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Lattice generalizations of fractional quantum Hall (FQH) systems, called fractional Chern insulators (FCIs), have been extensively investigated in strongly correlated systems. Despite many efforts, previous studies have not revealed all of the guiding principles for the FCI search. In this paper, we investigate a relationship between the topological band structure in the two-particle problem and the FCI ground states in the many-body problem. We first formulate the two-particle problem of a bosonic on-site interaction projected onto the lowest band of a given tight-binding Hamiltonian. We introduce a reduced Hamiltonian whose eigenvalues correspond to the two-particle bound-state energies. By using the reduced Hamiltonian, we define the two-particle Chern number and numerically check the bulk-boundary correspondence that is predicted by the two-particle Chern number. We then propose that a nontrivial two-particle Chern number of dominant bands roughly indicates the presence of bosonic FCI ground states at filling factor $\nu=1/2$. We numerically investigate this relationship in several tight-binding models with Chern bands and find that it holds well in most of the cases, albeit two-band models being exceptions. Although the two-particle topology is neither a necessary nor a sufficient condition for the FCI state as other indicators in previous studies, our numerical results indicate that the two-particle topology characterizes the degree of similarity to the FQH systems., Comment: 15pages,6 figures

Orientador: Ricardo Luís Doretto Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin Resumo: Nesse trabalho, estudamos uma fase ferromagnética de banda plana do modelo de Haldane-Hubbard na rede hexagonal através de um método de bosonização desenvolvido recentemente para isolantes de Chern e isolantes topológicos Z2. Em particular, procuramos determinar o espectro de excitação das ondas de spin. Consideramos configurações do modelo de Haldane-Hubbard onde as bandas eletrônicas (limite não-interagente) de mais baixa energia são quase-planas e com fator de preenchimento (limite não-interagente) igual a 1/4. Duas versões distintas do modelo de Haldane-Hubbard foram estudadas, uma que quebra a simetria de inversão temporal (isolante de Chern correlacionado) e uma segunda que preserva a simetria de inversão temporal (isolante topológico correlacionado). Através do método de bosonização, mostramos que o modelo de Haldane-Hubbard é mapeado em um modelo bosônico efetivo e interagente, cujo termo quadrático nos permite determinar o espectro de excitação das ondas de spin em uma aproximação harmônica. Para o isolante de Chern correlacionado, encontramos que o espectro de excitação é constituído por dois ramos, apresentando um modo de Goldstone e pontos Dirac no centro e nos pontos K e K' da primeira zona de Brillouin, respectivamente. Para o isolante topológico correlacionado, verificamos que o espectro de excitação também é formado por dois ramos, porém ambos apresentam energia de gap finita e ausência de pontos de Dirac. De fato, para o isolante topológico, verificamos que é possível definir operadores bosônicos associados a dois tipos diferentes de excitações de spin, isto é, excitações que alteram (mixed-lattice excitations) e que preservam (same-lattice excitations) os índices de subrede dos operadores de spin. Além disso, consideramos os efeitos no espectro das ondas de spin associados (i) a uma diferença nos valores das energias de repulsão locais e (ii) à presença de um termo de energia de um corpo local alternado (termo de massa), ambas as quantidades relacionadas com as duas subredes triangulares. Para ambas as perturbações, encontramos que o espectro das ondas de spin do isolante de Chern correlacionado apresenta um gap de energia nos pontos K e K', em contraste com o caso homogêneo que apresenta pontos de Dirac nessa região do espectro. Para o isolante topológico correlacionado, verificamos pequenas modificações no gap de energia dos pontos K e K', sendo que espectro mantém sua forma quando comparada com o caso não-perturbado. Importante, para o isolante de Chern, também encontramos algumas evidências de uma instabilidade da fase ferromagnética da banda plana na presença do termo de massa. Finalmente, comentamos sobre as diferenças na aplicação do método de bosonização para os isolantes de Chern e topológicos correlacionados para modelos de Hubbard topológicos nas redes quadrada e hexagonal Abstract: In this thesis, we study the flat-band ferromagnetic phase of the Haldane-Hubbard model on a honeycomb lattice within a bosonization scheme for both Chern and Z2 topological insulators, focusing on the calculation of the spin-wave excitation spectrum. We consider a spinfull Haldane-Hubbard model with the noninteracting lower bands in a nearly flat band limit, previously determined for the spinless Haldane model, and at 1/4-filling of its corresponding noninteracting limit. Two configurations of the Haldane-Hubbard model, one that breaks time-reversal symmetry (correlated Chern insulator) and a second one that preserves time-reversal symmetry (correlated Z2 topological insulator), are discussed. Within the bosonization scheme, the Haldane-Hubbard model is mapped into an effective interacting boson model, whose quadratic term allows us to determine the spin-wave spectrum at the harmonic approximation. For the correlated Chern insulator, we show that the excitation spectrum has two branches with a Goldstone mode and Dirac points at center and at the K and K' points of the first Brillouin zone, respectively. In contrast, for the correlated Z2 topological insulator, the excitation spectrum also has two branches, but both of them are gapped and there is no Dirac points. Indeed, for the correlated Z2 topological insulator, we found that it is possible to define boson operators associated with two distinct spin-flip excitations, one that changes (mixed-lattice excitations) and a second one that preserves (same-lattice excitations) the index related with the two sublattices. We also consider the effects on the spin-wave spectrum due to an energy offset in the on-site Hubbard repulsion energies and due to the presence of an staggered on-site energy term (mass term), both quantities associated with the two triangular sublattices. For both perturbations, we found that an energy gap opens at the K and K' points, dissolving the Dirac points found in the spin-wave spectrum of the correlated Chern insulator. For the correlated Z2 topological insulator, a mild modification of the gap at the same points were found, with the overall spectrum retaining its shape. Moreover, for the Chern insulator, we also found some evidences for an instability of the flat-band ferromagnetic phase in the presence of the staggered on-site energy term. Finally, we comment on the differences between the bosonization scheme implementation for the correlated Chern and Z2 topological insulators on both square and honeycomb lattices Doutorado Física Doutor em Ciências CNPQ 162323/2017-4

