landau-levels. In particular, our many-body momentum eigenstates can not be factorized into one-particle states. We derive explicitly the antisymmetric many-body ground state at fixed momentum and find the macroscopic quantization of momenta and the corresponding superconductivity mechanism, mentioned before. These protected states generate a real magnetic field inside the cavity of the torus, which we compute, and whose flux turns out to be quantized as J;- times the fundamental unit of flux. It is interesting to note that this is precisely the amount of the elementary fluxoid excitation entering the discussion in refs. [9,11]. Chapter six is devoted to the analysis of a system of anyons on a torus in a (real) external magnetic field, which we think of as one of the vortex excitation components appearing in the ground state of Haldane's hierarchy of the Fractional Quantum Hall Effect. Said in other words, we consider the Hall Effect of anyons. Our treatment differs, however, from the usual ones in that we impose the vanishing of the Lorentz force, as is appropriate for the classical Hall effect, by means of an effective Lagrangian (the Lorentz force has to be cancelled by the electric field). Accordingly this Lagrangian has then to contain a Chern-Simons action also f<?r the real electromagnetic field. Then topological components of the electromagnetic field appear naturally and, as one can expect, the overall translation invariance, which is broken by the introduction of the external magnetic field, is restored. In summary, the introduction of an electromagnetic Chern-Simons action in the Lagrangian imposes the vanishing of the Lorentz force and restores, at the same time, translation invariance. Using the results of the previous chapters we construct the Hilbert space, and we find, moreover, the exact ground state at fixed momentum (minimizing the Coulomb repulsion a la Laughlin). This Laughlinlike ground state turns out to exist only for particular values of the momentum of the total system (electrons plus vortices) and this explains, as said above, the vanishing of the diagonal resistance. Finally we repeat the steps, which lead to the fractional Hall hierarchy, on the torus.
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