In this thesis we study the scaling properties of unconventional metals that arise at quantum critical points using low-energy effective field theories. Due to high rate of scatterings between electrons and critical fluctuations of the order parameter associated with spontaneous symmetry breaking, Landau’s Fermi liquid theory breaks down at the critical points. The theories that describe these critical points generally flow into strong coupling regimes at low energy in two space dimensions. Here we develop and utilize renormalization group methods that are suitable for the interacting non-Fermi liquids. We focus on the critical points arising at excitonic, and commensurate spin and charge density wave transitions. By controlled analyses we find stable non-Fermi liquid and marginal Fermi liquid states, and extract the scaling behaviour. The field theories for the non-Fermi liquids are characterized by symmetry groups, local curvature of the Fermi surface, the dispersion of the order parameter fluctuations, and dimensions of space and Fermi surface. / Thesis / Doctor of Philosophy (PhD)

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The Kibble-Zurek mechanism (KZM) primarily characterizes scaling in the formation of topological defects when a system crosses a continuous phase transition. The KZM was first used to study the evolution of the early universe, describing the topology of cosmic domains and strings as the symmetry-breaking phase transitions acted on the vacuum fields during the initial cooling. A ferromagnetic spin-1 $^{87}$Rb Bose-Einstein condensate (BEC) exhibits a second-order gapless quantum phase transition due to a competition between the magnetic and collisional spin interaction energies. Unlike extended systems where the KZM is illustrated by topological defects, we focus our study on the temporal evolution of the spin populations and observe how the scaling of the spin dynamics depend on how fast the system is driven through the critical point. In our case, the excitations are manifest in the temporal evolution of the spin populations illustrating a Kibble-Zurek type scaling, where the dynamics of slow quenches through the critical point are predicted to exhibit universal scaling as a function of quench speed. The KZM has been studied theoretically and experimentally in a large variety of systems. There has also been a tremendous interest in the KZM in the cold atoms community in recent years. It has been observed not only in ion chains and in atomic gases in optical lattices, but also in Bose gases through the formation of vortices or solitons. The KZM in the context of crossing the quantum phase transition in a ferromagnetic BEC has been theoretically studied, but this thesis is the first experimental investigation of this phenomenon.

Neste trabalho estudamos um sistema bidimensional composto de duas espécies atômicas condensadas, uma delas contendo uma rede de vórtices. Analogamente ao modelo desenvolvido para tratar de átomos ultrafrios em redes ópticas, mapeamos o Hamiltoniano do nosso sistema com o Hamiltoniano do modelo Bose-Hubbard (BH), com o potencial periódico da rede advindo da interação de campo médio entre as duas espécies. A variação do comprimento de espalhamento atômico permite alterar as propriedades do potencial confinante, com a indução da transição de fase quântica na espécie aprisionada nos vórtices. O novo aspecto trazido pela rede de vórtices advém dos seus modos de excitação de baixa energia, os modos de Tkachenko. Consideramos os efeitos da dinâmica própria desse potencial sobre a espécie aprisonada através de um modelo BH efetivo, com novos valores para interação local e tunelamento, além de um termo adicional de interação de longo alcance, mediada pelos modos da rede. Além de complementar os estudos com redes ópticas estáticas, a proposta teórica desenvolvida apresenta grande viabilidade experimental no contexto das técnicas atuais para manipulação de átomos ultrafrios. / In this work we consider a two dimensional system composed of two condensed atomic species, one containing a vortex lattice. Analogously to the model used to describe ultracold atoms in optical lattices, we mapped our system Hamiltonian in the Hamiltonian of the Bose-Hubbard (BH) model, with the periodic lattice potential arising from the meanfield interaction between the two species. The variation of the atomic scattering length allow us to change the properties of the confining potential, to induce the quantum phase transition in the species trapped in the vortices. The new aspect brought by the vortex lattice comes with its low energy normal modes, the Tkachenko modes. We considered the effects of such dynamic potential over the confined species thought an effective BH model, with new values for the local interaction and tunneling parameters, besides an additional long-range interaction term mediated by the lattice modes. Our theoretical proposal goes beyond the studies with static optical lattice. Additionally, it has great feasibility in the current context of ultra-cold atoms experimental techniques.

Quantum phase transitions (QPT) on the border of magnetism have provided a fertile hunting ground for the discovery of new states of matter, for example; the marginal Fermi Liquid and non Fermi Liquid states as well high T$_C$ cuprate and magnetically mediated superconductivity. In this thesis I present work on three materials in which it may be possible to tune the system through a magnetic QPT with the application of hydrostatic pressure. Although the details of the underlying physics are different in each of the materials, they are linked by the possibility of finding new states on the border of magnetism. Applying hydrostatic pressure, we have suppressed the ferromagnetic (FM) transition in metallic Fe$_2$P to very low temperature and to a potential QPT. Counter-intuitive broadening of the magnetic hysteresis leading up to the FM-AFM QPT may well be a crucial clue as to the nature of the model needed to understand this phase transition. A sharp increase in the quasi-particle scattering cross-section as well as the residual resistivity accompany a departure from the quadratic temperature dependence of the resistivity. This possible deviation from Fermi liquid behaviour is stable over a significant range of temperature. The unexplained upturn in the resistivity of CeGe that accompanies the AFM transition was studied under pressure. Pressure increased the residual resistivity as well as decreasing the relative size of the upturn, but had a moderate effect on the Neel temperature. The insensitivity of the N$\acute e$el temperature to pressure has been compared to its relative sensitivity to applied feld. The existence of the upturn and its evolution with pressure and applied feld can reasonably be argued to be due to the details of the electron band structure in the system. By applying pressure we have drastically reduced the resistivity of the insulating antiferromagnet NiPS$_3$. Concurrent work on FePS$_3$ has shown metallisation under pressure. It seems reasonable to speculate that NiPS$_3$ may also metallise at higher pressure. The energy gap is narrowed in both materials as pressure is increased. Magnetisation measurements have revealed a low temperature upturn indicating some possible ferromagnetic component or proximity to another magnetic state. A peak in the magnetisation is also seen at 45K in zero-feld cooled measurements. Both of these features point to a system with a complex magnetic ground state.