We report on the possibility of detecting hinge spin polarization in magnetic topological insulators by resistance measurements. By implementing a three-dimensional model of magnetic topological insulators into a multiterminal device with ferromagnetic contacts near the top surface, local spin features of the chiral edge modes are unveiled. We find local spin polarization at the hinges that inverts the sign between the top and bottom surfaces. At the opposite edge, the topological state with inverted spin polarization propagates in the reverse direction. A large resistance switch between forward and backward propagating states is obtained, driven by the matching between the spin polarized hinges and the ferromagnetic contacts. This feature is general to the ferromagnetic, antiferromagnetic, and canted antiferromagnetic phases, and enables the design of spin-sensitive devices, with the possibility of reversing the hinge spin polarization of the currents., We acknowledge the European Union Horizon 2020 research and innovation programme under Grant Agreement No. 824140 (TOCHA, H2020-FETPROACT-01-2018). ICN2 is funded by the CERCA Programme/Generalitat de Catalunya, and is supported by the Severo Ochoa program from Spanish MINECO (Grant No. SEV-2017-0706).

We consider the generic problem of a two-level quantum emitter near a two-dimensional Chern insulator in the dipole approximation, and study how the frequency-dependent response and electronic density of states of the insulator modifies the transition rate of the emitter between the ground and excited levels. To this end, we obtain the full real-frequency behavior of the conductivity tensor by performing a tight-binding calculation based on the Qi-Wu-Zhang model and using a Kubo formula, and derive the full electromagnetic Green tensor of the system, which breaks Onsager reciprocity. This enables us to find that for frequencies smaller than the maximum band gap, the system is sensitive to time reversal symmetry-breaking, whereas for much larger frequencies the system becomes insensitive, with implications for the discrimination of the state of a circularly polarised dipole emitter. We also study the impact of a van Hove singularity on the surface-induced correction to the transition rate, finding that it can enhance its amplitude by a few orders of magnitude compared to the case where the conductivity is set to its static value. By considering configurations in which the dipole is circularly polarised or parallel with the surface of the Chern insulator, we find that the surface correction to the transition rate can exhibit a novel decay with sine integral-like oscillations., 23 pages, 11 figures

