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Physical review. B, Condensed matter

Application of tridiagonalization to the many-body problem

Jay D. Mancini, Daniel Charles Mattis

Published Nov 15, 1983• 30 citations
Publisher

Abstract

The problem of a single magnetic, Wolff-model impurity in an otherwise ideal metallic host is investigated using the nonperturbative Lanczos method. Convergence is very rapid. The many-body ground-state energy is investigated and comparisons are made with Tomonaga operator theory and other weak-coupling schemes. We believe that this is the first application of tridiagonalization to the many-body problem.

DOI
10.1103/physrevb.28.6061

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The problem of a single magnetic, Wolff-model impurity in an otherwise ideal metallic host is investigated using the nonperturbative Lanczos method. Convergence is very rapid. The many-body ground-state energy is investigated and comparisons are made with Tomonaga operator theory and other weak-coupling schemes. We believe that this is the first application of tridiagonalization to the many-body problem.
In a previous paper of the same title, we obtained the ground-state energy of a magnetic (Wolff-model) impurity in a nonmagnetic metal. In the present Brief Report, we calculate the impurity's contribution to the density of states and heat capacity of the metal at low temperatures. Here, the Lancz\"os reduction ("tridiagonalization") converges less rapidly, so that our results are of qualitative merit only. Nevertheless, we confirm the validity of perturbation theory in the weak-coupling regime, and find, at strong coupling, that the interaction introduces an extra fraction 2N\frac{2}{N}N2​ of states at the Fermi level.
We report preliminary results of convergence properties for nonperturbative resolvent approximations to Anderson-like models of magnetic ions in metals. Our study is initially focused on the spin- 1/2 Anderson model for magnetic impurities, but the methods studied can include multiplet and crystal-field effects which are needed for more accurate descriptions of real systems. We will compare the nonperturbative Lanczos method (tridiagonalization) and similar truncation schemes to exact ground-state energies for the impurity model and assess the efficacy of these nonperturbative approaches to understanding the Anderson lattice, heavy fermions, and other strongly interacting electronic systems.
A closed form expression for the ground-state energy density of the general extensive many-body problem is given in terms of the Lanczos tridiagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, ${\mathrm{\ensuremath{\alpha}}}_{\mathit{n}}$(N) and ${\mathrm{\ensuremath{\beta}}}_{\mathit{n}}$(N), asymptotic forms \ensuremath{\alpha}(z) and \ensuremath{\beta}(z) can be defined in terms of a parameter z\ensuremath{\equiv}n/N (n is the Lanczos iteration and N is the size of the system). By application of theorems on the zeros of orthogonal polynomials we find the ground-state energy density in the bulk limit to be given, in general, by E0{\mathit{E}}_{0}E0​=inf[\ensuremath{\alpha}(z)-2\ensuremath{\beta}(z)]. \textcopyright{} 1996 The American Physical Society.
The basic idea of the Lanczos method is to construct a special basis where the Hamiltonian has a tridiagonal representation. Once in this form, the matrix can be diagonalized easily using standard library subroutines. This approach is known as the standard Lanczos technique (SLT) which still has the problem of diagonalizing large matrix sizes emanating from increase of the size of the Hilbert space with the size of the system. In this current presentation, we develop a simplified formulation of the SLT and then use it to study the rapid convergence to the ground state energy and wavefunction of some finite systems.
The classical formalism of the Moment Problem has been combined with a cumulant approach and applied to the extensive many-body problem. This has yielded many new exact results for many-body systems in the thermodynamic limit - for the ground state energy, for excited state gaps, and for arbitrary ground state averages. The method applies to any extensive Hamiltonian system, for any phase or symmetry arising in the model, whether on a lattice or in the continuum, and for any dimensionality. The theorems are of a nonperturbative nature with respect to any couplings occuring in the model.
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.