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Physical review. B, Condensed matter

Conserving approximations for strongly fluctuating electron systems. II. Numerical results and parquet extension

N. E. Bickers, Steven R. White

Published Apr 1, 1991• 319 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.43.8044

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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.
A variety of self-consistent-field techniques have recently been used to calculate correlation functions of the two-dimensional Hubbard model. Such techniques extend Hartree–Fock theory by including a large class of fluctuation processes. The computational requirements of the self-consistent-field approach decrease in lower dimensional systems. For this reason, a thorough test of techniques may be performed by examining the Anderson impurity model, the zero-dimensional limit of the Hubbard model. Using the fluctuation exchange and pseudopotential parquet approximations applied previously in two dimensions, we calculate static and dynamic correlation functions for the Anderson model and compare with quantum Monte Carlo results.
This thesis presents the development of new numerical methods for the treatment of strongly correlated electron systems based on self-consistent approaches at both the one and the two-particle level such as the parquet formalism. The parquet formalism was solved for the first time on a two-dimensional cluster. When the fully irreducible vertex is approximated by the bare vertex, we obtain the parquet approximation. Its validity was investigated by comparing results that it produces to those of other conserving approximations such as the FLuctuation EXchange (FLEX) approximation or the Second Order Perturbation Theory (SOPT). We found that it provides a significant improvement of FLEX or SOPT and a satisfactory agreement with Quantum Monte Carlo results despite instabilities in the self-consistency at low temperatures and for strong Coulomb interaction. We use the parquet formalism to study the Quantum Critical Point at finite doping in the Hubbard model by decomposing the vertex into its contributions from different channels. We apply this decomposition to the pairing channel and we find that the dominant contribution to the vertex originates in the spin channel even at the quantum critical doping. Furthermore, we explore the divergence of the two parts of the pairing matrix at optimal doping and observe that the irreducible vertex decreases monotonically as the doping is increased while the bare susceptibility exhibits an algebraic divergence at the quantum critical doping supporting the Quantum Critical BCS scenario proposed by She and Zaanen. To circumvent the instabilities in the iteration of the parquet formalism, we explored the dual fermion approach introduced by Rubtsov et al. Here, we extended the formalism to the Dynamical Cluster Approximation, in the process introducing a small parameter in the dual fermions perturbation theory. We demonstrate the quality of the resulting Dual Fermion DCA through a systematic study of the cluster size dependence and of the different perturbative approximations. These efforts represent the initial steps in the development of the Multi-Scale Many Body approach that appropriately treats correlations at different length scales.
We present a numerical solution of the parquet approximation, a conserving diagrammatic approach which is self-consistent at both the single-particle and the two-particle levels. The fully irreducible vertex is approximated by the bare interaction thus producing the simplest approximation that one can perform with the set of equations involved in the formalism. The method is applied to the Hubbard model on a half-filled 4x4 cluster. Results are compared to those obtained from determinant quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. This comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation.
We present a general framework for investigating the stability of solutions being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we use explicit equations for one- and two-particle Green functions simultaneously. We formulate a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density or anomalous type. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices become unstable with respect to spin flips, before the Mott-Hubbard metal-insulator transition in the paramagnetic phase or the Kondo strong-coupling regime can be reached. Further on, we use the parquet approximation and turn it into a workable scheme with dynamical vertex corrections. We obtain a theory with two-particle self-consistence, the complexity of which is comparable with fluctuation-exchange-type approximations. We show that it is the simplest consistent theory free of spurious phases being able to correctly describe qualitatively, quantum critical points and the transition from weak to strong coupling in correlated electron systems.
We present the finite-difference parquet method that greatly improves the applicability and accuracy of two-particle correlation approaches to interacting electron systems. This method incorporates the nonperturbative local physics from a reference solution and builds all parquet diagrams while circumventing potentially divergent irreducible vertices. Its unbiased treatment of different fluctuations is crucial for reproducing the strong-coupling pseudogap in the underdoped Hubbard model, consistent with diagrammatic Monte Carlo calculations. We reveal a strong-coupling spin-fluctuation mechanism of the pseudogap with decisive vertex corrections that encode the enhanced, energy-dependent scattering amplitude between electrons and antiferromagnetic spin fluctuations.
We have implemented the dynamical vertex approximation $(\mathrm{D}\ensuremath{\Gamma}\mathrm{A})$ in its full parquet-based version to include spatial correlations on all length scales and in all scattering channels. The algorithm is applied to study the electronic self-energies and the spectral properties of finite-size one-dimensional Hubbard models with periodic boundary conditions (nanoscopic Hubbard rings). From a methodological point of view, our calculations and their comparison to the results obtained within dynamical mean-field theory, plain parquet approximation, and the exact numerical solution allow us to evaluate the performance of the $\mathrm{D}\ensuremath{\Gamma}\mathrm{A}$ algorithm in the most challenging situation of low dimensions. From a physical perspective, our results unveil how nonlocal correlations affect the spectral properties of nanoscopic systems of various sizes in different regimes of interaction strength.