High-order perturbation theory for strongly correlated Boson systems

Perturbative calculations soon tend to become messy in higher orders, or even intractable. Fortunately, the non-recursive 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 lattice-type many-Boson 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 high-performance computational facilities, this technique enables us to treat even three-dimensional 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 Hanse-Wissenschaftskolleg in Delmenhorst.

 

Atom-atom correlation functions for various filling factors

  • We have systematically applied the process chain approach to the d-dimensional Bose-Hubbard model, and computed ground-state energies, atom-atom correlation functions, density-density correlations, and occupation number fluctuations for the Mott-insulator 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.

 

Mott lobes for the three-dimensional Bose-Hubbard model with various filling factors

  • 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 mean-field limit by treating dimensionalites d > 3.
    See Phys. Rev. B 79, 100503(R) (2009) for details.

 

Diagrams of order zero to three in the hopping parameter for a triangular lattice

 

Diagrams of order zero to three in the hopping parameter for a hexagonal lattice

  • The technique can be applied to arbitrary lattice types. The above figure shows the lowest-order 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.