Trotter decomposition and a series expansion of the partition function for hund's exchange coupling in a quantum monte carlo algorithm

We have studied numerically exact auxiliary field quantum Monte Carlo scheme for the spin and orbital rotational invariant Hamiltonian by combining series expansion and Trotter decomposition. We have used algorithm to solve the dynamical field theory equations. We have employed Boltzmann operator with respect to the total interaction. The method significantly relaxes the sign problem and allows in principle for more than two orbitals. The computational effort increases with the perturbation orders appearing in the Mote Carlo samples. The order become very large in multiorbital systems. To remove this difficulty we have used the Hirsch-Fye quantum Monte Carlo and parturbation series expansion quantum Monte Carlo methods. This algorithm enabled us not only to handle three or more orbitals but also to reach much stronger coupling than in Hirsch-Fye quantum Monte Carlo calculations, even for the two orbital Hubbard model.