This work explores the effects of weak nonlinearity on harmonic oscillators.Two particular systems are studied experimentally: A superconductingresonator formed from a coplanar waveguide that oscillates at microwave frequencies,and the cantilever of an atomic force microscope (AFM) vibratingat ultrasonic frequencies. Both of these systems are described in the introduction,followed by a theory chapter giving a general theoretical framework for nonlinear oscillators. Basic properties of nonlinear oscillators, such asbifurcation and intermodulation, are explained using simple models. Experimental methods, including cryogenic and microwave measurement techniques,are described in some detail. The nonlinear superconducting resonator is studied for use as a parametric amplifier. A strong drive tone, called the pump, drives the oscillator nearthe point of bifurcation. A second, much weaker drive signal that is slightlydetuned from the pump, will cause energy to move from the pump to the signal, giving signal amplification. We have measured a signal gain greaterthan 22 dB in a bandwidth of 30 kHz, for a resonator pumped at 7.6 GHz.This type of amplifier is phase-sensitive, meaning that signals in phase withthe pump will be amplified, but signals in quadrature phase of the pump will be deamplified. Phase-sensitivity has important implications on the amplifier’snoise properties. With a parametric amplifier, a signal can be amplified without any additional noise being added by the amplifier, something that is fundamentally impossible for a standard amplifier. The vibrating AFM cantilever becomes a nonlinear oscillator when it is interacting with a surface. When driven with two frequencies, the amplitudeand phase of the cantilever’s response will develop mixing products, or intermodulation products, that are very sensitive to the exact form of the nonlinearity. Very small changes in the surface properties will be detectable when measuring the intermodulation products. Simultaneously measuring many intermodulation products, or acquiring an intermodulation spectrum,allows one to reconstruct the tip-surface interaction. Intermodulation AFM increases the sensitivity of the measurement or the contrast of the acquiredimages, and provides a means of rapidly measuring the nonlinear tip-surface interaction. The method promises to enhance the functionality of the AFM beyond simple topography measurement, towards quantitative analysis of the chemical or material properties of the surface. / <p>QC 20100812</p>

Chapter 1 to 4 give a short introduction to superconductivity, microscopic theory, phase transitions, and Monte-Carlo simulations. Chapter 2 is about Cooper pairing in different settings, but I also give a short introduction to the Hofstadter problem of lattice fermions on a square lattice in a perpendicular magnetic field. The purpose is to clarify some points in Paper-I. Chapter 3 is about phase transitions, and introduces the important concepts of spontaneous symmetry breaking, scaling, and renormalization. In the last section I stress some of the main differences between first order and second order phase transitions. Chapter 4 starts with a short elementary introduction to Monte-Carlo simulations and proceeds with the important, but somewhat more advanced topic of reweighting. Chapter 5 to 7 are more closely related to the specific projects I have worked on, and are meant to illuminate and clarify some aspects in Paper-II and Paper-III. Chapter 5 introduce the Ginzburg-Landau model in various parametrizations, present some perturbative (mean-field) results, and introduce the concept of topological defects (vortices) and duality. Chapter 6 is closely related to Paper-II and introduce the concept of fractal dimension and the relation between the vortex excitations of the original theory and the dual field theory. Chapter 7 is closely related to Paper-III where we studied the order of the metal to superconductor phase transition. To do this we had to do infinite volume and continuum limit extrapolations. We also had to consider ultraviolet renormalization since the Ginzburg-Landau theory is a continuum field theory with no inherent short scale cut-off. To reduce auto-correlation times we added several improvements to the standard Metropolis algorithm in the Monte-Carlo simulations, the most important being an overrelaxation algorithm for the scalar field and a global update of the scalar amplitude.

