Rigorous lower bounds on the ground-state energy of correlated Fermi systems.

Abstract

We show that the thermodynamic potential E — pN of an itinerant interacting Fermi system is bounded below by the sum of the thermodynamic potentials of small subsystems into which the infinite system is partitioned. Exact lower bounds for the energy in the thermodynamic limit are therefore easily obtainable by diagonalization of small clusters. In one dimension, these bounds are remarkably close to Bethe-ansatz results for the Hubbard model. We present lower bounds for the two-dimensional Hubbard and t-J Hamiltonians. Such bounds may serve as rigorous tests for approximate treatments of this class of problems. Interest in correlated Fermi systems has been heightened in recent years by the suggestion' that strong local electronic interactions could play an important role in the unusual normal-state properties of the oxide superconductors, and possibly in the pairing mechanism itself. Unfortunately, few rigorous results are available to us for the simplest microscopic models thought to describe such systems: the Hubbard model and its strong-coupling limit, the t-J model. In one spatial dimension the Betheansatz solution of Lieb and Wu provides the simplest information on equilibrium thermodynamic and magnetic properties. This approach has recently been extended by many authors to calculate correlation functions. Anderson has suggested that the solution of the twodimensional (2D) problem may possess many features of the Lieb-Wu result, in particular an algebraic singularity in the momentum distribution function at the Fermi surface (Z =0).