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arXiv:cond-mat/9602091 · Physical review. B, Condensed matter

Analytic solution for the ground-state energy of the extensive many-body problem

Lloyd C. L. Hollenberg, Nicholas S. Witte

Published Dec 15, 1996• 22 citations

Abstract

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 ${\mathit{E}}_{0}$=inf[\ensuremath{\alpha}(z)-2\ensuremath{\beta}(z)]. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physrevb.54.16309

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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

{\mathit{E}}_{0}

=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.

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.

On the grounds of a Feynman–Kac-type formula for Hamiltonian lattice systems, we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result, we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by solving a simple scalar equation. Furthermore, explicit solutions of this equation are obtained in the noninteracting case.

We establish rigorously the scaling properties of the Lanczos process applied to an arbitrary extensive many-body system which is carried to convergence n→∞ and the thermodynamic limit N→∞ taken. In this limit the solution for the limiting Lanczos coefficients are found exactly and generally through two equivalent sets of equations, given initial knowledge of the exact cumulant generating function. The measure and the orthogonal polynomial system associated with the Lanczos process in this regime are also given explicitly. Some important representations of these Lanczos functions are given, including Taylor series expansions, and theorems controling their general properties are proven.