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arXiv:cond-mat/0410410 · Physical Review B

Interpolative approach for solving the Anderson impurity model

Sergey Yu. Savrasov, V. S. Oudovenko, Kristjan Haule, D. Villani, Gabriel Kotliar

Published Mar 31, 2005• 21 citations
arXivHTMLPDF

Abstract

A rational representation for the self-energy is explored to interpolate the solution of the Anderson impurity model in the general orbitally degenerate case. Several constraints such as Friedel's sum rule and the positions of the Hubbard bands, as well as the value of the quasiparticle residue, are used to establish the equations for the coefficients of the interpolation. We employ two fast techniques, the slave-boson mean-field and the Hubbard I approximations, to determine the functional dependence of the coefficients on doping, degeneracy, and the strength of the interaction. The obtained spectral functions and self-energies are in good agreement with the results of the numerically exact quantum Monte Carlo method.

DOI
10.1103/physrevb.71.115117

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A rational representation for the self-energy is explored to interpolate the solution of the Anderson impurity model in the general orbitally degenerate case. Several constraints such as Friedel's sum rule and the positions of the Hubbard bands, as well as the value of the quasiparticle residue, are used to establish the equations for the coefficients of the interpolation. We employ two fast techniques, the slave-boson mean-field and the Hubbard I approximations, to determine the functional dependence of the coefficients on doping, degeneracy, and the strength of the interaction. The obtained spectral functions and self-energies are in good agreement with the results of the numerically exact quantum Monte Carlo method.
A rational representation for the self-energy is explored to interpolate the solution of the Anderson impurity model in general orbitally degenerate case. Several constrains such as the Friedel's sum rule, high--frequency moments and the value of quasiparticle residue are used to establish the equations for the coefficients of the interpolation. We test two fast techniques, the slave--boson mean--field and the Hubbard I approximation to determine the coefficients. The obtained self--energies are compared with the results of numerically exact Quantum Monte Carlo method. We find that using the slave--boson mean--field approach we can construct an accurate self--energy for all frequencies via the proposed interpolation procedure.
We develop a method for calculating the self-energy of a quantum impurity coupled to a continuous bath by stochastically generating a distribution of finite Anderson models that are solved by exact diagonalization, using the noninteracting local spectral function as a probability distribution for the sampling. The method enables calculation of the full analytic self-energy and single-particle Green's function in the complex frequency plane, without analytic continuation, and can be used for both finite and zero temperature at arbitrary fillings. Results are in good agreement with imaginary frequency data from continuous-time quantum Monte Carlo calculations for the single-impurity Anderson model and the two-orbital Hubbard model within dynamical mean-field theory (DMFT) as well as real frequency data for self-energy of the single-band Hubbard model within DMFT using the numerical renormalization group. The method should be applicable to a wide range of quantum impurity models and particularly useful when high-precision real frequency results are sought.
An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to correlated-electron models. The method is based on performing an exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descent treatment of the charge part. We test this method by applying it to one- or two-site approximations to the single-band Hubbard model with different band structures, and comparing the results to quantum Monte Carlo and simplified exact diagonalization calculations. The resulting electron self-energies, densities of states, and magnetic transition temperatures show reasonable agreement with the quantum Monte Carlo simulation over wide parameter ranges, suggesting that the semiclassical method is useful for obtaining a reasonable picture of the physics in situations where other techniques are too expensive.
We introduce a representation of electron operators as a product of a spin-carrying fermion and of a phase variable dual to the total charge (slave quantum rotor). Based on this representation, a method is proposed for solving multiorbital Anderson quantum impurity models at finite interaction strength U. It consists in a set of coupled integral equations for the auxiliary field Green's functions, which can be derived from a controlled saddle point in the limit of a large number of field components. In contrast to some finite-UUU extensions of the noncrossing approximation, the method provides a smooth interpolation between the atomic limit and the weak-coupling limit, and does not display violation of causality at low frequency. We demonstrate that this impurity solver can be applied in the context of dynamical mean-field theory, at or close to half-filling. Good agreement with established results on the Mott transition is found, and large values of the orbital degeneracy can be investigated at low computational cost.
We develop a projective quantum Monte Carlo algorithm of the Hirsch-Fye type for obtaining ground state properties of the Anderson impurity model. This method is employed to solve the self-consistency equations of dynamical mean field theory. It is shown that the approach converges rapidly to the ground state so that reliable zero-temperature results are obtained. As a first application, we study the Mott-Hubbard metal-insulator transition of the frustrated one-band Hubbard model, reconfirming the numerical renormalization group results.
A slave-boson functional-integral method has been developed for the Hubbard model with arbitrary, orbital degeneracy DDD. Its saddle-point mean-field theory is equivalent to the Gutzwiller approximation, as in the case of single-band Hubbard model. Our theory is applied to the doubly degenerate (D=2D = 2D=2) model, and numerical calculations have been performed for this model in the paramagnetic states. The effect of the exchange interaction on the metal-insulator (MI) transition is discussed. The critical interaction for the MI transition is analytically calculated as functions of orbital degeneracy and electron occupancy.