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

• Published in: Annalen der Physik Vol. 14; no. 9-10; pp. 642 - 657
• Main Authors: Kollar, M., Eckstein, M., Byczuk, K., Blümer, N., van Dongen, P., Radke de Cuba, M.H., Metzner, W., Tanasković, D., Dobrosavljević, V., Kotliar, G., Vollhardt, D.
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag 01-09-2005; WILEY‐VCH Verlag
• Subjects: Bethe lattice; dynamical mean-field theory; frustration; Green function
• ISSN: 0003-3804; 1521-3889
• DOI: 10.1002/andp.200510152

Dedicated to Bernhard Mühlschlegel on the occasion of
his 80th birthday
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.