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.

Strong electron-electron correlations and lattice frustration are two physical interactions that pose serious challenges to condensed matter physics. A variety of exotic physical phenomena, for example, charge ordering, spin liquid, and unconventional superconductivity, are believed to arise from the interplay of the two interactions. In this dissertation, I examine two families of systems which exhibit both electron-electron correlations and lattice frustration – charge transfer solids and layered cobaltates. The half-filled band Hubbard model on the triangular lattice has been proposed by mean-field theories as the minimal model for the superconductivity in the charge transfer solids. In the first part of this dissertation, by using exact calculations, I prove the absence of superconductivity in this model. This result calls for a new theoretical approach to describe the rich physics in charge transfer solids. In the second part of this dissertation, I study charge transfer solids by focusing on its real bandfilling ¼. I show that a new kind of insulating phase, paired electron crystal, emerges from antiferromagnetism as the frustration is increased. The paired electron crystal state can explain the various insulating states adjacent to the superconducting state, thus provides a new avenue towards the understanding of the unconventional superconductivity in charge transfer solids and other ¼ filled systems. In the third part of this dissertation, I investigate the carrier concentration-dependent electronic behavior in layered cobaltates. I provide a natural yet simple explanation for this behavior. I show that it can be described within correlated-electron Hamiltonians with finite on-site and significant nearest neighbor hole-hole Coulomb repulsions. I also point out the similarities between organic charge transfer solids and layered cobaltates, which may involve superconductivity.