Ph.D. (Theoretical Physics) / In this thesis the thermodynamic and magnetic properties of the non-degenerate Hubbard model are investigated. The underlying lattice is the bcc-lattice. The results obtained will therefore be especially applicable to systems with a single, narrow conduction band. As a check the thermodynamic properties of the model system are first calculated in two limiting cases, namely the free electron 'gas and the strong coupling limit. In this process, use is made of results related to Wick's Theorem, which are developed in an appendix. Another check is provided by the calculation of the ground state spectrum of a finite, fourpoint system. These results are obtained using standard group theory techniques. The ground state for the non-degenerate Hubbard model is solved approximatively by a variational method. Once again the necessary version of Wick's theorem is developed in an appendix. The ,results for the neutral case (i.e. a half-filled band) is in agreement with other studies on AB-lattices: It is found that the system is anti ferromagnetic for all values of the coupling constant. The quarter and three-quarter filled cases, hitherto not studied because of numerical complexity, yield a completely different picture. For increasing values of the coupling constant second order phase transitions are found, first from the para- to the ferromagnetic phase and then from the ferro- to the anti ferromagnetic phase. The only results available in the literature related to this case were obtained for an almost half-filled band in the strong-coupling limit and qualitatively support the findings of the present study. It is proposed that the simple theory used in this study be extended for use in physical systems such as Cr.

We study two interacting particles in a random potential chain by a transfer matrix method which allows a correct handling of the symmetry of the two- particle wave function, but introduces an artificial ¨bag¨ interaction. The dependence of the two-particle localization length lambta 2on disorder, interaction strength and range is investigated. Our results demonstrate that the recently proposed enhancement of lambta 2 as compared to the results for single particles is vanishingly small for a Hubbard interaction. For longer-range interactions, we observe a small enhancement but with a different disorder dependence than proposed previously.

This dissertation describes a theoretical study of strongly correlated electron systems. We present a variational quantum Monte Carlo approach based on matrix-product states, which enables us to naturally extend our work into higher-dimensional tensor-network states as well as to determine the ground state and the low-lying excitations of quasi-onedimensional electron systems. Our results show that the ground state of the quarterilled zigzag electron ladder is expected to exhibit a bond distortion whose pattern is not affected by the electron-electron interaction strength. This dissertation also presents a new method that combines a quantumMonte Carlo technique with a class of tensor-network states. We show that this method can be applied to two-dimensional fermionic or frustrated models that suffer from a sign problem. Monte Carlo sampling over physical states reveals better scaling with the size of matrices under periodic boundary conditions than other types of higher-dimensional tensor-network states, such as projected entangled-pair states, which lead to unfavorable exponential scaling in the matrix size.

<p> In this thesis, we investigate the possibility that a purely electronic mechanism is the cause of superconductivity in C60 materials. Several computational methods are adopted to calculate the pair-binding energy. They are perturbation theory, exact diagonalization, Gutzwiller projection, and auxiliary field Monte Carlo. Results from these different methods are compared with each other both in a C60 molecule and in other smaller molecules in order to test conclusions about whether or not a purely electronic mechanism can lead to an attractive interactions between electrons in C60 molecules.</p> <p> Besides this test of the superconductivity mechanism, we also explain in detail how to apply these different computational methods to C60 for the specific geometry of C60. Clearly illustrating these computational methods is the second goal of this thesis.</p> <p> Our final conclusion is that for both small and large Hubbard interaction U, there is pair binding in a single C60 molecule. For intermediate Hubbard interaction strength, there is no clear evidence for pair binding for the range of temperatures we explored. We suggest that the truncation of the Coulomb interaction, which is implicit in the Hubbard Hamiltonian, may suppress pair-binding of electrons in C60 and that it may be necessary to consider a model that includes the long range character of Coulomb interaction. This is a subject for further study.</p> / Thesis / Master of Science (MSc)

Almost a century after its discovery, superconductivity (SC) is still the most challenging and fascinating topic in condensed matter physics. Organic superconductors show exotic phases and phase transitions with a change in temperature or pressure. In this dissertation, we studied the phases and phase-transitions in one-dimensional (1D) and two-dimensional (2D) organic materials. This dissertation itself is a group of three sub-projects. In project (i), we studied the properties of a novel state “paired electron crystal” (PEC) in the quarterfilled Hubbard model to understand the phases and properties of 2D organic materials. We also studied the effects of charge and spin frustration on the 2D strongly correlated quarterfilled band. Our conclusions are based on exact diagonalization (ED) studies that include electron-electron and adiabatic electron-phonon interactions. For moderate to strong frustration, the dominant phase is a novel spin-singlet PEC. We discuss the implications of the PEC concept for understanding several classes of quarterilled band materials that display unconventional superconductivity. In project (ii), we studied the thermodynamics of a zigzag ladder model, applicable to quasi-1D organic materials. Using the quantum Monte Carlo (QMC) method, we studied the thermodynamics of charge ordering in quarterilled quasi-1D organic charge transfer solids (CTS). Previous theoretical studies on these CTS have focused on ground state properties or purely 1D systems. In the zigzag ladder, no separate high-temperature ordering is expected; instead the ladder is metallic at high temperature, and as temperature decreases, a single transition to the PEC state with a spin-gap takes place. In project (iii), we studied superconducting pairing correlation and metal-insulator transitions in the halfilled Hubbard model. We employed the Hubbard model and used the path integral renormalization group (PIRG) method for this study. Antiferromagneticmediated SC was suggested for small to large frustration in anisotropic triangular lattices. Previous work on the halfilled Hubbard model using the ED method was successful in showing the absence of d-wave SC on a small anisotropic triangular lattice. We extended this study to larger lattices to investigate the existence of long-range order of superconducting pair-pair correlations. We also show the absence of d-wave SC in this model on larger lattices.

