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Cooperative interactions permeate through nature, bringing about emergent behavior and complexity. Using a simple cooperative model, I illustrate the mean field dynamics that occur at the critical point of a second order phase transition in the framework of Langevin equations. Through this formalism I discuss the response, both linear and nonlinear, to external forces. Emphasis is placed on how information is transferred from one individual to another in order to facilitate the collective response of the cooperative network to a localized perturbation. The results are relevant to a wide variety of systems, ranging from nematic liquid crystals, to flocks and swarms, social groups, and neural networks.

Non-stationary motion of a cracked rotating shaft with accelerating or decelerating angular velocity Ω through a critical speed is studied. The shaft has a breathing transverse crack and a disk. There are two parts, which are the investigation of flexural response, neglecting the torsional vibrations, and the investigation of flexural-torsional response. In both studies the longitudinal vibration and the influence of shear deformation are neglected. The boundary conditions of the supports are simply supported for the transverse displacements and fixed-free in relation to torsion (for the flexural-torsional response only). The transverse surface crack, which causes a geometric discontinuity, is replaced by generalized moments at the crack location. The equations of motion follow the formulation of Wauer. Galerkin’s method and numerical integration are used to obtain approximate solutions. The maximum responses are determined. The effects of the acceleration and deceleration rate and the different parameters of the breathing cracked rotating shaft, such as crack depth, crack location, disk location, disk eccentricity, disk eccentricity angle, and disk mass, are studied. The influence of internal damping, external damping, and torsional external damping are investigated. Comparisons with an open cracked rotating shaft and an uncracked rotating shaft are also presented. The influence of torsional deformation is analyzed. The results are presented in tables and figures. / Master of Science

Em decorrência do grande avanço alcançado pela mecânica estatística devido a introdução das idéias de invariância conforme às teorias de escala para sistemas finitos, retomamos, neste trabalho, o estudo do modelo XY em campo transverso. A princípio, apresentamos uma análise detalhada do comportamento \"crossover\" característico do modelo, onde incluímos cálculos melhorados dos expoentes da susceptibilidade e do gap de energia anteriormente apresentados por dos Santos e Stinch-combe. Em seguida, uma análise numérica do espectro foi desenvolvida, considerando-se condições livres de contorno, e comparada com as previsões da invariância conforme. Finalmente, as correções à energia do estado fundamental de cadeias finitas foram utilizadas para obter o parâmetro que caracteriza as classes de universalidade (a carga central c). / In view of the great advance attached from statistical mechanics due to the conformal invariance ideas introduced to the scale theories, we take over at this work, the study of the XY model in a transverse field. At first, we present a detailed analysis on the sample\'s typical crossover behavior. An improved calculation of the susceptibility and gap exponents early presented by dos Santos and Stinchcombe is included. Nest, a numerical analysis of the spectrum, regarding free boundary condi¬tions was developed and compared with conformal invariance predictions. Finally, the fundamental state energy corrections of finite chains were used to obtain the parameter which ,distingoishes the universality classes (the central charge c).

In this work we investigate themes related to many-body systems in which multiple ground states are accessible, a condition known as frustration. Frustration can arise in a number of contexts, and we consider the consequences of this situation with some examples from condensed-matter physics. In some magnetic materials interactions between spins are such that no single spin configuration provides a unique ground state. In the class of frustrated magnets where the number of ground states is extensive, thermal fluctuations are strong even at temperatures significantly below the interaction strength. At such temperatures spins are highly correlated, and small perturbations may have profound consequences. In this thesis we provide an example of this. By considering classical n-component spins with nearest-neigbour exchange on a frustrated octahedral lattice we show that – in the limit where exchange interactions are large – the system is in a disordered, correlated phase where correlations have the form of a dipole field. This is termed a Coulomb phase. From this phase we induce an ordering transition, lifting the degeneracy with weak, additional short-range interactions. By studying the transition in the solvable limit of n → ∞, we discover that the transition has identical thermodynamics to that of a magnetic system interacting through long-range, dipolar forces. Finally, we provide a more apposite characterisation of the transition, where the high-temperature side of the transition is described through the fluctuations of solenoidal fields, and the ordering corresponds to a condensation of these fields. In a separate part of the thesis, we investigate the influence of disorder on frustrated lattices. We study a two-dimensional tight-binding model with nearest-neighbour hopping and on-site disorder. Restricting the allowed states to being those from the low-lying manifold of ground states, the disorder feeds through to act as effective disorder in the hopping terms, which decay algebraically with distance. The quasi-long range nature of this effective hopping leads to a situation in which the resultant single-particle eigenstates are critical, and we probe their behaviour numerically with a transfer matrix calculation.

