Nesta dissertação estudamos sistemas de spins em redes de baixa dimensionalidade e em temperatura nula, analisando suas transições de fases quânticas. Mais precisamente, estu- damos as propriedades do estado fundamental e as possíveis transições de fase do modelo de Heisenberg quântico antiferromagnético de spin-1/2, com interações entre os primeiros e segundos vizinhos, em diversas redes, e em particular na rede triangular, que é o foco de nosso estudo. Para a obtenção do estado fundamental aproximado, usamos um método variacional em que a rede é particionada num conjunto de plaquetas de sítios. O estado fundamental é escrito como um produto tensorial dos estados das plaquetas. Para a rede triangular, escolhemos um triângulo como uma plaqueta. Quatro fases foram encontra- das: a fase antiferromagnética de Néel, a colinear, a fase de Néel modificada e aquela que denominamos de ligação covalente ressonante. Obtivemos as energias e as magnetizações de subrede em função da razão entre as interações de primeiros e segundos vizinhos. En- tre as fases de Néel e a colinear, podemos observar a fase de ligação covalente ressonante caracterizada como um singleto quanto ao spin de cada plaqueta. / In this thesis we study spin systems in low-dimensional lattices at zero temperature, analyzing their quantum phase transitions. More precisely, we study the properties of the ground state and the possible phase transitions in the antiferromagnetic spin-1/2 quan- tum Heisenberg model with interaction between the first and second neighbors, in several lattices, and in particular in the triangular lattice, which is the focus of our study. To obtain the approximate ground state, we use a variational method in which the lattice is partitioned into a set of plates of sites. The ground state is written as a tensor product of the states of plates. For the triangular lattice, we choose a triangle as a plate. Four phases were found: the antiferromagnetic Néel phase, the collinear, the modified Néel phase and that we call resonating valence bond. We obtained the energy and the magnetization as a function of the ratio of the interactions between the first and second neighbor sites. Between the Néel and collinear phases, we can observe the spin resonating valence bond phase, characterized as a singlet with respect to the spin of each plate.

Nesta dissertação estudamos sistemas de spins em redes de baixa dimensionalidade e em temperatura nula, analisando suas transições de fases quânticas. Mais precisamente, estu- damos as propriedades do estado fundamental e as possíveis transições de fase do modelo de Heisenberg quântico antiferromagnético de spin-1/2, com interações entre os primeiros e segundos vizinhos, em diversas redes, e em particular na rede triangular, que é o foco de nosso estudo. Para a obtenção do estado fundamental aproximado, usamos um método variacional em que a rede é particionada num conjunto de plaquetas de sítios. O estado fundamental é escrito como um produto tensorial dos estados das plaquetas. Para a rede triangular, escolhemos um triângulo como uma plaqueta. Quatro fases foram encontra- das: a fase antiferromagnética de Néel, a colinear, a fase de Néel modificada e aquela que denominamos de ligação covalente ressonante. Obtivemos as energias e as magnetizações de subrede em função da razão entre as interações de primeiros e segundos vizinhos. En- tre as fases de Néel e a colinear, podemos observar a fase de ligação covalente ressonante caracterizada como um singleto quanto ao spin de cada plaqueta. / In this thesis we study spin systems in low-dimensional lattices at zero temperature, analyzing their quantum phase transitions. More precisely, we study the properties of the ground state and the possible phase transitions in the antiferromagnetic spin-1/2 quan- tum Heisenberg model with interaction between the first and second neighbors, in several lattices, and in particular in the triangular lattice, which is the focus of our study. To obtain the approximate ground state, we use a variational method in which the lattice is partitioned into a set of plates of sites. The ground state is written as a tensor product of the states of plates. For the triangular lattice, we choose a triangle as a plate. Four phases were found: the antiferromagnetic Néel phase, the collinear, the modified Néel phase and that we call resonating valence bond. We obtained the energy and the magnetization as a function of the ratio of the interactions between the first and second neighbor sites. Between the Néel and collinear phases, we can observe the spin resonating valence bond phase, characterized as a singlet with respect to the spin of each plate.

In recent years there has been significant work aimed at understanding what effect the variation of electronic doping has on material properties. In the high transition-temperature (high-T$_c$) cuprate superconductors hole doping has an impact on the superconducting transition temperature. In the underdoped regime, the cuprates exhibit anomalous properties due to a pseudogap which forms and is thought to be related to Mott insulating physics. While there is no general consensus as to the mechanism underlying high temperature superconductivity, the resonating valence bond (RVB) theory proposed by Anderson in 1987 with a Gutzwiller projected d-wave BCS wave function could give a first picture of the high-T$_c$ cuprates. We have calculated properties of the cuprates using the assumption that the pseudogap state acts as a normal state to an otherwise standard BCS mean field theory. We find that the phenomenological RVB spin liquid model proposed by Yang, Rice and Zhang (YRZ) is highly successful at describing the doping dependent features of the cuprates. Through application of the YRZ model and the tools of many-body theory we present results on anomalous properties observed in: electronic specific heat; Raman and angle-resolved photoemission spectroscopy (ARPES) data; effective mass renormalization; and thermal broadening seen in ARPES. We verify that the YRZ ansatz qualitatively describes these anomalies along with their doping dependent variations. We conclude from this work that the physics underlying the pseudogap, while distinct in origin from superconductivity, is likely to arise from an RVB wavefunction that is closely related to the BCS state. In graphene, variation in doping modifies the polarization function which describes a screened electron-electron interaction. This leads to additional features in the spectral function which are due to electron-plasmon coupling. In this work, we calculated the electronic density of states including this interaction along with its doping dependence with and without an electron-phonon interaction. We find clear features of electron-electron interactions in the density of states. These features are related to the energies of plasmaron bands in the spectral function and can be modified through doping so as to be distinct from the phonon energy scales.