Phase transitions and their critical scaling properties, especially in systems with disorder, are important both for our theoretical understanding of our environment, but also for their practical use in applications and materials in our everyday life. This thesis presents results from finite size scaling analysis of critical phenomena in systems with disorder, using high-precision Monte Carlo simulations and state of the art numerical methods. Specifically, theoretical models suitable for simulations in the presence of uncorrelated or correlated disorder are studied. Uncorrelated strong disorder, as present in the two dimensional gauge glass model to study the vortex glass phase of high temperature superconductors in an applied magnetic field is shown to lack a finite temperature phase transition. Further, results from dynamic quantities, such as resistance and autocorrelation functions, indicate the existence of two distinct diverging correlation times, one associated with local relaxation and one associated with vortex phase slips. Correlated disorder is studied both in the superfluid transition of helium-4 and in the anisotropic critical scaling of a transverse Meissner-like transition in an experimental setup of a high temperature superconductor. For the superfluid helium transition, it is shown that the presence of fractally correlated disorder presumably alters the universality class of the pure model. Also, a comparison with experimental data suggests that the critical scaling theory describing the heat capacity of helium-4 may need to be modified in the presence of the disorder. In the case of superconductors, analyzing experimental data from resistance measurements in a system with columnar defects together with an anisotropy in the applied magnetic field, reveals a fully anisotropic scaling regime. Finally, a data analysis is presented from simulations of a charged particle gas system in three dimensions, where the normal Coulomb interaction between charges is changed into a logarithmic interaction. Previous work indicates the possibility of a transition similar to the Kosterlitz-Thouless transition in certain two dimensional systems. On the contrary, our simulations seem to favor a system whose critical scaling behavior is consistent with a transition occurring only at zero critical temperature. Overall, disorder in the model systems studied leads to important modifications of the critical scaling properties of pure systems, and thereby also to possible changes of the corresponding universality classes. This results in interesting predictions with experimentally relevant consequences. / QC 20100811

In this thesis, specific heat, magnetic susceptibility and resistivity studies on the iron-pnictide superconductors LiFeAs, NaFe1-xCoxAs, AFe2As2 (A = K, Ca, Ba), M1-xNaxFe2As2 (M = Ca, Ba), and Ca(Fe1-xCox)2Fe2As2 are presented, from which different intrinsic physical properties are resolved. The combined first-order spindensity wave/structural transition which occurs in the parent compounds of the 122 pnictide systems is shown to gradually shift to lower temperature for low doping levels. Upon higher doping, this transition is completely suppressed and simultaneously, superconductivity appears at lower temperature. In contrast, the phase diagram in Ca(Fe1-xCox)2Fe2As2 is shown to exhibit a pronounced region of coexistence of magnetism and superconductivity. Further important results reported in this work concern the electronic properties and superconducting-gap characteristics. In LiFeAs, the zero-field temperature dependence of the electronic specific heat can be well described by two s-wave gaps, whose magnitudes are in agreement with ARPES results. Our gap analysis in KFe2As2, Ca0.32Na0.68Fe2As2, and Ba0.65Na0.35Fe2As2 single crystals also implies the presence of two s-wave-like gaps. The magnetic phase diagram of LiFeAs and KFe2As2 for magnetic fields along both principal orientations has been constructed and an anisotropy of Hc2(T) of 3 and 5, respectively, has been obtained.

The quest for obtaining higher Tc superconductivity led to the discovery of cuprates about 20 years ago. Since then, they continue to puzzle the scientific community with their bizarre properties like non-BCS superconductivity, pseudo gap, Fermi arcs, linear T resistivity etc. Since these materials show unusually high Tc, a novel mechanism is at play and strong correlations are believed to play an important role. The theme of this thesis work is to study physics of such strongly correlated systems in two dimensions at T = 0 along with development of new theoretical tools necessary for the study. The focus of the thesis is on the ground state studies of strongly correlated models like t-J and Hubbard models using variational Monte Carlo (VMC) and renormalized mean field theory (RMFT). The general method is to propose a variational wave function, motivated by the physics ideas, to be a candidate ground state of the system. Methods to efficiently evaluate the ground state energy and minimizing it with respect to the variational parameters are developed in this work. Antiferromagnetism-superconductivity competition and electron-hole asymmetry in the extended t-J model is investigated. The main result of this work is that increasing the magnitude of the next neighbor hopping (t') on hole doped side strengthen superconductivity while it stabilizes antiferromagnetism on the electron doped side. It is also shown that it is possible to characterize the T = 0 phase diagram with just one parameter called as Fermi Surface Convexity Parameter (FSCP). Next, the possibility of phase separation in the t-J model on a square lattice is investigated using local RMFT technique. It is found that for certain doping, the system phase separates into regions with antiferromagnetic and superconducting orders. Next, the role played by crystalline anisotropy of orthorhombic YBCO cuprates on their properties is examined using anisotropic tx-ty-J model and this ground state study suggests that the anisotropies seen in their properties are plausible solely due to the crystalline anisotropy. A new general method to study strongly correlated systems with singlet ground states is developed and tested in this thesis work. The last part of the thesis explores the possibility of high Tc superconductivity in graphene which is a intermediate coupling resonating valence bond (RVB) system. It is found that undoped graphene is not a superconductor, consistent with the experiments. On doping, the ground state of graphene is found to be a superconductor with “d+id” symmetry whose strength shows a dome as a function of doping which is reminiscent of RVB physics.