We investigate the Hall conductivity in a Sierpinski carpet, a fractal of Hausdorff dimension $d_f=\ln(8)/\ln(3) \approx 1.893$, subject to a perpendicular magnetic field. We compute the Hall conductivity using linear response and the recursive Green function method. Our main finding is that edge modes, corresponding to a maximum Hall conductivity of at least $\sigma_{xy}=\pm \frac{e^2}{h}$, seems to be generically present for arbitrary finite field strength, no mater how one approaches the thermodynamic limit of the fractal. We discuss a simple counting rule to determine the maximal number of edge modes in terms of paths through the system with a fixed width. This quantized edge conductance, as in the case of the conventional Hofstadter problem, is stable with respect to disorder and thus a robust feature of the system., Comment: V1: 5 pages, 6 figures, 1 table; V2: Changed title

This dissertation presents studies of topological superconductivity in Chern insulator systems. In particular, when a Chern insulator such as a quantum Hall or a quantum anomalous Hall system is proximity coupled to a trivial s-wave superconductor. While quantum anomalous Hall based system is investigated in detail in both one and two dimensions, the focus is solely on two dimensions in quantum Hall bases system. Both quantum Hall and quantum anomalous Hall break time reversal (T)-symmetry. Hence throughout this thesis, we focus on the T-symmetry broken topological superconductor systems, which fall in the class [double struck D] of the Altland-Zirnbauer classification. After a brief introduction to thesis in chapter 1 and review of topological superconductivity in chapter 2, in chapter 3, a realistic system motivated from the experimental observation of the quantum anomalous Hall effect is considered. The focus of this chapter is on one dimensional (1D) topological superconducting phase in thin ribbon geometries [Phys. Rev. B 97, 081102(R)]. It identifies the quantum anomalous Hall based system as highly controllable platform for Majorana zero modes, which can be potentially used as Majorana braiding device. In chapter 4, two dimensional (2D) topological superconducting phase in the quantum Hall based system is studied [arXiv:1903.12249]. Because of the inherent requirement of the external magnetic field to achieve quantum Hall physics, this system requires consideration of vortex lattice phase in the parent superconductor. It is shown that the topological superconducting phase is determined by the type of vortex lattice. Hence, making quantum Hall based system different than quantum anomalous Hall based system, giving much richer phase diagram. Experimental protocol to engineer and observe Majorana edge modes in this system is also discussed. In chapter 5, reentrant superconductivity under magnetic fields beyond semiclassical critical magnetic field is discussed. It is argued that recently discovered superconductivity in magic angle twisted bilayer graphene (MATBG) can be a promising ground to observe this phase. In chapter 6, recent transport experiments in quantum anomalous Hall/superconductor devices are discussed. The two main experiments on this system have contradictory results and there is widespread debate on possible explanations of the experimental observations, which can born out of chiral Majorana mode (i.e. topological) or disorder (i.e. trivial). The main arguments on both sides are summarized and some strategies to resolve the debate are proposed.

We investigate spin transport in 2-dimensional insulators, with the long-term goal of establishing whether any of the transport coefficients corresponds to the Fu-Kane-Mele index which characterizes 2d time-reversal-symmetric topological insulators. Inspired by the Kubo theory of charge transport, and by using a proper definition of the spin current operator, we define the Kubo-like spin conductance $G_K^{s_z}$ and spin conductivity $\sigma_K^{s_z}$. We prove that for any gapped, periodic, near-sighted discrete Hamiltonian, the above quantities are mathematically well-defined and the equality $G_K^{s_z} = \sigma_K^{s_z}$ holds true. Moreover, we argue that the physically relevant condition to obtain the equality above is the vanishing of the mesoscopic average of the spin-torque response, which holds true under our hypotheses on the Hamiltonian operator. This vanishing condition might be relevant in view of further extensions of the result, e.g. to ergodic random discrete Hamiltonians or to Schr\"odinger operators on the continuum. A central role in the proof is played by the trace per unit volume and by two generalizations of the trace, the principal value trace and it directional version., Comment: 35 pages, 2 figures

