Consider the variational inequality (VI) x* ∈C, ‹Fx*, x - x* ›≥0, x∈C (*) where C is a nonempty closed convex subset of a real Hilbert space H and F : C¡÷ H is a monotone operator form C into H. It is known that if F is strongly monotone and Lipschitzian, then VI (*) is equivalently turned into a fixed point problem of a contraction; hence Banach's contraction principle applies. However, in the case where F is inverse strongly monotone, VI (*) is equivalently transformed into a fixed point problem of a nonexpansive mapping. The purpose of this paper is to present some results which apply iterative methods for nonexpansive mappings to solve VI (*). We introduce Mann's algorithm and Halpern's algorithm and prove that the sequences generated by these algorithms converge weakly and respectively, strongly to a solution of VI (*), under appropriate conditions imposed on the parameter sequences in the algorithms.

Strong effective photon-photon interactions mediated by atom-photon couplings have been routinely achievable in QED setups for some time now. Recently, there have been several proposals to push the physics of interacting photons into many- body distributed architectures. The essential idea is to coherently couple together arrays of QED resonators, such that photons can hop between resonators while interacting with each other inside each resonator. These proposed structures have attracted intense theoretical attention while simultaneously inspiring experimental efforts to realise this novel regime of strongly-correlated many-body states of light. A central challenge of both theoretical and practical importance is to understand the physics of such coupled resonator arrays (CRAs) beyond equilibrium, when unavoidable (or sometimes even desired) photon loss processes are accounted for. This thesis presents several studies whose purpose can roughly be divided in two aims. The first part studies just what constitutes a valid physical and computational representation of non-equilibrium driven-dissipative CRAs. Addressing these ques- tions constitutes essential groundwork for further investigations of CRA phenomena, as numerical experiments are likely to guide and interpret near-future experimen- tal array observations. The relatively small body of existing work on CRAs out of equilibrium has often truncated their full, rich physics. It is important to establish the effects and validity of these approximations. To this end we introduce powerful numerical algorithms capable of efficiently simulating the full dynamics of CRAs, and use them to characterise the non-equilibrium steady states of arrays reached under the combined influence of dissipation and pumping. Having established the rigour necessary to realistically describe CRAs, we exam- ine two novel phenomena observable in near-future small arrays. Firstly we relate a counter-intuitive ‘super bunching’ in the statistics of photons emitted from arrays engineered to demonstrate strong effective photon-photon repulsion at the single and two-photon level, to an interplay between the underlying eigen-structure and details of the non-equilibrium operation. Secondly we characterise a dynamical phenomenon in which domains of ‘frozen’ photons remain trapped in sufficiently nonlinear arrays. Finally we present a preliminary characterisation of a previously unexplored phase diagram of arrays under coherent two-photon pumping. Com- petition between the coherence injected by the pumping, photon interactions and delocalisation processes lead to interesting new physical signatures.