Recent studies of the transition metal dichalcogenide niobium diselenide have led to debate in the scientific community regarding the mechanism of the charge density wave (CDW) instability in this material. Moreover, whether or not CDW boosts or competes with superconductivity (SC) is still unknown, as there are experimental measurements which supports both scenarios. Motivated by these measurements we study the interplay of charge density modulations and superconductivity in the context of the Bogoliubov de-Gennes (BdG) equations formulated on a tight-binding lattice. As the BdG equations require large numerical demand, software which utilizes parallel algorithms have been developed to solve these equations directly and numerically. Calculations were performed on a large-scale Beowulf-class PC cluster at the University of Saskatchewan.<p> We first study the effects of inhomogeneity on nanoscale superconductors due to the presence of surfaces or a single impurity deposited in the sample. It is illustrated that CDW can coexist with SC in a finite-size s-wave superconductor. Our calculations show that a weak impurity potential can lead to significant suppression of the superconducting order parameter, more so than a strong impurity. In particular, in a nanoscale d-wave superconductor with strong electron-phonon coupling, the scattering by a weakly attractive impurity can nearly kill superconductivity over the entire sample.<p> Calculations for periodic systems also show that CDW can coexist with s-wave superconductivity. In order to identify the cause of the CDW instability, the BdG equations have been generalized to include the next-nearest neighbour hopping integral. It is shown that the CDW state is strongly affected by the magnitude of the next-nearest neighbour hopping, while superconductivity is not. The difference between the CDW and SC states is a result of the anomalous, or off-diagonal, coupling between particle and hole components of quasiparticle excitations. The Fermi surface is changed as next-nearest neighbour hopping is varied; in particular, the perfect nesting and coincidence of the nesting vectors and the vectors connecting van Hove singularities (vHs) for zero next-nearest neighbor hopping is destroyed, and vHs move away from the Fermi energy. It is found that within our one-band tight-binding model with isotropic s-wave superconductivity, CDW and SC can coexist only for vanishing nearest neighbor hopping and for non-zero hopping, the homogeneous SC state always has the lowest ground-state energy. Furthermore, we find in our model that as the magnitude of the next-nearest neighbor hopping parameter increases, the main cause of the divergence in the dielectric response accompanying the CDW transition changes from nesting to the vHs mechanism proposed by Rice and Scott. It is still an open question as to the origin of CDW and its interplay with SC in multiple-band, anisotropic superconductors such as niobium diselenide, for which fundamental theory is lacking. The work presented in this thesis demonstrates the possible coexistence of charge density waves and superconductivity, and provides insight into the mechanism of electronic instability causing charge density waves.

This thesis considers various strongly correlated quantum phases in solid state and cold atom spin systems. In the first part we focus on phases emerging in multi-orbital materials. We study even-parity spin-triplet superconductivity originating from Hund's coupling between t2g orbitals and investigate the effect of spin-orbit interaction on spin-triplet and spin-singlet pairing. Various aspects of the pairing state are discussed against the backdrop of the spin-triplet superconductor Sr2RuO4. Motivated by the remarkable phenomena observed in the bilayer compound Sr3Ru2O7, which point to the formation of an electronic nematic phase in the presence of critical fluctuations, we investigate how such a broken symmetry state emerges from electronic interactions. Since the broken x-y symmetry is revealed experimentally by applying a small in-plane magnetic field component, we examine nematic phases in a bilayer system and the role of the in-plane magnetic field using a phenomenological approach. In addition, we propose a microscopic mechanism for nematic phase formation specific to Sr3Ru2O7. The model is based on a realistic multi-orbital band structure and local and nearest neighbour interactions. Considering all t2g-orbital derived bands on an equal footing, we find a nematic quantum critical point and a nearby meta-nematic transition in the phase diagram. This finding harbours important implications for the phenomena observed in Sr3Ru2O7. The second part is devoted to the study of the anisotropic bilinear biquadratic spin-1 Heisenberg model, where the existence of an unusual direct phase transition between a spin-nematic phase and a dimerized valence bond solid phase in the quasi-1D limit was conjectured based on Quantum Monte Carlo simulations. We establish the quasi-1D phase diagram using a large-N Schwinger boson approach and show that the phase transition is largely conventional except possibly at two particular points. We further discuss how to realize and to detect such phases in an optical lattice.

