<div>Over the last decade, Dirac matter has become one of the most prominent fields of research in contemporary material science due to the incredibly rich physics of the Dirac equation. Notable examples are the Dirac cones in graphene, Weyl points in TaAs, and gapless edge states in Bi₂Te₃. These unique phases of matter are intimately related to the topological structure of Dirac fermions. However, it remains an open question if the topological structure of Maxwell's equations predicts yet new phases of matter. This thesis will conclusively answer this question.</div><div><br></div><div>Topological electrodynamics is concerned with the geometry of electromagnetic waves in condensed matter. At the microscopic level, photons couple to the dipole-carrying excitations of a material, such as plasmons and excitons, which hybridize to form new normal modes of the system. The interaction between these bosonic oscillators is the origin of temporal and spatial dispersion in optical response functions like the conductivity tensor. Our main achievement is motivating a global interpretation of these response functions, over all frequencies and wavevectors. This theory led us to the conclusion that there are topological invariants associated with the conductivity tensor itself. In this thesis, we show exactly how to calculate these electromagnetic invariants, in both continuum and lattice theories, to identify unique Maxwellian phases of matter. Magnetohydrodynamic electron fluids in strongly-correlated 2D materials like graphene are the first candidates of this new class of topological phase. The fundamental physical mechanism that gives rise to a topological electromagnetic classification is Hall viscosity which adds a nonlocal component to the Hall conductivity. To study the topological electrodynamics, we propose viscous Maxwell-Chern-Simons theory -- a Lagrangian framework that naturally generates the equations of motion, nonlocal Hall response and the boundary conditions. We demonstrate that nonlocal Hall conductivity is the spin-1 photonic equivalent of dispersive mass and induces precession of bulk photonic skyrmions. Nontrivial photonic skyrmions are associated with Dirac monopoles in the bulk momentum space and a singular Berry gauge. A singular gauge occurs when the photonic mass changes sign. Remarkably, the boundary of this medium supports gapless chiral edge states that are spin-1 helically-quantized and satisfy open boundary conditions.</div>

[pt] Nesta tese estudamos dois importantes sistemas de Isoladores Topológicos (TIs), onde nos concentramos particularmente no papel das interações e percolação nos estados de borda topológicos. Primeiro, analisamos o papel das interações vizinhas mais próximas em um protótipo de TI unidimensional, o modelo Su-Schrieffer-Heeger (SSH). Com base em um formalismo de função de Green, aplicamos a equação de Dyson em combinação com a aproximação da matriz-T para verificar a correspondência bulk-edge na presença de interações. Os expoentes críticos próximos às transições de fase topológicas são os mesmos do modelo SSH não interagente, indicando que o sistema permanece na mesma classe de universalidade, apesar da presença de interações. O segundo sistema é um TI bidimensional simétrico na inversão de tempo, ou seja, o modelo de Bernevig-Hughes-Zhang (BHZ) em conjunto com um metal ferromagnético com quebra de reversão do tempo (FMM), onde investigamos a percolação do estado Hall de spin quântico do modelo BHZ para o FMM por meio de um modelo de ligações fortes (tight-binding). Demonstramos que dependendo de se o estado de borda do cone de Dirac submerge nas sub-bandas do FMM e da direção da magnetização do FMM, a percolação do estado de borda e seu spin-momentum-locking são afetados de maneiras diferentes. Surpreendentemente, descobrimos que a corrente de spin de borda de equilíbrio no modelo BHZ, naturalmente esperada dos estados de borda de propagação do spin polarizado, está de fato ausente devido ao cancelamento das bandas de valência. No entanto, fluxos laminares de correntes de carga e spin persistente à temperatura ambiente são produzidos perto da interface da junção BHZ / FMM. Usando teoria de resposta linear, investigamos a polarização de spin induzida pela corrente causada pela percolação do estado de borda, que serve como um torque de rotação que é encontrado ser predominantemente field-like. Além disso, a polarização do spin é dramaticamente aumentada perto das impurezas na borda do modelo BHZ. / [en] In this thesis we studied two important Topological Insulators (TIs), where we focused particularly on the role of interactions and percolation on the topological edge states. First, we analyzed the role of nearest-neighbor interactions in a prototype one-dimensional TI, namely the Su-Schrieffer-Heeger (SSH) model. Based on a Green s function formalism, we applied Dyson s equation in combination with T-matrix approximation to verify the bulk-edge correspondence in the presence of interactions. The critical exponents near topological phase transitions are found to be the same as the noninteracting SSH model, indicating that the system stays in the same universality class despite the presence of interactions. The second system is a two-dimensional timereversal symmetric TI, namely the Bernevig-Hughes-Zhang (BHZ) model in conjunction with a time-reversal breaking ferromagnetic metal (FMM), where we investigated the percolation of the quantum spin Hall state from the TI layer to the FMM by means of a tight-binding model. We demonstrated that depending on whether the edge state Dirac cone submerges into the FMM subbands and the direction of the magnetization of the FMM, the percolation of the edge state and its spin-momentum locking are affected in different ways. Surprisingly, we uncover that the equilibrium edge spin current in the BHZ model, naturally expected from the spin polarized propagating edge states, is in fact absent due to the cancellation from the valence bands. Nevertheless, laminar flows of room temperature persistent charge and spin currents are produced near the interface of the BHZ/FMM junction. Using a linear response theory, we investigate the current-induced spin polarization caused by the percolation of the edge state, which serves as a spin torque that is found to be predominantly field-like. Moreover, the spin polarization is dramatically enhanced near the impurities at the edge of the BHZ model.