Current Research

Introduction
Besides its charge, the spin of the electron represents another important degree of freedom and in general there is a complex interplay between charge and spin excitations. In some insulating materials, however, the low energy properties are completely determined by their spin excitations. These materials can be treated as spin systems. In special cases the interaction between the spins can lead to a long range ordering of the spins, i.e., to ferromagnetism or antiferromagnetism (depending on the sign of the exchange interaction). In general, however, the properties of a spin system depend on the spin quantum number and the spatial dimensionality of the system. The smaller the spin quantum number and the lower the dimensionality of the system, the more dominant the quantum nature of the spin will be.

Motivation
In particular we are interested[1] in understanding the magnetic properties of the two-dimensional undoped cuprates, i.e., the parent compounds of the high-T_c-superconductors, which are thought to be a nearly ideal realization of a two-dimensional spin-½ Heisenberg model with an antiferromagnetic exchange coupling J between nearest neighbor spins S_i, S_j:
H=J ∑_<i,j>S_iS_j
In order to gain further insight into the two-dimensional systems (d=2), we also investigate spin ladder systems, which provide a crossover from d=1 to d=2.

Methods
To investigate these systems we employ a bosonic representation of the spin operators for d≥2 and a fermionic representation for d<2[2]. In addition to these analytic approaches we use the Density Matrix Renormalization Group (DMRG)[3], which is a very powerful numerical technique for quasi one-dimensional systems, for the spin-ladder systems.

Comparision with Experiment
In the two-dimensional cuprates and in the spin-ladder systems the magnetic excitations can be observed experimentally in the mid-infrared range of the optical conductivity[1][4]. These measurements are performed in the group of Markus Grüninger and Axel Freimuth at the University of Cologne. We carry out caclulations for the optical conductivity in these compounds and in collaboration with the group from Cologne we succeeded to interpret the experimental data.

Results
We have identified two resonances in the mid-infrared range of the optical conductivity of the spin ladder compound (La,Ca)₁₄Cu₂₄O₄₁ as singlet bound states of two triplet excitations[4]. The dispersion of the bound state (see Fig 1) gives rise to two peaks in the optical conductivity (see Fig 2). Furthermore we found, that it is necessary to consider a 4-spin ring exchange[5] besides the normal two-spin exchange interaction, in order to understand the magnetic properties of cuprate spin ladders more thoroughly.

Present Work
At the moment we investigate 4-leg spin ladders. In this way we hope to gain insight into the crossover from 2-leg spin ladders to two-dimensional systems.