This thesis includes both experimental and theoretical investigations, presented in a series of eight papers. The experimental part ranges from the construction procedures of an apparatus for Bose-Einstein condensates, to full scale experiments using three different set-ups for ultracold atoms in optical lattices. As one of the main themes of the thesis, an experimental apparatus for production of Bose-Einstein Condensates is under construction. A magneto-optically trapped sample, hosting more than 200 million 87Rb atoms, have successfully been loaded into a magnetic trap with high transfer rate. The lifetime of the sample in the magnetic trap is in the range of 9 s, and the atoms have been shown to respond to evaporative cooling. The experiment is ready for optimization of the magnetic trap loading, and evaporative cooling parameters, which are the final steps for reaching Bose-Einstein condensation. The set-up is designed to host experiments including variable geometry optical lattices, and includes the possibility to align laser beams with high angular precision for this purpose. The breakdown of Bloch waves in a Bose-Einstein condensate is studied, attributed to the effect of energetic and dynamical instability. This experimental study is performed using a Bose-Einstein condensate in a moving one-dimensional optical lattice at LENS, Florence Italy. The optical lattice parameters, and the thermal distribution of the atomic sample required to trigger the instabilities, are detected, and compared with a theoretical model developed in parallel with the experiments. In close connection with these one-dimensional lattice studies, an experimental survey to characterize regimes of superradiant Rayleigh scattering and Bragg scattering is presented. Tunneling properties of repulsively bound atom pairs in double well potentials are characterized in an experiment at Johannes Gutenberg University, Mainz Germany. A three-dimensional optical lattice, producing an array of double wells with tunable properties is let to interact with a Bose-Einstein condensate. Pairs of ultracold atoms are produced on one side in the double wells, and their tunneling behavior, dependent on potential barrier and repulsion properties, is studied. A theoretical study of the crossover between one- and two-dimensional systems has been performed. The simulations were made for a two-dimensional array of atoms, where the behavior for different tunneling probabilities and atom-atom repulsion strengths was studied. Scaling relations for systems of variable sizes have been examined in detail, and numerical values for the involved variables have been found.

This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an $n-$cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.

This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an $n-$cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.

This thesis focuses on the analysis of Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) evolution in ultracold superfluid Fermi gases when the interaction between atoms is varied. The tuning of attractive interactions permits the ground state of the system to evolve from a weak fermion attraction BCS limit of loosely bound and largely overlapping Cooper pairs to a strong fermion attraction limit of tightly bound small bosonic molecules which undergo BEC. This evolution is accompanied by anomalous behavior of many superfluid properties, and reveals several quantum phase transitions. This thesis has two parts: In the first part, I analyze zero and nonzero orbital angular momentum pairing effects, and show that a quantum phase transition occurs for nonzero angular momentum pairing, unlike the $s$-wave case where the BCS to BEC evolution is just a crossover. In the second part, I analyze two-species fermion mixtures with mass and population imbalance in continuum, trap and lattice models. In contrast with the crossover physics found in the mass and population balanced mixtures, I demonstrate the existence of phase transitions between normal and superfluid phases, as well as phase separation between superfluid (paired) and normal (excess) fermions in imbalanced mixtures as a function of scattering parameter and mass and population imbalance.

Neste trabalho estudamos um sistema bidimensional composto de duas espécies atômicas condensadas, uma delas contendo uma rede de vórtices. Analogamente ao modelo desenvolvido para tratar de átomos ultrafrios em redes ópticas, mapeamos o Hamiltoniano do nosso sistema com o Hamiltoniano do modelo Bose-Hubbard (BH), com o potencial periódico da rede advindo da interação de campo médio entre as duas espécies. A variação do comprimento de espalhamento atômico permite alterar as propriedades do potencial confinante, com a indução da transição de fase quântica na espécie aprisionada nos vórtices. O novo aspecto trazido pela rede de vórtices advém dos seus modos de excitação de baixa energia, os modos de Tkachenko. Consideramos os efeitos da dinâmica própria desse potencial sobre a espécie aprisonada através de um modelo BH efetivo, com novos valores para interação local e tunelamento, além de um termo adicional de interação de longo alcance, mediada pelos modos da rede. Além de complementar os estudos com redes ópticas estáticas, a proposta teórica desenvolvida apresenta grande viabilidade experimental no contexto das técnicas atuais para manipulação de átomos ultrafrios. / In this work we consider a two dimensional system composed of two condensed atomic species, one containing a vortex lattice. Analogously to the model used to describe ultracold atoms in optical lattices, we mapped our system Hamiltonian in the Hamiltonian of the Bose-Hubbard (BH) model, with the periodic lattice potential arising from the meanfield interaction between the two species. The variation of the atomic scattering length allow us to change the properties of the confining potential, to induce the quantum phase transition in the species trapped in the vortices. The new aspect brought by the vortex lattice comes with its low energy normal modes, the Tkachenko modes. We considered the effects of such dynamic potential over the confined species thought an effective BH model, with new values for the local interaction and tunneling parameters, besides an additional long-range interaction term mediated by the lattice modes. Our theoretical proposal goes beyond the studies with static optical lattice. Additionally, it has great feasibility in the current context of ultra-cold atoms experimental techniques.