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Annals of Physics

Conserving approximations for strongly fluctuating electron systems. I. Formalism and calculational approach

N. E. Bickers, Douglas J. Scalapino

Published Jul 1, 1989• 423 citations
Publisher
DOI
10.1016/0003-4916(89)90359-x

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We describe an iterative technique for solution of the de Dominicis--Martin parquet equations for lattice electrons and state the relationship of this approach to conserving extensions of Hartree-Fock theory. We propose a physically motivated and computationally feasible pseudopotential approximation, which allows the solution of the parquet equations for the two-dimensional Hubbard model. We present calculations of static and dynamic Hubbard-model correlation functions based on the pseudopotential parquet and a simpler conserving approximation. For a wide parameter range the pseudopotential parquet results are in nearly quantitative agreement with finite-lattice quantum Monte Carlo results.
This thesis contains a series of numerical studies of strongly correlated electron systems. In these systems, interesting emergent properties are brought about by strong electron correlations, leading to unusual fluctuations and phase transitions. We start with a review of basic concepts in many-body physics from a field theory point of view, including second quantization and path integral formalisms. We then introduce extensions of the single-particle and two-particle Green’s function formalisms in the singlet superconducting state, which enable us to extract physical information of the systems in the symmetry-broken phase. Next, we proceed to the models and numerical methods we use to study the strongly correlated systems. We introduce a low energy effective model -- the Hubbard model, which contains a subset of the electron degrees of freedom and can be solved numerically using advanced many-body methods. With the dynamical cluster approximation and continuous-time quantum Monte Carlo impurity solver, we are able to study the competing fluctuations in the paramagnetic state and analyze the fluctuations behind superconductivity in the singlet superconducting state. We introduce self-consistent diagrammatic methods used in realistic material calculations in the last part of this thesis, with an outline of group theory concepts that can be used to optimize the simulations. Numerical representations and methods for effectively solving equations in realistic material calculations are discussed afterwards. We briefly review some of the developments in this field, and then introduce spectral methods that are based on mathematical properties of Legendre polynomials for solving both the imaginary- and real-time Dyson equations.
We report a superconducting transition in the fully conserving fluctuation exchange approximation for a two-dimensional attractive Hubbard model. The transition temperature agrees well with quantum Monte Carlo calculations, but the superfluid density and specific heat near Tc{T}_{c}Tc​ have anomalous temperature dependences for lattice sizes up to $64\ifmmode\times\else\texttimes\fi{}64$.