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Orientador: Eduardo Miranda / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-10-31T13:24:18Z (GMT). No. of bitstreams: 1 HoyosNeto_JoseAbel_D.pdf: 1434769 bytes, checksum: 70acbb99e5c8d9636d4209b0919b56ca (MD5) Previous issue date: 2005 / Resumo: O propósito desse trabalho é estudar o papel da desordem em sistemas de spins fortemente interagentes de baixa dimensionalidade. Do ponto de vista teórico, cadeias de spin são extremamente atrativas por apresentarem uma nova física de baixas energias que vem da competição entre o ordenamento magnético e as .utuações quânticas. A introdução de desordem, onipresente no contexto experimental, é um elemento que pode desestabilizar as fases puras dando origem a uma nova física. Essa é a motivação principal do estudo de seu papel. Neste trabalho nós estudamos 4 sistemas de spins antiferromagnéticos desordenados:(i ) as escadas de spins-1/2 dos tipos 2 pernas e zig-zag, (ii ) as cadeias isotrópicas de spins SU(N), (iii ) a cadeia anisotrópica de spins SU(4), e (iv ) revisitamos a cadeia de spins-1/2. O estudo destes sistemas foi realizado aplicando generalizações da técnica do grupo de renormalização no espaço real para desordem forte. No caso do primeiro sistema, nós mostramos que as escadas de spins sempre renormalizam em cadeias de spins muito bem conhecidas. A escada de 2 pernas renormaliza para uma cadeia de spins-1/2 dimerizada antiferromagnética desordenada e, portanto, possui duas fases. Para dimerização forte ou equivalentemente desordem fraca, o sistema se encontra na fase de Haldane onde há um "gap" e a desordem é irrelevante. Para dimerização fraca ou equivalentemente desordem forte, o "gap" de Haldane se fecha e o sistema se encontra numa fase de Griffiths onde as quantidades termodinâmicas são controladas por um expoente não universal denominado expoente dinâmico z . Em contraste, a escada zig-zag renormaliza ou para uma cadeia de spins-1/2 antiferromagnética desordenada ou para uma cadeia de spins com acoplamentos ferro e antiferromagnéticos desordenada. Se a desordem e a frustração são suficientemente fracas, a escada renormaliza para a primeira cadeia, caso contrário esta pertence à mesma classe de universalidade da segunda. Além disso, relacionamos o expoente dinâmico da cadeia de spins com acoplamentos ferro e antiferromagnéticos com a distribuição de ponto fixo desses acoplamentos. Finalmente, através de argumentos simples, consideramos dizimações de acoplamentos correlacionados. Nesse caso, torna-se bem claro que a frustração é responsável pelo surgimento de acoplamentos ferromagnéticos que põem a escada na bacia de atração do ponto fixo das cadeias com acoplamentos ferro e antiferromagnéticos. Com relação à cadeia SU(N), desenvolvemos uma generalização do método do grupo de renormalização para desordem forte para uma cadeia isotrópica antiferromagnética de spins que pertencem à representações irredutíveis totalmente anti-simétricas do grupo SU(N), com N maior ou igual a 2. Conseguimos resolver as equações de fluxo analiticamente e descobrimos que essas cadeias pertencem a uma nova classe de universalidade cujos pontos fixos são de desordem infinita e, por tal motivo, nossos resultados se tornam assintoticamente exatos. Próximo a esses pontos fixos, os expoentes característicos são universais, i. e., independentes da desordem inicial da cadeia, e dependem somente do posto N do grupo de simetria. Devido às similaridades entre as regras de aglomeração de spins quando da dizimação de uma cadeia de spins com acoplamentos ferro e antiferromagnéticos e da dizimação da cadeia isotrópica de spins SU(N) no limite N ® µ , fomos capazes de calcular analiticamente, através de expansões de 1/N , a função correlação da primeira cadeia.Com relação à cadeia de spins SU(4), modificamos a generalização do método de grupo de renormalização para levar em conta a anisotropia dos acoplamentos. Conseguimos determinar o diagrama de fases através de cálculos analíticos e numéricos. Todos os pontos fixos encontrados são universais e de desordem infinita, entretanto, os expoentes característicos dependem de uma maneira não trivial da anisotropia do sistema. Por fim, revisitamos a cadeia de spins-1/2 antiferromagnética. Calculamos a amplitude da função de correlação média e a relacionamos com a amplitude da entropia de emaranhamento da mesma. Além disso, argumentamos em favor da universalidade dessas amplitudes / Abstract: The purpose of this thesis is the study of the role of quenched disorder in low-dimensional strongly interacting quantum spin systems. From the theoretical point of view, spin chains are extremely attractive due to their unconventional behavior that originates in the competition between magnetic ordering and quantum fluctuations. The introduction of disorder, ubiquitous in experimental realizations, is an element that can destabilize the clean phases giving rise to new physical behavior. That is the main motivation of this study. In this thesis, we study 4 random antiferromagnetic spin systems: (i ) the spin-1/2 two-leg and zigzag ladders, (ii ) the isotropic SU(N) spin chains, (iii ) the anisotropic SU(4) spin chain, and (iv ) we also revisit the spin-1/2 chain. For such a task, we use generalizations of the strong disorder real-space renormalization group method. Concerning the first systems, we show that the ladders are always renormalized to well-known spin chains. The two-leg ladder is renormalized to a random dimerized antiferromagnetic spin-1/2 chain, hence exhibiting two phases. For strong dimerization or equivalently weak disorder the system is in the gapful Haldane phase where disorder is irrelevant. Otherwise, the Haldane gap closes and the system is driven into a nonuniversal Griffiths phase, where the thermodynamical quantities are controled by the dynamical exponent z. In contrast, the zigzag ladder is renormalized either to a random antiferromagnetic spin-1/2 chain or to a random spin chain with both ferro- and antiferromagnetic couplings. If the randomness and frustration are sufficiently weak, the ladder is renormalized to the former chain, but otherwise it belongs to the same universality class of the latter one. In addition, we related the dynamical exponent of the ferro- and aniferromagnetic spin chain with its fixed point coupling constant distributions. Moreover, through simple qualitative arguments, we determined the phase diagram of the zigzag ladder with correlated disorder. That calculation clearly showed that frustration is responsible for the appearance of ferromagnetic couplings, which place the system in the basin of attraction of the fixed point of the ferro- and antiferromagnetic spin chains. With respect to theSU(N) spin chain, we developed a generalization of the strong-disorder renormalization group method to the case of an antiferromagnetic isotropic spin chain whose spins belong to the totally antisymmetric irreducible representations of the SU(N) group, with N greater than or equal to 2. We solved the flow equations analytically and found that such chains belong to a new universality class whose fixed point distributions are characterized by infinite disorder, rendering our results asymptotically exact. The characteristic exponents of these fixed points are universal, i. e., independent of the bare disorder, and depend only on the symmetry group rank. Due to the similarities of the spin clustering rules between the ferro- and antiferromagnetic spin chain and the isotropic SU(N) spin chain in the limit of N ® µ, we were able to analytically calcu- late, through a 1/N expansion, the mean correlation function of the former chain. In the case of the SU(4) spin chain, we modified the generalization of the renormalization group method to take into account the coupling anisotropy. We determined the phase diagram through analytical and numerical calculations. All fixed points found are universal and of infinite-randomness type. However, the characteristic exponents depend in a nontrivial fashion on the anisotropy. Finally, we revisited the antiferromagnetic spin-1/2 chain. We calculated the amplitude of the mean correlation function and related it with the amplitude of the entanglement entropy of the chain. In addition, we gave arguments in favor of the universality of these amplitudes / Doutorado / Física da Matéria Condensada / Doutor em Ciências

