Density-Matrix Renormalization Group and Model Reduction Studies of Two-Dimensional Doped and Frustrated Systems

The density-matrix renormalization group (DMRG) invented by Steven R. White is a variational algorithm to search for the ground states of quantum many-body systems. Using the entanglement entropy as its organizing principle, DMRG stands as one of the most powerful methods in simulating two-dimensional (2D) quantum systems, and is especially useful for investigating strongly correlated systems that are otherwise challenging for analytical approaches. This thesis presents the applications and developments of DMRG and related tensor network methods in studying a variety of 2D doped and frustrated systems as well as their model reductions. Chapter 1 lays out the fundamentals of DMRG and tensor network states, along with multiple techniques for studying 2D systems. Chapter 2 presents our DMRG studies of the ground state phase diagram of the extended $t$-$J$ models. We found that while the models are consistent with the cuprates in antiferromagnetism and charge order, superconductivity nevertheless appears absent or marginal in hole-doped systems. Motivated by this discrepancy between the models and the cuprates, in Chapter 3 we carried out a DMRG-based downfolding of the parental three-band Hubbard model, seeking possible fixes to the previously studied single-band models. An effective model was derived via Wannier construction, which includes novel density-assisted hopping terms that appear to be important in enhancing hole-doped superconductivity. In Chapter 4, we examined the quantum spin nematic phase in the paradigmatic $S=1/2$ square-lattice $J_1$-$J_2$ ferro-antiferromagnetic Heisenberg model, employing a combination of DMRG and analytics. Our findings revealed that many-body effects induce significant contraction of the nematic phase compared to the na\"{i}ve expectation. Chapter 5 presents a study of the anisotropic $J_1^\Delta$-$J_3$ model on the honeycomb lattice, which is believed to be the fundamental model for many Kitaev material candidates upon adding bond-dependent terms. This chapter also includes ...