Analytic Solution for the Ground State Energy of the Extensive Many-Body Problem

TL;DR

This paper derives an analytical expression for the ground state energy density of extensive many-body systems using the Lanczos method and orthogonal polynomial theorems, providing a general formula applicable in the bulk limit.

Contribution

It introduces a closed-form solution for the ground state energy density based on the asymptotic behavior of Lanczos matrix elements, extending the analytical tools for many-body problems.

Findings

Provides a general formula for ground state energy density in the bulk limit.

Connects Lanczos matrix elements with orthogonal polynomial zeros.

Offers a theoretical framework for analyzing extensive many-body systems.

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 tri-diagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, $\alpha_n(N)$ and $\beta_n(N)$, asymptotic forms $\alpha(z)$ and $\beta(z)$ can be defined in terms of a new parameter $z\equiv n/N$ ($n$ 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 ${\cal E}_0 = {\rm inf}\,\left[\alpha(z) - 2\,\beta(z)\right]$.