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Green functions for nearest- and next-nearest-neighbor hopping on the Bethe lattice

M. Kollar·M. Eckstein·K. Byczuk·N. Blümer·P. van Dongen·M. H. Radke de Cuba·W. Metzner·D. Tanaskovic·V. Dobrosavljevic·G. Kotliar·D. Vollhardt
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Abstract

We calculate the local Green function for a quantum-mechanical particle with hopping between nearest and next-nearest neighbors on the Bethe lattice, where the on-site energies may alternate on sublattices. For infinite connectivity the renormalized perturbation expansion is carried out by counting all non-self-intersecting paths, leading to an implicit equation for the local Green function. By integrating out branches of the Bethe lattice the same equation is obtained from a path integral approach for the partition function. This also provides the local Green function for finite connectivity. Finally, a recently developed topological approach is extended to derive an operator identity which maps the problem onto the case of only nearest-neighbor hopping. We find in particular that hopping between next-nearest neighbors leads to an asymmetric spectrum with additional van-Hove singularities.

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M. Kollar, M. Eckstein, K. Byczuk, N. Blümer, P. van Dongen, M. H. Radke de Cuba, W. Metzner, D. Tanaskovic, V. Dobrosavljevic, G. Kotliar, D. Vollhardt. 2005-08-29. Green functions for nearest- and next-nearest-neighbor hopping on the Bethe lattice. https://doi.org/10.1002/andp.200510152

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