Thesis advisor: Ying Ran / In this thesis, I investigated and implemented various numerical and simulation methods, including mean field theory, functional renormalization group method (fRG), density matrix renormalization group (DMRG) method etc., to find different quantum phases and quantum phase diagrams on models of correlated electronic systems. I found different phase diagrams with phases such as magnetism, superconductivity. By summarizing the strength and limitations of these methods, I investigated the projected entangled paired states (PEPS) with symmetry quantum number to sharply distinguish phases into crude classes and applied a variation of fast full update (FFU) prototype[58] to simulate different phases numerically. This method provides a promising, powerful and efficient way to simulate unconventional quantum phases and quantum phase diagrams in correlated electronic systems. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

In this dissertation we study the phases and phase transition properties of quantum ferromagnets and related magnetic materials. We first investigate the effects of an external magnetic field on the Goldstone mode of a helical magnet, such as MnSi. The field introduces a qualitatively new term into the dispersion relation of the Goldstone mode, which in turn changes the temperature dependences of the contributions of the Goldstone mode to thermodynamic and transport properties. We then study how the phase transition properties of quantum ferromagnets evolve with increasing quenched disorder. We find that there are three distinct regimes for different amounts of disorder. When the disorder is small enough, the quantum ferromagnetic phase transitions is generically of first order. If the disorder is in an intermediate region, the ferromagnetic phase transition is of second order and effectively characterized by mean-field critical exponents. If the disorder is strong enough the ferromagnetic phase transitions are continuous and are characterized by non-mean-field critical exponents.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: We present a theoretical approach that allows for the description of trapped Bose-Fermi mixtures with a tunable interspecies interaction in the vicinity of a Feshbach resonance magnetic field.The many-body physics of the system is treated at equilibrium using the well-established mean-field and local density approximations. This reduces the physics locally to that of a homogeneous system. We observe a rich local phase structure exhibiting both first and second order phase transitions between the normal and BEC phases. We also consider the global properties of the mixture at a fixed number of particles and investigate how the density profiles and the populations of the various particle species depend on the detuning and trap profile. / AFRIKAANSE OPSOMMING: Ons beskou ’n teoretiese beskrywing van gevangde Bose-Fermi mengsels met ’n verstelbare interspesie wisselwerking in die teenwoordigheid van ’n magneties-geïnduseerde Feshbach resonansie. Die veeldeeltjiefisika van die sisteem word by ekwilibrium binne die welbekende gemiddelde-veld en lokale-digtheid benaderings hanteer. Sodoende word die fisika lokaal tot die van ’n homogene sisteem gereduseer. Ons neem ’n ryk fase-struktuur waar met beide eerste- en tweede-orde fase-oorgange tussen die normale en BEK fases. Ons beskou ook die globale eienskappe van die mengsel by ’n vaste totale aantal deeltjies en ondersoek hoe die digtheidsprofiele en deeltjiegetalle van die afstemming en die profiel van die val afhang.

Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.

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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Singularities in quantum spectra - ground state and excited-state quantum phase transitions - are often connected with singularities in the classical limit of the system and have influence on other properties, such as quantum entanglement, as well. In the first part of the thesis we study quantum phase transitions within the U(2)-based Lipkin model. The relation between quasistationary points of the classical potential and the respective singularities in the spectrum is shown. In the second part, a system of two-level atoms interacting with electromagnetic field in an optical cavity is studied within two simplified models (non-integrable Dicke model and its integrable approximation known as Jaynes-Cummings model). The behaviour of quantum entanglement in these models is shown with a focus on the vicinity of the singular points.

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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.