Experimentally pure Lifshitz critical behavior has only been reported in the orthorhombic magnetic system MnP which exhibits uniaxial Ising-Type Lifshitz points LP (d=3, n=l, m=l). Two LPs have been previously identified in MnP. One when the external magnetic field H was applied along the intermediate b axis, the other when IH was along the hard anisotropy axis a. Both LPs occurs roughly at the same temperature (TL ~121K) but at different values of H due to the difference in anisotropy. In this work we report on measurements made with H applied within the ab plane. Two configurations were studied: (1) with H making an angle of 20° with the b axis and (2) with H at 45° in the b (and a) axis. The observed phase diagram obtained using an AC susceptibility technique shows, near the confluence of the modulated and ferromagnetic phases with the paramagnetic phase, the same characteristics of the phase boundaries found when H is along a and also along b . The temperature at the multicritical point is TL~120K. From an analysis of the shape of the phase boundaries near TL we obtained a crossover exponent =0.61±0.02, a value that is consistent with the found for IH along the a and the b axis. This value is also consistent with the predicted theoretically from Renormalization-Group calculation. These data suggest that there is a line of LPs connecting the multicritical points found when H is along the b and a axis. / Experimentally pure Lifshitz critical behavior has only been reported in the orthorhombic magnetic system MnP which exhibits uniaxial Ising-Type Lifshitz points LP (d=3, n=l, m=l). Two LPs have been previously identified in MnP. One when the external magnetic field H was applied along the intermediate b axis, the other when IH was along the hard anisotropy axis a. Both LPs occurs roughly at the same temperature (TL ~121K) but at different values of H due to the difference in anisotropy. In this work we report on measurements made with H applied within the ab plane. Two configurations were studied: (1) with H making an angle of 20° with the b axis and (2) with H at 45° in the b (and a) axis. The observed phase diagram obtained using an AC susceptibility technique shows, near the confluence of the modulated and ferromagnetic phases with the paramagnetic phase, the same characteristics of the phase boundaries found when H is along a and also along b . The temperature at the multicritical point is TL~120K. From an analysis of the shape of the phase boundaries near TL we obtained a crossover exponent =0.61±0.02, a value that is consistent with the found for IH along the a and the b axis. This value is also consistent with the predicted theoretically from Renormalization-Group calculation. These data suggest that there is a line of LPs connecting the multicritical points found when H is along the b and a axis.

Thesis (Ph.D.)--M.I.T, Dept. of Physics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Includes bibliographical references. / by Cyrus Reza Safinya. / Ph.D.

Foi desenvolvido o aparato experimental necessário à realização de experimentos de RMN em baixas temperaturas, com o objetivo de investigar fenômenos críticos nos sistemas antiferromagnéticos diluídos FexZn1-xF2. Estudou-se a dependência do segundo momento da linha de ressonância do F0 com a temperatura, desde T &#8776 TN até T &#8776 250K, através de medidas experimentais e simulações numéricas baseadas num modelo de campo médio. Também foi estudada a dependência angular da meia largura e da forma da linha. Os resultados das simulações numéricas concordam qualitativamente, mas não quantitativamente, com os resultados experimentais. Encontramos também a indicação de uma distribuição em T2 ao longo da linha de ressonância do F0, quando H0 é orientado perpendicularmente ao eixo C. O expoente crítico da magnetização do REIM foi medido diretamente a partir do segundo momento da linha do F0, após um esfriamento da amostra sem campo aplicado (Zero Field Cooling). O resultado obtido &#946=0.36 &#177 0.01 concorda com o valor esperado &#946= 0.35 / The experimental apparatus required for Nuclear Magnetic Resonance low temperatures experiments was developed for the investigation of critical phenomena in the diluted antiferromagnetic FexZn1-xF2. The dependence of the second moment of the F0 resonance line with the temperature was studied from T &#8776 TN to T &#8776 250K, both by experimental measurements and numerical mean field simulations. Angular dependence of the line width and line shape were also investigated. The numerically simulated results qualitatively agreed, but quantitatively didn´t agree with the experimental results. We found an indication of a distribution in T2 through the F0 resonance line, when H0 is oriented perpendicularly to the C axis. The magnetization critical exponent &#946 of the REIM was measured directly from the second moment of the F0 resonance line, after a zero field cooling procedure. We obtained &#946= 0.36 &#177 0.01 which is to be compared with the theoretical value &#946= 0.35

