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Agradecimentos: L.S.G.L. kindly acknowledges the financial support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under the Grant No. 162323/2017-4 Abstract: We study the flat-band ferromagnetic phase of the Haldane-Hubbard model on a honeycomb lattice within a bosonization scheme for flat-band Chern insulators, focusing on the calculation of the spin-wave excitation spectrum. We consider the Haldane-Hubbard model with the noninteracting lower bands in a nearly flat band limit, previously determined for the spinless model, and at 1/4 filling of its corresponding noninteracting limit. Within the bosonization scheme, the Haldane-Hubbard model is mapped onto an effective interacting boson model, whose quadratic term allows us to determine the spin-wave spectrum at the harmonic approximation. 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. 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 a staggered on-site energy term, both quantities associated with the two triangular sublattices. In both cases, we find that an energy gap opens at the K and K' points. Moreover, we also find some evidence for an instability of the flat-band ferromagnetic phase in the presence of the staggered on-site energy term. We provide some additional results for the square lattice topological Hubbard model previously studied within the bosonization formalism and comment on the differences between the bosonization scheme implementation for the correlated Chern insulators on both square and honeycomb lattices CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQ Fechado

We study a system of (N) over bar interacting triplons (the elementary excitations of a valence-bond solid) described by an effective interacting boson model derived within the bond operator formalism in order to determine the stability and the properties of many-triplon states. In particular, we consider the square lattice spin-1/2 J(1)-J(2) antiferromagnetic Heisenberg model, focus on the intermediate parameter region where a quantum paramagnetic phase sets in, and consider the columnar valence-bond solid as a reference state. Within the bond operator theory, the Heisenberg model is mapped into an effective boson model in terms of triplet operators t. The effective boson model is studied at the harmonic approximation and the energy of the triplons and the expansion of the triplon operators b in terms of the triplet operators t are determined. Such an expansion allows us to perform a second mapping and, therefore, determine an effective interacting boson model in terms of the triplon operators b. We then consider systems with a fixed number (N) over bar of triplons and study the stability of many-triplon states within a mean-field approximation. We show that many-triplon states are stable, the lowest-energy ones are constituted by a small number of triplons, and the excitation gaps are finite. For J(2) = 0.48J(1) and 0.52J(1), we also calculate spin-spin and dimer-dimer correlation functions, dimer order parameters, and the bipartite von Neumann entanglement entropy within our mean-field formalism in order to determine the properties of the many-triplon state as a function of the triplon number (N) over bar. We find that the spin and the dimer correlations decay exponentially and that the entanglement entropy obeys an area law, regardless of the triplon number (N) over bar. Moreover, only for J(2) = 0.48J(1), the spin correlations indicate that the many-triplon states with large triplon number (N) over bar might display a more homogeneous singlet pattern than the columnar valence-bond solid. We also comment on possible relations between the many-triplon states with large triplon number (N) over bar and gapped spin-liquid states FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP Fechado

Agradecimentos: We thank E. Miranda for helpful discussions. L.S.G.L. kindly acknowledges the financial support of the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 Abstract: We calculate the bipartite von Neumann and second Renyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to L' = 1000. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but they seem to not diverge as the dimerization decreases and the system approaches the Neel-VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPES Fechado

Agradecimentos: R.L.D. kindly acknowledges Faepex/PAPDIC and FAPESP, Project No. 2010/00479-6, for the financial support Abstract: We study the flat-band ferromagnetic phase of a topological Hubbard model within a bosonization formalism and, in particular, determine the spin-wave excitation spectrum. We consider a square lattice Hubbard model at 1/4-filling whose free-electron term is the p-flux model with topologically nontrivial and nearly flat energy bands. The electron spin is introduced such that the model either explicitly breaks time-reversal symmetry (correlated flat-band Chern insulator) or is invariant under time-reversal symmetry (correlated flat-band Z(2) topological insulator). We generalize for flat-band Chern and topological insulators the bosonization formalism [R. L. Doretto, A. O. Caldeira, and S. M. Girvin, Phys. Rev. B 71, 045339 (2005)] previously developed for the two-dimensional electron gas in a uniform and perpendicular magnetic field at filling factor nu = 1. We show that, within the bosonization scheme, the topological Hubbard model is mapped to an effective interacting boson model. We consider the boson model at the harmonic approximation and show that, for the correlated Chern insulator, the spin-wave excitation spectrum is gapless while, for the correlated topological insulator, gapped. We briefly comment on the possible effects of the boson-boson (spin-wave-spin-wave) coupling FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP Aberto