This thesis considers various strongly correlated quantum phases in solid state and cold atom spin systems. In the first part we focus on phases emerging in multi-orbital materials. We study even-parity spin-triplet superconductivity originating from Hund's coupling between t2g orbitals and investigate the effect of spin-orbit interaction on spin-triplet and spin-singlet pairing. Various aspects of the pairing state are discussed against the backdrop of the spin-triplet superconductor Sr2RuO4. Motivated by the remarkable phenomena observed in the bilayer compound Sr3Ru2O7, which point to the formation of an electronic nematic phase in the presence of critical fluctuations, we investigate how such a broken symmetry state emerges from electronic interactions. Since the broken x-y symmetry is revealed experimentally by applying a small in-plane magnetic field component, we examine nematic phases in a bilayer system and the role of the in-plane magnetic field using a phenomenological approach. In addition, we propose a microscopic mechanism for nematic phase formation specific to Sr3Ru2O7. The model is based on a realistic multi-orbital band structure and local and nearest neighbour interactions. Considering all t2g-orbital derived bands on an equal footing, we find a nematic quantum critical point and a nearby meta-nematic transition in the phase diagram. This finding harbours important implications for the phenomena observed in Sr3Ru2O7. The second part is devoted to the study of the anisotropic bilinear biquadratic spin-1 Heisenberg model, where the existence of an unusual direct phase transition between a spin-nematic phase and a dimerized valence bond solid phase in the quasi-1D limit was conjectured based on Quantum Monte Carlo simulations. We establish the quasi-1D phase diagram using a large-N Schwinger boson approach and show that the phase transition is largely conventional except possibly at two particular points. We further discuss how to realize and to detect such phases in an optical lattice.

No description available.

Over the past 20 years a great deal of progress has been made towards understanding the physics of the high-temperature (high-Tc) cuprate superconductors. Much of the low- energy physics of these materials appears to be captured by two-dimensional Hubbard or t-J models which have provided significant insight into a number of properties such as the pseudogap, antiferromagnetism and superconductivity itself. However, intrinsically planar models are unable to account for the large variations in Tc observed across materials nor do they capture the electron-phonon (el-ph) interaction, the importance of which a number of experimental probes now indicate. This thesis examines the el-ph interaction in cuprates using a combination of analytical and numerical techniques. Starting from the microscopic mechanism for coupling to in-plane and c-axis polarized oxygen phonons, the theory of el-ph coupling is presented. The el-ph self-energy is derived in the context of Migdal-Eliashberg theory and then applied to understanding the detailed temperature and doping dependence of the renormalizations observed by Angle-resolved photoemission spectroscopy. The qualitative signatures of el- boson coupling in the density of states of a d-wave superconductor are also examined on general grounds and a model calculation is presented for el-ph coupling signatures in the density of states. Following this, the theory is extended to include the effects of screening and the consequences of this theory are explored. Due to the quasi-2D nature of the cuprates, screening is found to anomalously enhance the el-ph contribution to d-wave pairing. This result is then considered in light of the material and doping dependence of Tc and a framework for understanding the materials variations in Tc is presented. From these studies, a detailed picture of the role of the el-ph interaction in the doped cuprates emerges where the interaction, working in conjunction with a dominant pairing interaction, provides much of the materials variations in Tc observed across the cuprate families. Turning towards numerical techniques, small cluster calculations are presented which examine the effects of a local oxygen dopant in an otherwise ideal Bi2Sr2CaCu2O8+δ crystal. Here, it is demonstrated that the dopant locally enhances electronic properties such as the antiferromagnetic exchange energy J via local el-ph coupling to planar local oxygen vibrations. Finally, in an effort to extend the scope of this work to the underdoped region of the phase diagram, an examination of the properties of the single-band Hubbard and Hubbard-Holstein model is carried out using Determinant Quantum Monte Carlo. Here focus is placed on the spectral properties of the model as well as the competition between the the antiferromagnetic and charge-density-wave orders. As with the small cluster calculations, a strong interplay between the magnetic and lattice properties is observed.