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Physical Review A

Approximations to the ground-state energy of the single-impurity Kondo model using variational and connected-moments expansions

Jay D. Mancini, Janice D. Prie, William J. Massano

Published Feb 1, 1991• 18 citations
Publisher

Abstract

We have studied the ground-state energy of the single-impurity Kondo model. Results using a strict 4\ifmmode\times\else\texttimes\fi{}4 Lanczos truncation are shown to be equivalent to a relaxed 26\ifmmode\times\else\texttimes\fi{}26 Lanczos basis. An application of the connected-moments expansion to obtain the ground-state energy of this model many-body Hamiltonian is shown to be limited to certain regions of parameter space.

DOI
10.1103/physreva.43.1777

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We have studied the ground-state energy of the single-impurity Kondo model. Results using a strict 4\ifmmode\times\else\texttimes\fi{}4 Lanczos truncation are shown to be equivalent to a relaxed 26\ifmmode\times\else\texttimes\fi{}26 Lanczos basis. An application of the connected-moments expansion to obtain the ground-state energy of this model many-body Hamiltonian is shown to be limited to certain regions of parameter space.
By means of a projector operator formalism we study the ground state properties of the Anderson Impurity Hamiltonian. The non-perturbative treatment of the model agrees with the previous one, obtained by Inagaki [Prog. Theor. Phys. 62, 1441 (1979)] by means of a perturbation expansion with respect the hybridization term. We go beyond the Inagaki's formalism to the next leading order. It provides a very accurate calculation of the energy spectrum in the total spin sector and, in particular, the ground state energy in the whole parameter space. For a one body spinless system, the dependence of the ground state energy as a function of the impurity level obtained by this procedure remarkably agrees with analytical results. For the many body case the occupancy of the impurity as a function of the parameters is studied and it agrees with the corresponding one obtained by using the Bethe ansatz and the Numerical Renormalization Group solution of the Hamiltonian. The magnetization and susceptibility of the impurity is analyzed by studying the response of the system to an external magnetic field, from which it is possible to extract the Kondo temperature. The dependence of the Kondo scale with the parameters of the model is in excellent agreement with well-known results.
The Kondo model describes a magnetic impurity in a metallic host. Here, the authors use variational approaches (Lanczos, Yosida, Gutzwiller) and quasi-exact numerical techniques (numerical renormalization group, density matrix renormalization group) to study the ground-state energy and magnetization, and compare these properties to Bethe ansatz results. The Gutzwiller wave function is an excellent trial state for strong coupling. For weak coupling, the numerical renormalization group gives the best description of the magnetic properties. A combination of Bethe ansatz and perturbation theory provides new benchmark results for the zero-field susceptibility.
An analysis of a general non-perturbative technique for calculating ground-state properties of extensive lattice many-body systems is presented, in order to extract accurate numerical values characterizing the ground-state spectrum. This technique, the plaquette expansion, employs an expansion about the thermodynamic limit of the coefficients that are generated by the Lanczos process. For the ground-state energy this error analysis, using theorems on the error bounds for the Lanczos method and the truncation in the plaquette expansion, allows for an accurate estimate when the approximation is taken to a given order. As an example we analyse the one-dimensional antiferromagnetic Heisenberg model, and find that the best ground-state energy density is within of the exact value, although the systematic error is . We also find, for this model, systematic improvement with each new order included in the expansion and have not observed any asymptotic tendencies. At equivalent orders of truncation we achieve far better results than for the other moment methods, such as the t-expansion or the connected-moment expansion.