Campos multivalentes e estados topológicos da matéria

Abstract

In this thesis we investigate topological aspects of states of matter as well as the role of disorder operator in field theory. First we introduce some aspects of, duality, condensation of defects and topology that will be our theoretical basis in order to describe topological insulators and topological superconductors. To understand the case of the topological insulators we review the Integer and Fractional Quantum Hall Effect which are the base for this new state of matter, and we show how the effective field theory is obtained. In the topological superconductors case we review the mechanism to obtain the effective field theory based on dimensional reduction of a Chern-Simons theory in (4+1)-dimensional space into a axion field theory in (3+1)-dimensional space. We then propose a new way of obtain the effective field theory considering multiple Fermi surfaces. For the case of one Fermi surface we re-obtain the result that the superconductor is more precisely described as a topological state of matter. Studying the case of more than one Fermi surface, we obtain the effective theory describing a time reversal symmetric topological superconductor. These results are obtained by employing a general procedure to construct effective low energy actions describing states of electromagnetic systems interacting with charges and defects. The procedure consists in taking into account the proliferation or dilution of these charges and defects and its consequences for the low energy description of the electromagnetic response of the system. We find that the main ingredient entering the low energy characterization of the system with more than one Fermi surface is a non-conservation of the canonical supercurrent caused by particular vortex configurations thats don t possess electromagnetic flux. In the disorder operator we review the original Kadanoff and Ceva prescription based on the 2-dimensional Ising model as well as the generalization to the continuum field system. We them investigate the role of multivalued fields in the formulation of disorder operators and its connection with topological defects. In quantum field theory it is known that certain states describe collective modes of the fundamental fields and are created by operators that are often non-local, being defined over lines or over higher dimensional surfaces, and for this reason are sensitive to global, topological, properties of the system and depends on nonperturbative data. Such operators are generally known as disorder operators because its nonzero expectation values define a disordered vacuum associated with a condensate of the collective modes, also known sometimes as defects. We investigate the definition of these operators and its relation with the multivalued properties of the fundamental fields. We study some examples of scalar field theories and generalize the discussion to p-forms. By splitting the fields in their regular and singular parts we identify a ambiguity that can be explored, much like gauge symmetry, in order to define the relevant observables. We also use these ideas to re-obtain the vortex in the topological superconductors.