This thesis is dedicated to the description of both single-particle and bosonic quantum fluid Physics in topological systems. After introductory chapters on these subjects, I first discuss single-particle topological phenomena in honeycomb lattices. This allows to compare two theoretical models leading to quantum anomalous Hall effect for electrons and photons and to discuss the photonic quantum valley Hall effect at the interface between opposite staggered cavity lattices.In a second part, I present some phenomena which emerge due to the interplay of the linear topological effects with the presence of interacting bosonic quantum fluid described by mean-field Gross-Pitaevskii equation. First, I show that the spin-anisotropic interactions lead to density-driven topological transitions for elementary excitations of a condensate loaded in the polariton quantum anomalous Hall model (thermal equilibrium and out-of-equilibrium quasi-resonant excitation configurations). Then, I show that the vortex excitations of a scalar condensate in a quantum valley Hall system, contrary to linear wavepackets, can exhibit a robust chiral propagation along the interface, with direction given by their winding in real space, leading to an analog of quantum spin Hall effect for these non-linear excitations. Finally, coming back to linear geometrical effects, I will focus on the anomalous Hall effect exhibited by an accelerated wavepacket in a two-band system. In this context, I present a non-adiabatic correction to the known semiclassical equations of motion which can be expressed in terms of the quantum geometric tensor elements. We also propose a protocol to directly measure the tensor components in radiative photonic systems.; Cette thèse est consacrée à la description de la physique à une particule ainsi qu'à celle de fluides quantiques bosoniques dans des systèmes topologiques. Les deux premiers chapitres sont introductifs. Dans le premier, nous introduisons des éléments de théorie des bandes et les quantités géométriques et topologiques associées : tenseur métrique quantique, courbure de Berry, nombre de Chern. Nous discutons différents modèles et réalisations expérimentales donnant lieu à des effets topologiques. Dans le second chapitre, nous introduisons les condensats de Bose-Einstein ainsi que les excitons-polaritons de cavité.La première partie des résultats originaux discute des phénomènes topologiques à une particule dans des réseaux en nid d'abeilles. Cela permet de comparer deux modèles théoriques qui mènent à l'effet Hall quantique anormal pour les électrons et les photons dû à la présence d'un couplage spin-orbite et d'un champ Zeeman. Nous étudions aussi l'effet Hall quantique de vallée photonique à l'interface entre deux réseaux de cavités avec potentiels alternés opposés.Dans une seconde partie, nous discutons de nouveaux effets qui émergent due à la présence d'un fluide quantique interagissant décrit par l’équation de Gross-Pitaevskii dans ces systèmes. Premièrement, il est montré que les interactions spin anisotropes donnent lieu à des transitions topologiques gouvernées par la densité de particules pour les excitations élémentaires d’un condensat spineur d’exciton-polaritons.Ensuite, nous montrons que les tourbillons quantifiés d'un condensat scalaire dans un système avec effet Hall quantique de vallée, manifestent une propagation chirale le long de l'interface contrairement aux paquets d'ondes linéaires. La direction de propagation de ces derniers est donnée par leur sens de rotation donnant lieu à un transport de pseudospin de vallée protégé topologiquement, analogue à l’effet Hall quantique de spin.Enfin, revenant aux effets géométriques linéaires, nous nous sommes concentrés sur l’effet Hall anormal. Dans ce contexte, nous présentons une correction non-adiabatique aux équations semi-classiques décrivant le mouvement d’un paquet d’ondes qui s’exprime en termes du tenseur géométrique quantique. Nous proposons un protocole expérimental pour mesurer cette quantité dans des systèmes photonique radiatifs.

