Variational Discrete Action Theory

This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians. VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2 provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework. VDAT will have broad applications in condensed matter and materials physics. In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability in lattice model Hamiltonians, in addition to real materials systems. The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should find broad applications in both model Hamiltonians and real materials