A Local Approach for Interacting Electrons: Application to the Heisenberg Model

Peter Streitenberger

V. V. França, Klaus Capelle

A combined analytical and numerical study is performed of the mapping between strongly interacting fermions and weakly interacting spins, in the framework of the Hubbard, t-J, and Heisenberg models. While for spatially homogeneous models in the thermodynamic limit the mapping is thoroughly understood, we here focus on aspects that become relevant in spatially inhomogeneous situations, such as the effect of boundaries, impurities, superlattices, and interfaces. We consider parameter regimes that are relevant for traditional applications of these models, such as electrons in cuprates and manganites, and for more recent applications to atoms in optical lattices. The rate of the mapping as a function of the interaction strength is determined from the Bethe-Ansatz for infinite systems and from numerical diagonalization for finite systems. We show analytically that if translational symmetry is broken through the presence of impurities, the mapping persists and is, in a certain sense, as local as possible, provided the spin-spin interaction between two sites of the Heisenberg model is calculated from the harmonic mean of the onsite Coulomb interaction on adjacent sites of the Hubbard model. Numerical calculations corroborate these findings also in interfaces and superlattices, where analytical calculations are more complicated.

Shu Zhang

This dissertation reports the results of three loosely connected research projects. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis for the investigation of the dynamical properties at zero temperature of a particular quantum many-body system, including models of localized electron spins and models of strongly correlated itinerant electrons. We have studied these four models: two-dimensional (2D) s = 1/2 Heisenberg antiferromagnet (square lattice), the 2D s = 1/2 Heisenberg model with competing nearest-neighbor ($J\\sb1$) and next-nearest neighbor ($J\\sb2$) interactions, the 1D t-J model, and the 1D t-$J\\sb{z}$ model.

V. E. Korepin, Fabian H. L. Eßler

A. K. Rajagopal

Takashi Yanagisawa

In this article we discuss the reflection positivity in spin space for the Heisenberg model and the Kondo Hamiltonian. We show that the ground state is unique for J ≠0 where J is the strength of the exchange interaction in the Heisenberg model or the exchange coupling between the localized spins and conduction electrons in the Kondo model. We introduce the Coulomb interaction U >0 between the conduction lectrons for the Kondo Hamiltonian to show the uniqueness of the ground state. We write the Hamiltonian in terms of fermions to express the exchange interactions and impose constraints. We employ the Lagrange-multiplier method to deal with the constraints in this work.