This thesis is dedicated to the description of both single-particle and bosonic quantum fluid Physics in topological systems. After introductory chapters on these subjects, I first discuss single-particle topological phenomena in honeycomb lattices. This allows to compare two theoretical models leading to quantum anomalous Hall effect for electrons and photons and to discuss the photonic quantum valley Hall effect at the interface between opposite staggered cavity lattices.In a second part, I present some phenomena which emerge due to the interplay of the linear topological effects with the presence of interacting bosonic quantum fluid described by mean-field Gross-Pitaevskii equation. First, I show that the spin-anisotropic interactions lead to density-driven topological transitions for elementary excitations of a condensate loaded in the polariton quantum anomalous Hall model (thermal equilibrium and out-of-equilibrium quasi-resonant excitation configurations). Then, I show that the vortex excitations of a scalar condensate in a quantum valley Hall system, contrary to linear wavepackets, can exhibit a robust chiral propagation along the interface, with direction given by their winding in real space, leading to an analog of quantum spin Hall effect for these non-linear excitations. Finally, coming back to linear geometrical effects, I will focus on the anomalous Hall effect exhibited by an accelerated wavepacket in a two-band system. In this context, I present a non-adiabatic correction to the known semiclassical equations of motion which can be expressed in terms of the quantum geometric tensor elements. We also propose a protocol to directly measure the tensor components in radiative photonic systems.; Cette thèse est consacrée à la description de la physique à une particule ainsi qu'à celle de fluides quantiques bosoniques dans des systèmes topologiques. Les deux premiers chapitres sont introductifs. Dans le premier, nous introduisons des éléments de théorie des bandes et les quantités géométriques et topologiques associées : tenseur métrique quantique, courbure de Berry, nombre de Chern. Nous discutons différents modèles et réalisations expérimentales donnant lieu à des effets topologiques. Dans le second chapitre, nous introduisons les condensats de Bose-Einstein ainsi que les excitons-polaritons de cavité.La première partie des résultats originaux discute des phénomènes topologiques à une particule dans des réseaux en nid d'abeilles. Cela permet de comparer deux modèles théoriques qui mènent à l'effet Hall quantique anormal pour les électrons et les photons dû à la présence d'un couplage spin-orbite et d'un champ Zeeman. Nous étudions aussi l'effet Hall quantique de vallée photonique à l'interface entre deux réseaux de cavités avec potentiels alternés opposés.Dans une seconde partie, nous discutons de nouveaux effets qui émergent due à la présence d'un fluide quantique interagissant décrit par l’équation de Gross-Pitaevskii dans ces systèmes. Premièrement, il est montré que les interactions spin anisotropes donnent lieu à des transitions topologiques gouvernées par la densité de particules pour les excitations élémentaires d’un condensat spineur d’exciton-polaritons.Ensuite, nous montrons que les tourbillons quantifiés d'un condensat scalaire dans un système avec effet Hall quantique de vallée, manifestent une propagation chirale le long de l'interface contrairement aux paquets d'ondes linéaires. La direction de propagation de ces derniers est donnée par leur sens de rotation donnant lieu à un transport de pseudospin de vallée protégé topologiquement, analogue à l’effet Hall quantique de spin.Enfin, revenant aux effets géométriques linéaires, nous nous sommes concentrés sur l’effet Hall anormal. Dans ce contexte, nous présentons une correction non-adiabatique aux équations semi-classiques décrivant le mouvement d’un paquet d’ondes qui s’exprime en termes du tenseur géométrique quantique. Nous proposons un protocole expérimental pour mesurer cette quantité dans des systèmes photonique radiatifs.

The Haldane model on a honeycomb lattice is a paradigmatic example of a system featuring quantized Hall conductivity in the absence of an external magnetic field, that is, a quantum anomalous Hall effect. Recent theoretical work predicted that the anomalous Hall conductivity of massive Dirac fermions can display Shubnikov-de Haas (SdH) oscillations, which could be observed in topological insulators and honeycomb layers with strong spin--orbit coupling. Here, we investigate the electronic transport properties of Chern insulators subject to high magnetic fields by means of accurate spectral expansions of lattice Green's functions. We find that the anomalous component of the Hall conductivity displays visible SdH oscillations at low temperature. \textcolor{black}{The effect is shown to result from the modulation of the next-nearest neighbour flux accumulation due to the Haldane term,} which removes the electron--hole symmetry from the Landau spectrum. To support our numerical findings, we derive a long-wavelength description beyond the linear ('Dirac cone') approximation. Finally, we discuss the dependence of the energy spectra shift for reversed magnetic fields with the topological gap and the lattice bandwidth.

We review recent results concerning the localization of gapped periodic systems of independent fermions, as, e.g., electrons in Chern and Quantum Hall insulators. We show that there is a “localization dichotomy” which shows some analogies with phase transitions in Statistical Mechanics: either there exists a system of exponentially localized composite Wannier functions for the Fermi projector, or any possible system of composite Wannier functions yields a diverging expectation value for the squared position operator. This fact is largely model-independent, covering both tight-binding and continuous models. The results are discussed with emphasis on the main ideas and the broader context, avoiding most of the technical details.

Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of $|\mathbf{x}|^2$ diverges. Intermediate regimes are forbidden. Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane (Phys. Rev. Lett. 61, 2017 (1988)). We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in (Commun. Math. Phys. 359, 61-100 (2018)), and finally we announce further generalizations to non-periodic models., 20 pages, 5 figures. Extended version of the paper published in the special issue of Rendiconti di Matematica appeared on the occasion of Gianfausto Dell'Antonio's 85th birthday. In comparison with the published version, we added here some details and the whole Chapter 5

From an experimental point of view, quasielectrons and quasiholes play very similar roles in the fractional quantum Hall effect. Nevertheless, the theoretical description of quasielectrons is known to be much harder than the one of quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here, we demonstrate that the same problem does not arise in lattice fractional quantum Hall models. This result allows us to make detailed investigations of the properties of quasielectrons, including their braiding statistics and density distribution on lattices on the plane and on the torus. We show that some of the states considered have high overlap with certain fractional Chern insulator states. We also derive few-body Hamiltonians, for which various states containing quasielectrons are exact ground states., Comment: 13 pages, 4 figures, v3: accepted version

2012/2013 La geometria, e la topologia in particolare, rivestono un profondo ruolo in molti campi della fisica ed in particolare in materia condensata ove è possibile identificare diversi stati quantistici della materia attraverso proprietà topologiche. L'invariante di Chern è un invariante topologico che caratterizza lo stato isolante dei cristalli. Esso è definito attraverso la descrizione in spazio reciproco di un cristallo perfetto, per cui è necessario considerare un sistema infinito oppure finito ma con condizioni periodiche al bordo. In questa tesi il concetto di invariante di Chern viene generalizzato definendo un opportuno marcatore locale di Chern in spazio reale. Infatti se si considera un cristallo perfetto infinito oppure finito e con condizioni periodiche al bordo, la media sulla cella elementare di questo marcatore restituisce il consueto invariante di Chern. Tuttavia, grazie al suo carattere locale, il marcatore di Chern è ben definito e può essere utilizzato per identificare il carattere locale di Chern anche di un sistema microscopicamente disordinato o macroscopicamente disomogeneo (ad esempio etorogiunzioni di diversi cristalli) e con qualsiasi tipo di condizioni al bordo (periodiche o aperte). Nella seconda parte della tesi l'invariante locale di Chern viene utilizzato per fornire una descrizione locale in spazio reale della magentizzazione orbitale. Questa descrizione è utilizzabile sia con condizioni al bordo aperte che periodiche e quindi unifica i due separati approcci utilizzati in questi due casi. La nuova formula permette, inoltre, di ottenere anche una migliore comprensione del ruolo che gli stati di bordo rivestono nella magnetizzazione di un sistema. In entrambi i casi vengono presentati i risultati di simulazioni numeriche che confermano i risultati teorici derivati. The geometry and the topology play a profound role in many fields of physics and in particular in condensed matter where it is possible to identify different quantum states of matter through their topological properties. The Chern invariant is a topological invariant which characterizes the insulating state of crystals. It is defined through the description in the reciprocal space of a perfect crystal, which then has to be considered as an infinite system or a finite size system with periodic boundary conditions. In this thesis the concept of Chern invariant is generalized by defining a local Chern marker in the real space. For an infinite crystal or a finite crystal with periodic boundary conditions, the average of this marker over an elementary unit cell returns the usual invariant Chern. However, thanks to its local character, the Chern marker is well defined and can be used to identify the local Chern character also of microscopically disordered systems or macroscopically inhomogeneous systems (e.g. heterojunctions of different crystals) and with any kind of boundary conditions adopted (periodic boundary conditions or open bounday conditions as well). In the second part of the thesis the local Chern invariant is used to provide a local description in the real space of the orbital magnetization. This description can be used both with open and periodic boundary conditions, so it unifies the two separate approaches used in these different cases. Moreover, the new formula makes it possible to get a better understanding of the role that the edge states play in the magnetization of a system. In both cases we present the results of numerical simulations that confirm the theoretical results. XXVI Ciclo 1979