Agradecimentos: We thank M. Vojta, E. Miranda, and A. O. Caldeira for helpful discussions and FAPESP, Project No. 2010/00479-6, for the financial support Abstract: We study the plaquette valence-bond solid phase of the spin-1/2J(1)-J(2) antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four S = 1/2 spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the J(1)-J(2) model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region 0.34 < J(2)/J(1) < 0.59, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings J(2) = 0.34J(1) and J(2) = 0.59J(1). Interestingly, for J(2) < 0.48J(1), the excitation gap corresponds to a singlet-triplet excitation at the Gamma point while, for J(2) > 0.48J(1), it is related to a singlet-singlet excitation at the X = (pi/2,0) point of the tetramerized Brillouin zone FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP Aberto

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

Orientador: Ricardo Luís Doretto Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin Resumo: Nesse trabalho, calculamos a entropia de emaranhamento de um antiferromagneto de Heisenberg dimerizado em uma rede quadrada. Dois padrões de dimerização distintos são considerados: colunar e alternado. Em ambos os casos, focamos na fase de sólidos de singletos (VBS) que é descrita pela representação dos operadores de ligação. Nesse formalismo, o hamiltoniano de spin original é mapeado em um modelo efetivo de bósons interagentes com excitações de tripleto. O hamiltoniano efetivo é estudado na aproximação harmônica e o espectro das excitações elementares e o diagrama de fase dos dois modelos dimerizados são determinados. Consideramos um subsistema unidimensional (cadeia) de comprimento $L$ dentro de uma rede quadrada com condições periódicas de contorno e calculamos a entropia de emaranhamento. Seguimos um procedimento analítico baseado na teoria de ondas de spin modificadas que havia sido desenvolvido originalmente para calcular a entropia de emaranhamento em fases magneticamente ordenadas. Em particular, esse procedimento nos permite considerar subsistemas unidimensionais compostos por até 200 sítios. Combinamos esse procedimento com o formalismo dos operadores de ligação na aproximação harmônica e mostramos que, para os dois modelos de Heisenberg dimerizados, a entropia de emaranhamento da fase VBS obedece uma lei de área. Tanto para a dimerização colunar quanto para a alternada, mostramos que a entropia de emaranhamento aumenta à medida que o sistema se aproxima da transição de fase quântica entre as fases Néel-VBS Abstract: Nesse trabalho, calculamos a entropia de emaranhamento de um antiferromagneto de Heisenberg dimerizado em uma rede quadrada. Dois padrões de dimerização distintos são considerados: colunar e alternado. Em ambos os casos, focamos na fase de sólidos de singletos (VBS) que é descrita pela representação dos operadores de ligação. Nesse formalismo, o hamiltoniano de spin original é mapeado em um modelo efetivo de bósons interagentes com excitações de tripleto. O hamiltoniano efetivo é estudado na aproximação harmônica e o espectro das excitações elementares e o diagrama de fase dos dois modelos dimerizados são determinados. Consideramos um subsistema unidimensional (cadeia) de comprimento $L$ dentro de uma rede quadrada com condições periódicas de contorno e calculamos a entropia de emaranhamento. Seguimos um procedimento analítico baseado na teoria de ondas de spin modificadas que havia sido desenvolvido originalmente para calcular a entropia de emaranhamento em fases magneticamente ordenadas. Em particular, esse procedimento nos permite considerar subsistemas unidimensionais compostos por até 200 sítios. Combinamos esse procedimento com o formalismo dos operadores de ligação na aproximação harmônica e mostramos que, para os dois modelos de Heisenberg dimerizados, a entropia de emaranhamento da fase VBS obedece uma lei de área. Tanto para a dimerização colunar quanto para a alternada, mostramos que a entropia de emaranhamento aumenta à medida que o sistema se aproxima da transição de fase quântica entre as fases Néel-VBS Mestrado Física Mestre em Física CAPES 1547615/2015