Grassmann integral and Balian-Brézin decomposition in Hartree-Fock-Bogoliubov matrix elements

We present a new formula to calculate matrix elements of a general unitary operator with respect to Hartree-Fock-Bogoliubov states allowing multiple quasi-particle excitations. The Balian-Brézin decomposition of the unitary operator [R. Balian, E. Brézin, Il Nuovo Cimento B 64 (1969) 37] is employed in the derivation. We found that this decomposition is extremely suitable for an application of Fermion coherent state and Grassmann integrals in the quasi-particle basis. The resultant formula is compactly expressed in terms of the Pfaffian, and shows the similar bipartite structure to the formula that we have previously derived in the bare-particles basis [T. Mizusaki, M. Oi, Phys. Lett. B 715 (2012) 219].