Motivation for the model studies in Project 8 is the observation that in most transition metal chalcogenides, the observed metal-insulator transition is combined with a structural distortion of the lattice. Thus the question arises whether in these systems the electronic correlations (typically of medium strength) or the coupling to the lattice are the "driving force" for the transition - or whether the transition in real materials can be understood by an interplay of both only. Unfortunately, this question cannot be answered directly with today's theoretical methods for real systems. In this project, as a first step, we studied one dimensional model systems, for which quasi-exact results can be obtained, in order to understand the various trends and possibilities. In particular, it was important to compare numerical with approximate analytical (mean field theory, self-consistent harmonic approximation) results; the latter are expected to be helpful when studying realistic models.

In particular, we studied in great detail the model of spinless fermions on a lattice in one dimension, which is equivalent to the Heisenberg XXZ spin model. An important quantity, numerically accessible within the Density Matrix Renormalization Group method, is the so-called phase sensitivity, which measures the dependence of the ground state energy on the boundary conditions. At zero temperature, this quantity is proportional to the Drude weight, and hence a measure of the character of the ground state wave function. For a delocalized state, the Drude weight is independent of system size, whereas it decreases (exponentially) with size for a localized ground state.

As a first result, we determined the phase diagram of the model with random impurities, confirming that for a certain range of an attractive interaction, the ground state remains extended, i.e. the Luttinger liquid behavior of the clean model persists despite the disorder. We then included a static dimerization, in the hopping as well as in the (nearest neighbor) interaction term. For an attractive interaction of medium strength, we could show that the dimerization is irrelevant, i.e. the wave function remains delocalized. However, the energy gain due to the dimerization, for attractive interaction, is not sufficient to stabilize a finite dimerization (and hence an energy gap). In contrast, for repulsive interaction, the coupling to the lattice leads to a finite lattice distortion and an energy gap (as in the non-interacting case, for which this Peierls mechanism is well known). Indeed, the energy gap is of Peierls type for small (prepulsive) interaction, whereas for strong interaction, the instability to the formation of a charge-density wave state, characteristic for this model at half filling, dominates the behavior.

In another sub-project, we investigated integrable models; these, despite their integrability, can show a quite complex behavior. The models studied at present are related to the metal-insulator transition observed in twodimensional electron systems in a strong perpendicular magnetic field, namely the integer quantum Hall effect (IQHE). In the IQHE, a finite metallic range seems to have been observed in the transition from one plateau to the next; in this range, the parallel conductivity is supposed to assume a universal value. Our studies start with the assumption that the observed, rather complex behavior can be described by an integrable model, which still has to be determined.

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