Experimentally pure Lifshitz critical behavior has only been reported in the orthorhombic magnetic system MnP which exhibits uniaxial Ising-Type Lifshitz points LP (d=3, n=l, m=l). Two LPs have been previously identified in MnP. One when the external magnetic field H was applied along the intermediate b axis, the other when IH was along the hard anisotropy axis a. Both LPs occurs roughly at the same temperature (TL ~121K) but at different values of H due to the difference in anisotropy. In this work we report on measurements made with H applied within the ab plane. Two configurations were studied: (1) with H making an angle of 20° with the b axis and (2) with H at 45° in the b (and a) axis. The observed phase diagram obtained using an AC susceptibility technique shows, near the confluence of the modulated and ferromagnetic phases with the paramagnetic phase, the same characteristics of the phase boundaries found when H is along a and also along b . The temperature at the multicritical point is TL~120K. From an analysis of the shape of the phase boundaries near TL we obtained a crossover exponent =0.61±0.02, a value that is consistent with the found for IH along the a and the b axis. This value is also consistent with the predicted theoretically from Renormalization-Group calculation. These data suggest that there is a line of LPs connecting the multicritical points found when H is along the b and a axis. / Experimentally pure Lifshitz critical behavior has only been reported in the orthorhombic magnetic system MnP which exhibits uniaxial Ising-Type Lifshitz points LP (d=3, n=l, m=l). Two LPs have been previously identified in MnP. One when the external magnetic field H was applied along the intermediate b axis, the other when IH was along the hard anisotropy axis a. Both LPs occurs roughly at the same temperature (TL ~121K) but at different values of H due to the difference in anisotropy. In this work we report on measurements made with H applied within the ab plane. Two configurations were studied: (1) with H making an angle of 20° with the b axis and (2) with H at 45° in the b (and a) axis. The observed phase diagram obtained using an AC susceptibility technique shows, near the confluence of the modulated and ferromagnetic phases with the paramagnetic phase, the same characteristics of the phase boundaries found when H is along a and also along b . The temperature at the multicritical point is TL~120K. From an analysis of the shape of the phase boundaries near TL we obtained a crossover exponent =0.61±0.02, a value that is consistent with the found for IH along the a and the b axis. This value is also consistent with the predicted theoretically from Renormalization-Group calculation. These data suggest that there is a line of LPs connecting the multicritical points found when H is along the b and a axis.

In this thesis, we study several different renormalization group (RG) methods, including the conventional Wilson renormalization group, Monte Carlo renormalization group (MCRG), exact renormalization group (ERG, or sometimes called functional RG), and tensor renormalization group (TRG). We use the two dimensional nearest neighbor Ising model to introduce many conventional yet important concepts. We then generalize the model to Dyson's hierarchical model (HM), which has rich phase properties depending on the strength of the interaction. The partition function zeros (Fisher zeros) of the HM model in the complex temperature plane is calculated and their connection with the complex RG flows is discussed. The two lattice matching method is used to construct both the complex RG flows and calculate the discrete β functions. The motivation of calculating the discrete β functions for various HM models is to test the matching method and to show how physically relevant fixed points emerge from the complex domain. We notice that the critical exponents calculated from the HM depend on the blocking parameter b. This motivated us to analyze the connection between the discrete and continuous RG transformation. We demonstrate numerical calculations of the ERG equations. We discuss the relation between Litim and Wilson-Polchinski equation and the effect of the cut-off functions in the ERG calculation. We then apply methods developed in the spin models to more complicated and more physically relevant lattice gauge theories and lattice quantum chromodynamics (QCD) like theories. Finite size scaling (FSS) technique is used to analyze the Binder cumulant of the SU(2) lattice gauge model. We calculate the critical exponent nu and omega of the model and show that it is in the same universality class as the three dimensional Ising model. Motivated by the walking technicolor theory, we study the strongly coupled gauge theories with conformal or near conformal properties. We compare the distribution of Fisher zeros for lattice gauge models with four and twelve light fermion flavors. We also briefly discuss the scaling of the zeros and its connection with the infrared fixed point (IRFP) and the mass anomalous dimension. Conventional numerical simulations suffer from the critical slowing down at the critical region, which prevents one from simulating large system. In order to reach the continuum limit in the lattice gauge theories, one needs either large volume or clever extrapolations. TRG is a new computational method that may calculate exponentially large system and works well even at the critical region. We formulate the TRG blocking procedure for the two dimensional O(2) (or XY ) and O(3) spin models and discuss possible applications and generalizations of the method to other spin and lattice gauge models. We start the thesis with the introduction and historical background of the RG in general.