Coulomb interaction. [@re:GW]

To solve the mapped single impurity problem within DMFT, one can use several computational schemes such as the quantum Monte Carlo method (QMC), [@re:Rozen-QMC; @re:Geo-QMC; @re:QMC-difficult; @re:PQMC; @re:CTQMC] the iterative perturbation theory (IPT), [@re:Kaj-IPT; @re:Fuji; @re:miura; @re:MOIPT] the non-crossing approximation (NCA) [@re:NCA] and the exact diagonalization (ED). [@re:Geo-ED]

Hirsch-Fye QMC [@re:Hirsch-Fye-QMC] is widely used as a solver for the mapped single impurity problem within DMFT. However, it is not applicable in the low temperature limit. Moreover, Hirsch-Fye QMC has serious difficulty in application of multi-orbital systems with spin-flip and pair-hopping terms of the exchange interactions, since one cannot apply the Hubbard-Stratonovich transformation to these systems. [@re:QMC-difficult] To go beyond such problems of Hirsch-Fye QMC, several QMC methods have been proposed. Projective QMC [@re:PQMC] has been developed to calculate in the low temperature limit and Continuous Time QMC (CTQMC) [@re:CTQMC] to calculate multi-orbital systems with spin-flip and pair-hopping terms.

In spite of development of those QMC, the exact calculations such as QMC and ED can only be suitable for simple Hamiltonians with relatively small size because of their computational costs. Several LDA+DMFT methods with QMC adopt the projected effective Hamiltonian of Wannier-like functions with a few adopted bands 