Condensed Matter Theory
Highorder perturbation theory for strongly correlated Boson systems
Perturbative calculations soon tend to become messy in higher orders, or even intractable. Fortunately, the nonrecursive formulation of the perturbation series given in 1949 by the Japanese mathematician Tosio Kato allows one to write down all contributions to this series in closed form, to any order.We are employing this formulation for studying strongly correlated latticetype manyBoson systems, motivated by recent experimental works on ultracold atoms in optical lattices. Our strategy combines a process chain approach, as pioneered by A. Eckardt, with a Cinderella algorithm: The contributions to Kato's series correspond to sequences of operations at individual lattice sites. We generate all possible paths on the respective lattice with a length corresponding to the desired order of perturbation theory, and compare the resulting diagrams to the Kato terms: The matching contributions are evaluated, the others discarded. Utilizing highperformance computational facilities, this technique enables us to treat even threedimensional systems with arbitray filling factors routinely in tenth order.
This research is performed in collaboration with A. Pelster, who currently is a Fellow of the HanseWissenschaftskolleg in Delmenhorst.

We have systematically applied the process chain approach
to the ddimensional BoseHubbard model, and computed groundstate
energies, atomatom correlation functions, densitydensity correlations,
and occupation number fluctuations for the Mottinsulator phase.
Considering arbitrary filling, we have discovered a phenomenological
scaling behavior which renders the data almost independent of the
filling factor.
See Phys. Rev. B 79, 224515 (2009) for details.

Moreover, we have computed the phase boundary between the
Mott insulating and the superfluid state. The combination of the process
chain approach with the Cinderella algorithm allows us to calculate accurate
critical parameters for any filling factor, and to monitor the approach to
the meanfield limit by treating dimensionalites d > 3.
See Phys. Rev. B 79, 100503(R) (2009) for details.

The technique can be applied to arbitrary lattice types.
The above figure shows the lowestorder diagrams required for calculating
the phase boundary for triangular (a) and hexagonal (b) lattices,
with dots (crosses) marking the creation (annihilation) of a particle,
and arrows indicating tunneling processes which connect different lattice
sites.
See EPL 91, 10004 (2010) for details.
Current developments concern applications of the process chain approach to the accurate calculation of further characteristics of the quantum phase transition, such as critical exponents.
This work is supported by the Deutsche Forschungsgemeinschaft under grant No. HO 1771/5.
Disclaimer Druckversion Martin Holthaus Last modified: Dec 30 2010