The first project of this dissertation focuses on an extension of the Path Integral Renormalization Group (PIRG) method to the extended Hubbard model (EHM) including on-site U and a nearest-neighbor interaction V. The PIRG method is an efficient numerical algorithm for studying ground state properties of strongly correlated electron systems. A major advantage of the PIRG is that it is free from the Fermion sign problem. Many observables can be calculated using Wick’s theorem. The EHM is particularly important in models of charge-transfer solids (CTS) and at 1/4illing the V interaction drives a charge-ordered state. We test the method with comparisons to small two-dimensional (2D) clusters and long one-dimensional (1D) chains. The second project of this dissertation focuses on the Coulomb enhancement of superconducting pair-pair correlations in frustrated quarterilled band lattice systems. A necessary condition for superconductivity (SC) driven by electron correlation is that electronelectron (e-e) interactions enhance long range superconducting pair-pair correlations relative to the noninteracting limit. We present high-precision numerical calculations within the 2D Hubbard model on up to 100 sites showing that long range superconducting pair correlations are enhanced only for electron density 0.5. At all other fillings e-e interactions suppress pair correlations. We argue that the enhancement of pairing is due to a tendency to form local spin singlets at density 0.5. Our work provides a key ingredient to the mechanism of SC in the 2D organic-CTS superconductors, as well as in many other unconventional superconductors with frustrated crystal lattices and density 0.5. In the third project we apply our proposed concept to a real material, kappa-(BEDTTTF)2X. We present numerical results for 32 and 64 site lattices using the Constrained Path Monte Carlo and PIRG methods over a wide range of carrier density. We show that superconducting pair-pair correlations in this model are enhanced by e-e interactions for d-wave pairing symmetry uniquely for a hole density close to quarterilling. Our results indicate that this enhancement of superconductivity is not related to the presence of antiferromagnetic order, but to the strong tendency to spin-singlet formation in the quarterilled band.

No description available.

<p> Effects of electron correlations on a two dimensional quantum spin Hall system are studied. We examine possible phases of a generalized Hubbard model on a bilayer honeycomb lattice with a spin-orbit coupling and short range electron-electron repulsions at half filling, based on the slave rotor mean-field theory. The phase diagram of the model is found for a special case where the interlayer Coulomb repulsion is comparable to the intralayer Coulomb repulsion.</p> <p> Besides the conventional quantum spin Hall phase and a broken-symmetry insulating phase, we find a new phase, a fractionalized quantum spin Hall phase, where the quantum spin Hall effect arises for fractionalized spinons which carry only spin but not charge. Experimental manifestations of the exotic phase and effects of fluctuations beyond the saddle point approximation are also discussed.</p> <p> We finally study a toy Bose-Hubbard model for the charge sector of the theory to gain some insight into the phase diagram away from the special Coulomb repulsion values.</p> / Thesis / Master of Science (MSc)

Combinatorial methods are utilized to examine preimage iterations of topologically glued polynomials. In particular, this paper addresses using finite subdivision rules and Hubbard trees as tools to model the dynamic behavior of mated quadratic functions. Several methods of construction of invariant structures on modified degenerate matings are detailed, and examples of parameter-based families of matings for which these methods succeed (and fail) are given. / Ph. D.

Ultracold atoms in optical lattices offer an important tool for studying dynamics in many-body interacting systems in a pristine environment. This thesis focuses on three theoretical works motivated by recent optical lattice experiments. In the first, we theoretically study the center of mass dynamics of states derived from the disordered Bose-Hubbard model in a trapping potential. We find that the edge states in the trap allow center of mass motion even with insulating states in the center. We identify short and long-time mechanisms for edge state transport in insulating phases. We also argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our zero temperature results offer important insights into mechanisms of transport of atoms in trapped optical lattices while putting bounds on center of mass dynamics expected at non-zero temperature. In the second work, we study the domain wall expansion dynamics of strongly interacting bosons in 2D optical lattices with disorder in a recent experiment {[}J.-y. Choi et al., Science 352, 1547 (2016)]. We show that Gutzwiller mean-field theory (GMFT) captures the main experimental observations, which are a result of the competition between disorder and interactions. Our findings highlight the difficulty in distinguishing glassy dynamics, which can be captured by GMFT, and many-body localization, which cannot be captured by GMFT, and indicate the need for further experimental studies of this system. The last work features our study of phase diagrams of the 2D Bose-Hubbard model in an optical lattice with synthetic spin-orbit coupling. We investigate the transitions between superfluids with different phase patterns, which may be detected by measuring the spin-dependent momentum distribution. / Ph. D. / Ultracold atoms in optical lattices, a periodic potential generated by laser beams, offer an important tool for quantum simulations in a pristine environment. Motivated by recent optical lattice experiments with the implementation of disorder and synthetic spin-orbit coupling, we utilize Gutzwiller mean-field theory (GMFT) to study the dynamics of disordered state in an optical lattice under the sudden shift of the harmonic trap, the domain wall expansion of strongly interacting bosons in 2D lattices with disorder, and spin-orbit-driven transitions in the Bose-Hubbard model. We argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our findings show that evidence for many-body localization claimed in experiments [J.-y. Choi et al., Science 352, 1547 (2016)] must lie in the differences between GMFT and experiments. We also find that strong spin-orbit coupling alone can generate superfluids with finite momentum and staggered phase patterns.