Density Matrix Embedding Theory and Strongly Correlated Lattice Systems

Abstract

This thesis describes the development of the density matrix embedding theory (DMET) and its applications to lattice strongly correlated electron problems. We introduced a broken spin and particle-number symmetry DMET formulation to study the high-temperature superconductivity and other low-energy competing states in models of the cuprate superconductors. These applications also relied on (i) the development and adaptation of approximate impurity solvers beyond exact diagonalization, including the density matrix renormalization group, auxiliary-field quantum Monte Carlo and active-space based quantum chemistry techniques, which expanded the sizes of fragments treated in DMET; and (ii) the theoretical development and numerical investigations for the finite size scaling behavior of DMET.

Using these numerical tools, we computed a comprehensive ground state phase diagram of the standard and frustrated Hubbard models on the square lattice with well-controlled numerical uncertainties, which confirms the existence of the d-wave superconductivity and various inhomogeneous orders in the Hubbard model. We also investigated the long-sought strong coupling, underdoped regime of the Hubbard model in great detail, using various numerical techniques including DMET, and determined the ground state being a highly-compressible, filled vertical stripe at 1/8 doping in the coupling range commonly considered relevant to cuprates. The findings show both the relevance and limitations of the one-band Hubbard model in studying the cuprate superconductivity.

Therefore, we further explored the three-band Hubbard model and downfolded cuprate Hamiltonians from first principles, in an attempt to understand the physics beyond the one-band model. We also extended the DMET formulation to finite temperature using the superoperator representation of the density operators, which is potentially a powerful tool to investigate finite-temperature properties of cuprates and other strongly correlated electronic systems.