We investigate various forms of link positivity: braid positivity, strong quasipositivity, and quasi- positivity. On the one hand, this investigation is undertaken in the context of braid simplification: we give sufficient conditions under which a given braid word is conjugate to a braid word with strictly fewer negative bands. On the other hand, we use the famous Bennequin inequality to define a new link invariant: the defect of the Bennequin inequality, or 3-defect, and give criteria in terms of the 3-defect under which a given link is (strongly) quasipositive. Moreover, we use the 4-dimensional analogue of the Bennequin inequality, the slice Bennequin inequality in order to define the analogous defect of the slice Bennequin inequality, or 4-defect. We then investigate the relationship between the 4-defect and the most complicated class of 3- braids, Xu’s NP-form 3-braids, and establish several bounds. We also conjecture a formula for the signature of NP-form 3-braids which uses a new and easily computable NP-form 3-braid invariant, the offset. Finally, the appendices provide lists of all quasipositive and strongly quasipositive knots with at most 12 crossings (with two exceptions, 12n239 and 12n512), along with accompanying quasipositive or strongly quasipositive braid words. Many of these knots did not have previously established positivities or braid words reflecting these positivities—these facts were discovered using various criteria (conjectural or proven) expressed throughout this thesis.

Ce travail de thèse présente un nouvel ordre magnétique dans l'énigmatique phase de pseudo-gap des cuprates supraconducteurs à haute température critique. L'étude des composés YBa2Cu3O6+δ, HgBa2CuO4+δ et La1.92Sr0.08CuO4 par diffusion élastique de neutrons polarisés a permis de mettre en évidence un paramètre d'ordre magnétique en dessous d'une température comparable à celle de l'ouverture du pseudo-gap de ces systèmes. Nous avons également montré pour la première fois l'existence dans la famille HgBa2CuO4+δ de deux modes collectifs magnétiques associés à la phase de pseudo-gap. Tous ces résultats indiquent qu'à l'ouverture du pseudo-gap est associée une vraie transition de phase, avec un paramètre d'ordre magnétique et une symétrie brisée: la symétrie par renversement du temps. Il est toutefois important de noter que la symétrie de translation du réseau est préservée: on parle alors d'ordre à Q=0. Dans le système YBa2Cu3O6+δ, nous avons établi que lorsque l'on s'approche du composé parent, ou lorsque l'on introduit des impuretés telles que du Zn, les fluctuations de spin incommensurables autour du vecteur d'onde antiferromagnétique (QAF) se développent au détriment du nouvel ordre à Q=0. De manière similaire, nous avons pu mettre en évidence une interaction entre l'instabilité magnétique autour de QAF et le nouvel ordre à Q=0 dans La1.92Sr0.08CuO4. L'ensemble de ces résultats apporte une pièce maitresse au puzzle que représente toujours la supraconductivité à haute température critique, malgré 25 ans de recherche.

