Campos de Gauge e matéria na rede - generalizando o Toric Code

Abstract

I owe a special thanks to all my friends in São Paulo, to Javier, Miguel and Jahiro for being great friends, to Oscar and Jorge for the weekly workout sessions, to Javier B. and the Colomboys for all the discussions and the fun we had during these two years.I would also like to thank my parents and my siblings, for the unconditional reliance. They were always supporting and encouraging me with their best wishes.I am especially grateful with my dear Alessandra, for all the patience and support during the two years we have been together. For being always with me, cheering me up and standing by me through the good and bad times.Finally, I would like to thank to the three ladies at DFMA's secretary, the people in CPG for all the help, and to CNPq for the financial support. AbstractTopological phases of matter are characterized for having a topologically dependent ground state degeneracy, anyonic quasi-particle bulk excitations and gapless edge excitations. Different topologically ordered phases of matter can not be distinguished by te usual Ginzburg-Landau scheme of symmetry breaking. Therefore, a new mathematical framework for the study of such phases is needed. In this dissertation we present the simplest example of a topologically ordered system, namely, the Toric Code (TC) introduced by A. Kitaev in [1]. Its ground state is 4-fold degenerate when embedded on the surface of a torus and its elementary excited states are interpreted as quasi-particle anyons. The TC is a particular case of a more general class of lattice models known as Quantum Double Models (QDMs) which can be interpreted as an implementation of (2 + 1) Lattice Gauge Theories in the Hamiltonian formulation with discrete gauge group G. We generalize these models by the inclusion of matter fields at the vertices of the lattice. We give a detailed construction of such models, we show they are exactly solvable and explore the case when the gauge group is set to be the abelian Z 2 cyclic group and the matter degrees of freedom to be elements of a 2-dimensional vector space V 2 . Furthermore, we show that the ground state degeneracy is not topologically dependent and obtain the most elementary excited states. ResumoFases topológicas da matéria são caracterizadas por terem uma degenerescên-cia do estado fundamental que depende da topologia da variedade em que o sistema físico é definido, além disso apresentam estados excitados no interior do sistema que são interpretados como sendo quase-partículas com estatística de tipo anyonica. Estes sistemas apresentam também excitações sem gap de energia em sua borda. Fases topologicamente ordenadas distintas não podem ser distinguidas pelo esquema usual de quebra de simetria de Ginzburg-Landau. Nesta dissertação apresentamos como exemplo o modelo mais simples de um sistema com Ordem Topológica, a saber, o Toric Code (TC), introduzido originalmente por A. Kitaev em [1]. O estado fundamental deste modelo apresenta degenerescência igual a 4 quando incorporado à superfície de um toro. As excitações elementares são inte...

Entropia topológica de emaranhamento em teorias de Higher Gauge Abelianas

“…Both types of operators in H QD are projectors and they all commute with each other. We refer the reader to [1,84,87] for details about the construction of this model.…”

“…In addition to the states localized on the links l ∈ K 1 and labeled by G 1 , we also place quantum degrees of freedom at the vertices v ∈ K 2 that are labeled by another Abelian group G 0 . This class of models corresponds to the Abelian version of the models constructed in [87,94]. It is the lattice version of a scalar field coupled to a gauge field.…”

Categorias monoidais e álgebras de Hopf

“…Esta seçãoé fortemente baseada na dissertação de mestrado de Juan Pablo Ibieta Jimenez [11]. Não serão feitos todos os detalhes do toric code, para mais detalhes veja [11] e [15]. Esta seção nãoé necessária para o que será tratado adiante.…”