<p>Using field theoretical methods models of strongly correlated electrons have been investigated. Application to electronic structure calculations has been made.</p><p>In this thesis an attempt is made to build a bridge between first-principle band structure calculations and a theory of systems with strongly correlated electrons, by making use of perturbation theory from the atomic limit. Analyzing the total non-relativistic Hamiltonian leads to the basic model of strongly correlated systems, the Hubbard-Anderson model. In this thesis these basic models have been tested. Conclusions on delocalization and many-body aspects have been extracted from the solutions. Specifically for the lanthanides a separation of the f-system into two subsystems has resolved the discrepancy between calculated equilibrium volumes and experimental ones. The calculations are done within the Hubbard-I approximation, where it is possible to define renormalized fermion operators. The calculation is a true many-body calculation.</p><p>Using perturbation theory a set of self consistent equations has been formulated, and solved, for praseodymium metal using the periodical Anderson model. The solution shows a self consistent decrease of the Hubbard U, and delocalization of the f-shell, when crucial parameters of the model are changed. The most salient feature of the models for strongly correlated electrons is the transfer of spectral weight from one energy region to another by adjusting pressure or other external parameters. The effects come from kinematic interactions that are important for strongly correlated systems.</p><p>Investigations of the degenerate Hubbard model applied to the metal to insulator transition has also been made. When the degeneracy is considered, the transition to the metallic state occurs at smaller Coulomb energies. </p><p>The validity of the Fermi liquid description for strongly correlated electrons has also been studied. The results show that the general behavior of the Fermi liquid state is quite robust.</p>

Using field theoretical methods models of strongly correlated electrons have been investigated. Application to electronic structure calculations has been made. In this thesis an attempt is made to build a bridge between first-principle band structure calculations and a theory of systems with strongly correlated electrons, by making use of perturbation theory from the atomic limit. Analyzing the total non-relativistic Hamiltonian leads to the basic model of strongly correlated systems, the Hubbard-Anderson model. In this thesis these basic models have been tested. Conclusions on delocalization and many-body aspects have been extracted from the solutions. Specifically for the lanthanides a separation of the f-system into two subsystems has resolved the discrepancy between calculated equilibrium volumes and experimental ones. The calculations are done within the Hubbard-I approximation, where it is possible to define renormalized fermion operators. The calculation is a true many-body calculation. Using perturbation theory a set of self consistent equations has been formulated, and solved, for praseodymium metal using the periodical Anderson model. The solution shows a self consistent decrease of the Hubbard U, and delocalization of the f-shell, when crucial parameters of the model are changed. The most salient feature of the models for strongly correlated electrons is the transfer of spectral weight from one energy region to another by adjusting pressure or other external parameters. The effects come from kinematic interactions that are important for strongly correlated systems. Investigations of the degenerate Hubbard model applied to the metal to insulator transition has also been made. When the degeneracy is considered, the transition to the metallic state occurs at smaller Coulomb energies. The validity of the Fermi liquid description for strongly correlated electrons has also been studied. The results show that the general behavior of the Fermi liquid state is quite robust.

We show that the spectral kernel function of the dbar-Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary.

Let G be a graph and f be a function that maps V to {0,1,2, ..., diam(G)}. Let V+ be the set of all vertices such that f(v) is positive. If for every vertex v not in V+ there exists a vertex w in V+ such that the distance between v and w is at most f(w), then f is called a dominating broadcast of G. The cost of the broadcast f is the sum of the values f(v) over all vertices v in V. The minimum cost of a dominating broadcast is called the broadcast domination number of G. A subset S of V is a multipacking if, for every v in V and for every integer k which is at least 1 and at most rad(G), the set S contains at most k vertices at distance at most k from v. The multipacking number of G is the maximum cardinality of a multipacking of G. In the first part of the thesis, we describe how linear programming can be used to give a cubic algorithm to find the broadcast domination number and multipacking number of strongly chordal graphs. Next, we restrict attention to trees, and describe linear time algorithms to compute these numbers. Finally, we introduce k-broadcast domination and k-multipacking, develop the basic theory and give a bound for the 2-broadcast domination number of a tree in terms of its order. / Graduate